Prime Indexed Primes

It struck me today that the indexing of primes is base-independent. It's obvious I know but for some reason I'd maintained this vague notion that the indexing was base-dependent. Today I turned 28103 days old and this number is the product of the 37th prime (157) and the 41st prime (179).

28103 is a member of OEIS A119658:

 A119658: area of consecutive prime-indexed prime rectangles or the product of consecutive prime-indexed primes.

The initial sequence members are:

15, 55, 187, 527, 1271, 2419, 3953, 5561, 9047, 13843, 19939, 28103, 34189, 40301, 50851, 66757, 78391, 93673, 116843, 129551, 147167, 172831, 198691, 234649, 278423, 307961, 330481, 351613, 369583, 437453, 523951, 571247, 616081, 684623, ...

As can be seen, the sequence members get large rather quickly. There are 575 prime-indexed primes in the range up to 40000 and they can be combined in various ways to generate new sequences such as OEIS A119658. Another example involving the sum instead of the product is OEIS A299644:

 

 A299644: a(\(n\)) = prime(prime(\(n+1\))) + prime(prime(\(n\))).

The initial sequence members are:

8, 16, 28, 48, 72, 100, 126, 150, 192, 236, 284, 336, 370, 402, 452, 518, 560, 614, 684, 720, 768, 832, 892, 970, 1056, 1110, 1150, 1186, 1216, 1326, 1448, 1512, 1570, 1656, 1736, 1796, 1886, 1958, 2022, 2094, 2150, 2240, 2324, 2372, 2418, 2514, 2706, 2842, 2880, 2918, ...

Another example is OEIS A318294 involving Sophie Germain primes:

 A318294: prime-indexed primes \(p\) such that \(2p + 1\) is also a prime-indexed prime.

The initial sequence members are:

5, 41, 1031, 3299, 3761, 6323, 7649, 7841, 10589, 12959, 18443, 37853, 38039, 44729, 58511, 59723, 60509, 77813, 105971, 134489, 137999, 190523, 194933, 200351, 239633, 250751, 266093, 288653, 315011, 366161, 376241, 377633, 393401, 410999, 420419, 420779, 450431, 474983

You can of course create sequences of your own that aren't in the OEIS database. For example, though the sum of two prime-indexed primes is always a composite number, what if we add 1 and ask whether the result is also a prime-indexed prime. The result is the following sequence which has 12 members below 40000:

17, 127, 1217, 3761, 4091, 5701, 8287, 10663, 21149, 35023, 37061, 39733

The first member is \(17_{\small{7} }= 5_{\small{3 }}+ 11_{\small{5 }}+ 1\) where the subscripts denote the prime index.

Some Interesting Properties of 39

My daughter-in-law turned 39 yesterday and so I was prompted to investigate some of its mathematical properties. One of its properties is its membership in OEIS A055233:

A055233: composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

The only members of this sequence in the range up to 40000 are 10, 39, 155 and 371. All are semiprimes and factorise as follows:

• \(10 = 2 \times 5 \text{ with } 2 + 3 + 5 = 10 \)
• \(39 = 3 \times 13 \text{ with } 3 + 5 + 7 + 11+13 = 39\)
• \(155 = 5 \times 31 \text{ with } 5 + 7 + \ldots + 29 + 31=155\)
• \(371 = 7 \times 53 \text{ with } 7 + 11 + \ldots + 47 + 53=371\)

Because they are semiprimes they are thus equal to the product of their smallest and largest prime factors. However, this is not the case for the next member of the sequence: 2935561623745. The reason is that it is not a semiprime.

• \(2935561623745= 5 \times 19 \times 53 \times 61 \times 9557887\)

The next member of the sequence 454539357304421 is a semiprime and thus follows the pattern of the first four members of the sequence:

• \(454539357304421 = 3536123 \times 128541727\)

So we see that 39 by virtue of its membership in OEIS A055233 is rather special. Of course, it has some other interesting qualities. For example, it can be constructed from the first three powers of 3:$$39=3+3^2+3^3$$Gemini also mentions the following number properties:

Beyond these patterns, 39 is also classified as a \( \textbf{Perrin number}\) and a \( \textbf{Størmer number}\), placing it within specialized mathematical sequences that are far from intuitive. 

The number also has an \( \textbf{aliquot sum}\) of 17, which is a prime number, a unique characteristic that links it to a specific aliquot sequence. 

In the realm of number partitions, 39 is notable as the smallest natural number to have three distinct partitions into three parts that all yield the same product, 1200. These partitions are:

• {25, 8, 6} 

• {24, 10, 5} 

• {20, 15, 4}. 

Lastly, in analytic number theory, the \( \textbf{Mertens function}\) returns a value of 0 when given 39, a property that suggests a form of numerical equilibrium or stability, a concept that finds intriguing parallels in other domains. See blog post Zeroes of the Mertens Function.

39 is also what's termed a \( \textbf{perfect totient number} \) because the sum of its iterated totients equals the number itself. Let's confirm this:$$ \begin{align} \phi(39) &=24 \\ \phi(24) &=8 \\ \phi(8) &=4 \\ \phi(4) &=2 \\ \phi(2) &=1 \end{align} $$The sum of these iterated totients equals 39:$$24 + 8 + 4 + 2 + 1 =39$$The perfect totient numbers are listed in OEIS A082897 (permalink):

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147, 208191, 441027, 531441, 1594323, 4190263, 4782969, 9056583, 14348907, 43046721

Primes from SOD and POD

I'm using the acronym SOD to stand for Sum Of Digits and POD to stand for Product Of Digits. In a blog June 2021 post titled SOD ET AL, I made mention of OEIS A047791:

A047791

Numbers  such that  plus digit sum of  (A007953) equals a prime.

In March of 2024, I made a blog post titled More Sequences Involving SOD and POD in which I looked at semiprimes and sphenic numbers that remain semiprimes and sphenic numbers when their sums of digits and products of digits are added to them. For some reason, I've never looked at numbers that become prime when their sum of digits is added to them and that also become prime when their product of digits is added to them. When considering the product of digits of a number, it's usual to exclude numbers containing a zero because then the product of digits is alway zero. However, the decision can be made to exclude any zero digits in the multiplication.

Let's consider the first approach where numbers containing a zero are excluded from consideration. How many numbers \(n\) in the range up to 40000, satisfy the following criteria:

• \(n\) does not contain the digit 0
• \(n\) + SOD(\(n )\) \( \rightarrow \) a prime number
• \(n\) + POD(\(n )\) \( \rightarrow \) a prime number

There are 351 such numbers (permalink):

1, 163, 233, 253, 293, 341, 343, 431, 473, 493, 499, 563, 611, 617, 743, 767, 923, 1241, 1421, 1423, 1483, 1489, 1849, 1867, 2147, 2231, 2233, 2273, 2327, 2369, 2413, 2543, 2569, 2633, 2639, 2783, 2839, 3287, 3449, 3649, 3661, 3829, 4121, 4211, 4217, 4273, 4387, 4459, 4529, 4547, 4729, 4921, 5263, 5461, 5627, 5663, 5861, 6121, 6127, 6181, 6527, 6529, 6617, 6637, 6653, 6743, 6761, 6857, 6949, 7223, 7429, 7681, 8273, 8431, 8491, 8569, 8671, 8789, 8837, 8839, 8899, 8983, 9263, 9641, 9649, 9689, 9869, 9881, 9889, 11233, 11237, 11251, 11297, 11341, 11567, 11581, 11611, 11657, 11677, 11897, 11899, 12151, 12379, 12443, 12553, 12557, 12667, 12797, 12977, 13163, 13211, 13231, 13321, 13453, 13457, 13523, 13547, 13673, 13697, 13729, 13837, 13877, 13879, 13891, 13969, 14137, 14311, 14353, 14423, 14429, 14467, 14483, 14533, 14537, 14623, 14641, 14647, 14689, 14957, 15121, 15167, 15217, 15257, 15277, 15361, 15413, 15451, 15491, 15619, 15721, 15727, 15769, 15781, 15859, 15947, 16319, 16391, 16427, 16513, 16577, 16799, 16993, 16997, 17183, 17299, 17329, 17837, 17879, 18197, 18229, 18287, 18517, 18559, 18751, 19381, 19411, 19457, 19523, 19549, 19583, 19673, 19691, 19741, 19831, 19859, 21311, 21379, 21467, 21511, 21577, 21593, 21737, 21751, 21977, 21991, 22121, 22123, 22259, 22369, 22387, 22549, 22657, 22747, 22837, 22921, 22981, 23117, 23179, 23269, 23353, 23519, 23573, 23599, 23719, 23731, 23791, 24259, 24343, 24499, 24527, 24761, 24949, 25111, 25153, 25223, 25283, 25319, 25333, 25441, 25517, 25519, 25577, 25681, 25771, 25847, 25913, 25991, 25997, 26233, 26251, 26273, 26323, 26611, 26699, 26927, 27157, 27263, 27317, 27427, 27511, 27887, 27931, 28181, 28213, 28853, 28877, 28981, 29113, 29117, 29153, 29171, 29179, 29281, 29357, 29597, 29621, 29731, 29887, 29933, 31141, 31213, 31217, 31493, 31613, 31651, 31697, 31789, 31837, 31853, 31879, 32111, 32173, 32281, 32357, 32447, 32539, 32687, 32689, 32957, 32971, 33383, 33413, 33527, 33547, 33581, 33587, 33769, 33851, 34247, 34313, 34339, 34463, 34577, 34667, 34681, 34793, 34861, 35143, 35251, 35257, 35323, 35417, 35521, 35569, 35699, 35783, 35831, 35873, 35981, 36229, 36287, 36469, 36559, 36661, 36919, 36953, 36991, 37321, 37369, 37547, 37613, 37637, 37871, 38221, 38351, 38683, 38689, 38719, 38887, 38939, 38951, 38959, 39161, 39361, 39493, 39521, 39587, 39653, 39691, 39943, 39947

The number associated with my diurnal age today, 27887, can be found in this list because:

• SOD(27887) = 32 and POD(27887) = 6272
• 27887 + 32 = 27919 which is a prime number
• 27887 + 6272 = 34159 which is a prime number

If we require that the initial number \(n\) be prime, then only 136 numbers qualify. These numbers thus meet the following criteria:

• \(n\) is prime and does not contain the digit 0
• \(n\) + SOD(\(n) \) \( \rightarrow \) a prime number 
• \(n\) + POD(\(n) \) \( \rightarrow \) a prime number 

Here are the numbers (permalink);

163, 233, 293, 431, 499, 563, 617, 743, 1423, 1483, 1489, 1867, 2273, 2543, 2633, 3449, 4211, 4217, 4273, 4547, 4729, 5861, 6121, 6529, 6637, 6653, 6761, 6857, 6949, 7681, 8273, 8431, 8837, 8839, 9649, 9689, 11251, 11657, 11677, 11897, 12379, 12553, 13163, 13457, 13523, 13697, 13729, 13877, 13879, 14423, 14533, 14537, 14957, 15121, 15217, 15277, 15361, 15413, 15451, 15619, 15727, 15859, 16319, 16427, 16993, 17183, 17299, 17837, 18229, 18287, 18517, 19381, 19457, 19583, 21379, 21467, 21577, 21737, 21751, 21977, 21991, 22123, 22259, 22369, 22549, 22921, 23117, 23269, 23599, 23719, 24499, 24527, 25111, 25153, 25577, 25771, 25847, 25913, 25997, 26251, 26699, 26927, 27427, 28181, 29153, 29179, 32173, 32687, 32957, 32971, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35251, 35257, 35323, 35521, 35569, 35831, 36229, 36469, 36559, 36919, 37321, 37369, 37547, 37871, 38351, 38959, 39161, 39521

If we decide to include numbers containing the digit 0 (but exclude them in the multiplication) then 444 numbers qualify in the range up to 40000. For example, 10 qualifies since SOD(10) = 1 and POD(101) = 1 and 11 is prime. The criteria to be satisfied are thus:

• \(n\) + SOD(\(n) \) \( \rightarrow \) a prime number 
• \(n\) + POD(\(n) \) \( \rightarrow \) a prime number with 0 excluded from multiplication

Here are the numbers (permalink):

1, 10, 100, 163, 233, 253, 293, 341, 343, 431, 473, 493, 499, 563, 611, 617, 743, 767, 923, 1241, 1421, 1423, 1483, 1489, 1601, 1603, 1849, 1867, 2053, 2147, 2231, 2233, 2273, 2327, 2369, 2413, 2543, 2569, 2633, 2639, 2783, 2839, 2903, 3170, 3190, 3287, 3449, 3607, 3649, 3661, 3829, 3970, 4121, 4211, 4217, 4273, 4387, 4459, 4529, 4547, 4729, 4903, 4909, 4921, 5063, 5263, 5461, 5627, 5663, 5861, 6037, 6053, 6059, 6073, 6103, 6121, 6127, 6181, 6509, 6527, 6529, 6617, 6637, 6653, 6703, 6743, 6761, 6857, 6949, 7043, 7190, 7223, 7310, 7403, 7429, 7681, 8059, 8273, 8431, 8491, 8569, 8671, 8789, 8837, 8839, 8899, 8983, 9043, 9263, 9310, 9403, 9409, 9641, 9649, 9689, 9869, 9881, 9889, 10061, 10243, 10261, 10421, 10447, 10669, 10843, 10847, 11233, 11237, 11251, 11297, 11341, 11567, 11581, 11611, 11657, 11677, 11897, 11899, 12041, 12061, 12151, 12379, 12401, 12443, 12553, 12557, 12601, 12667, 12797, 12809, 12977, 13163, 13211, 13231, 13321, 13453, 13457, 13523, 13547, 13673, 13697, 13700, 13729, 13837, 13877, 13879, 13891, 13969, 14021, 14041, 14137, 14311, 14353, 14407, 14423, 14429, 14467, 14483, 14533, 14537, 14623, 14641, 14647, 14689, 14807, 14957, 15121, 15167, 15217, 15257, 15277, 15361, 15413, 15451, 15491, 15619, 15721, 15727, 15769, 15781, 15859, 15947, 16067, 16319, 16391, 16427, 16513, 16577, 16799, 16993, 16997, 17183, 17299, 17329, 17837, 17879, 18197, 18203, 18209, 18229, 18287, 18517, 18559, 18751, 19381, 19411, 19457, 19523, 19549, 19583, 19673, 19691, 19700, 19741, 19831, 19859, 20093, 20141, 20273, 20323, 20327, 20369, 20491, 20729, 20923, 20927, 21311, 21379, 21467, 21511, 21577, 21593, 21601, 21737, 21751, 21803, 21977, 21991, 22121, 22123, 22259, 22369, 22387, 22549, 22657, 22703, 22747, 22837, 22921, 22981, 23003, 23083, 23117, 23179, 23269, 23353, 23519, 23573, 23599, 23719, 23731, 23791, 24011, 24077, 24101, 24259, 24343, 24499, 24503, 24527, 24761, 24949, 25003, 25043, 25111, 25153, 25223, 25283, 25319, 25333, 25441, 25517, 25519, 25577, 25681, 25771, 25847, 25913, 25991, 25997, 26093, 26101, 26233, 26251, 26273, 26323, 26611, 26699, 26927, 27043, 27157, 27263, 27317, 27407, 27427, 27511, 27887, 27931, 28181, 28213, 28303, 28853, 28877, 28981, 29113, 29117, 29153, 29171, 29179, 29281, 29357, 29597, 29621, 29731, 29887, 29933, 30130, 30449, 30481, 30487, 31070, 31141, 31213, 31217, 31493, 31613, 31651, 31697, 31789, 31837, 31853, 31879, 32111, 32173, 32281, 32357, 32447, 32539, 32687, 32689, 32957, 32971, 33383, 33413, 33527, 33547, 33581, 33587, 33769, 33851, 34247, 34313, 34339, 34409, 34463, 34577, 34609, 34667, 34681, 34793, 34861, 35143, 35251, 35257, 35323, 35417, 35521, 35569, 35699, 35783, 35831, 35873, 35981, 36023, 36047, 36229, 36287, 36469, 36559, 36607, 36661, 36919, 36953, 36991, 37070, 37300, 37321, 37369, 37547, 37613, 37637, 37700, 37871, 38023, 38221, 38351, 38683, 38689, 38719, 38801, 38807, 38887, 38939, 38951, 38959, 39161, 39361, 39493, 39521, 39587, 39653, 39691, 39943, 39947

If the initial number is required to be prime, then only 170 numbers satisfy the following criteria:

• \(n\) is prime
• \(n\) + SOD(\(n) \) \( \rightarrow \) a prime number 
• \(n\) + POD(\(n) \) \( \rightarrow \) a prime number with 0 excluded from multiplication

Here are the numbers (permalink):

163, 233, 293, 431, 499, 563, 617, 743, 1423, 1483, 1489, 1601, 1867, 2053, 2273, 2543, 2633, 2903, 3449, 3607, 4211, 4217, 4273, 4547, 4729, 4903, 4909, 5861, 6037, 6053, 6073, 6121, 6529, 6637, 6653, 6703, 6761, 6857, 6949, 7043, 7681, 8059, 8273, 8431, 8837, 8839, 9043, 9403, 9649, 9689, 10061, 10243, 10847, 11251, 11657, 11677, 11897, 12041, 12379, 12401, 12553, 12601, 12809, 13163, 13457, 13523, 13697, 13729, 13877, 13879, 14407, 14423, 14533, 14537, 14957, 15121, 15217, 15277, 15361, 15413, 15451, 15619, 15727, 15859, 16067, 16319, 16427, 16993, 17183, 17299, 17837, 18229, 18287, 18517, 19381, 19457, 19583, 20323, 20327, 20369, 21379, 21467, 21577, 21601, 21737, 21751, 21803, 21977, 21991, 22123, 22259, 22369, 22549, 22921, 23003, 23117, 23269, 23599, 23719, 24077, 24499, 24527, 25111, 25153, 25577, 25771, 25847, 25913, 25997, 26251, 26699, 26927, 27043, 27407, 27427, 28181, 29153, 29179, 30449, 32173, 32687, 32957, 32971, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35251, 35257, 35323, 35521, 35569, 35831, 36229, 36469, 36559, 36607, 36919, 37321, 37369, 37547, 37871, 38351, 38959, 39161, 39521

Special Primes

I'm surprised that I haven't written about these sorts of primes before. The primes form OEIS A092529:

A092529  primes \(p\) such that both the digit sum of \(p\) plus \(p\) and the digit product of \(p\) plus \(p\) are also primes (zeroes are not permitted).

Up to 40000, there are 136 such primes and they are (permalink):

163, 233, 293, 431, 499, 563, 617, 743, 1423, 1483, 1489, 1867, 2273, 2543, 2633, 3449, 4211, 4217, 4273, 4547, 4729, 5861, 6121, 6529, 6637, 6653, 6761, 6857, 6949, 7681, 8273, 8431, 8837, 8839, 9649, 9689, 11251, 11657, 11677, 11897, 12379, 12553, 13163, 13457, 13523, 13697, 13729, 13877, 13879, 14423, 14533, 14537, 14957, 15121, 15217, 15277, 15361, 15413, 15451, 15619, 15727, 15859, 16319, 16427, 16993, 17183, 17299, 17837, 18229, 18287, 18517, 19381, 19457, 19583, 21379, 21467, 21577, 21737, 21751, 21977, 21991, 22123, 22259, 22369, 22549, 22921, 23117, 23269, 23599, 23719, 24499, 24527, 25111, 25153, 25577, 25771, 25847, 25913, 25997, 26251, 26699, 26927, 27427, 28181, 29153, 29179, 32173, 32687, 32957, 32971, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35251, 35257, 35323, 35521, 35569, 35831, 36229, 36469, 36559, 36919, 37321, 37369, 37547, 37871, 38351, 38959, 39161, 39521

Let's just check the first number in this sequence, 163. The digit sum is 10 and the digit product is 18. Now 163 + 10 = 173 which is prime and 163 + 18 = 181 is also prime.

Interestingly if we consider subtraction instead of addition then no such primes exist. That is to say that, even up to one million, there are no primes such that \(p\) minus the digit sum of \(p\) and \(p\) minus the digit product of \(p\) are also primes (with zeroes not permitted). I'm not sure why this is so.

We can thin the ranks of the above primes if we require that the sum and product of the squares of the digits also form primes when added to the original prime. Only four numbers such satisfy all these criteria in the range up to 40000. These are 1423, 13697, 14533 and 33413 (permalink). Let's examine the first of these numbers 1423. The sum of digits is 10 and the product is 24. The digits squared are 1, 16, 4 and 9 with a sum of 30 and a product of 576. Thus we have:

• 1423 + 10 = 1433 (prime)
• 1423 + 24 = 1447 (prime)
• 1423 + 30 = 1453 (prime)
• 1423 + 576 = 1999 (prime)

If we extend the range up to one million, there are 48 numbers that satisfy:

1423, 13697, 14533, 33413, 53419, 57529, 61991, 71569, 91129, 125789, 128153, 132527, 132679, 143477, 149161, 159463, 223423, 238649, 275929, 284831, 288493, 297613, 316343, 337261, 343639, 367819, 375227, 441797, 447791, 498733, 512521, 573829, 574969, 582937, 613673, 626723, 722333, 723923, 728681, 735283, 746533, 748883, 752273, 762539, 766531, 836917, 869951, 872959

To thin this sequence further, let's impose additional criteria, specifically that the number plus the sum of the cubes of the digits and the number plus the product of the cubes of the digits must be prime as well. Here are the numbers up to one million: 125789, 132527, 573829 and 752273. Let's look at the first of these numbers, 125789:

• 125789 + 32 = 125821 (prime)
• 125789 + 5040 = 130829 (prime)
• 125789 + 224 = 126013 (prime)
• 125789 + 25401600 = 25527389 (prime)
• 125789 + 1718 = 127507 (prime)
• 125789 + 128024064000 = 128024189789 (prime)

Notice how the product of the squares and cubes of the digits are larger than the number itself. If we remove the requirement that the initial number be prime then a few more numbers satisfy the criteria for digits, digits squared and digits cubed. These are 13969, 65821, 125789, 132527, 349789, 537881, 545123, 573829 and 752273 (primes are shown in red).

Jordan-Polya Numbers

The number associated with my diurnal age today, 27648, is a Jordan-Polya number which Numbers Aplenty defines as a number that can be written as the product of factorial numbers. In the case of 27648 we have:$$\begin{align} 27648 &=2^{10} \times 3^3\\&=(4!)^3 \times 2! \end{align} $$There aren't that many of them in the range up to 40,000. Here are the initial numbers:

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, 2592, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5184, 5760, 6144, 6912, 7680, 7776, 8192, 8640, 9216, 10080, 10368, 11520, 12288, 13824, 14400, 15360, 15552, 16384, 17280, 18432, 20160, 20736, 23040, 24576, 25920, 27648, 28800, 30240, 30720, 31104, 32768, 34560, 36864

Here is what Wikipedia had to say about them:

In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, 480 is a Jordan–Pólya number because:$$480 =2! \times 2! \times 5!$$Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George Pólya, who both wrote about them in the context of symmetries of trees.

These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials.

A tree that (as an abstract graph) has 480 symmetries (automorphisms). There are 2 ways of permuting the two children of the upper left vertex, 2 ways of permuting the two children of the upper middle vertex, and 5! = 120 ways of permuting the five children of the upper right vertex, for 2 x 2 x 120 = 480 symmetries altogether.

George Pólya
13 December 1887 - 7 September 1985

George Pólya was one of the greatest mathematicians of the 20th century and there is a brief biography of his life to be found here. To quote from this article:

With no hesitation, George Pólya is my personal hero as a mathematician. ... [he] is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known - these are all components of his happy personality. As a mathematician, his depth, speed, brilliance, versatility, power and universality are all inspiring. Would that there were a way of teaching and learning these traits.

Camille Jordan
5th January 1838 - 22nd January 1922

Camille Jordan made most of his mathematical contributions in the 19th century and a brief biography can be found here. It can be noted that, quoting from the article:

The Journal de Mathématiques Pure et Appliquées was a leading mathematical journal and played a very significant part in the development of mathematics throughout the 19th century. It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836. Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.

Dancing Digits

Whenever I'm confronted with a number associated with my diurnal age that seems to have no interesting properties, I inevitably find something very special and interesting about that number. Yesterday's number, 27542, was a number of this sort and it took me a day to stumble upon what's interesting about it.

My starting point was that it's a sphenic number because:$$2542=2 \times 47 \times 293$$Such numbers can be viewed as sphenic bricks with the three prime factors corresponding to the length, width and height. The surface area of such a brick means that there is always a second number that is inextricably linked to the original sphenic number and I've written about this in earlier posts. In the case of 27542, this second number and the surface area of the brick is 28902. This second number however, is also sphenic since we have:$$28902=2 \times 3 \times 4817$$This means that we can find the surface area of this second brick. It is 48182 which is not sphenic. However, we now have a triplet of numbers formed:$$27542, 28902, 48182$$If we find the product of these three numbers, it turns out to be an interesting number:$$27542 \times 28902 \times 48182 = 38353781868888$$It's interesting because it's 14 digits long and the digit 8 comprises precisely half of them.

The question then is how common is it for such triplets of numbers, when multiplied, to generate a number in which a single digit comprises at least 50% of all the digits? Let's reflect on the criteria for such numbers:

• the number must be sphenic and constitutes the first sphenic brick: p
• the surface area of this brick must also be a sphenic number: q
• this second number constitutes the second sphenic brick
• the surface area of this second brick constitutes the third number: r
• the product of p, q and r must contain a digit that comprises at least 50% of the digits of the number.

In the case of the digit 8, there are only three other numbers that qualify in the range up to 100,000 and these can be viewed in Figure 1. The first number in the list is 27542.

Figure 1: plethora of the digit 8

So it turns out that 27542 is the first member of a rather special sequence indeed. What about other digits? Let's start with 0.  Figure 2 shows the results for the digit 0, again up to 100,000.

Figure 2: plethora of the digit 0

The results for the digit 1 are shown in Figure 3.

Figure 3: plethora of the digit 1

For digits 2 and 3 there are no numbers and the results for digit 4 are shown in Figure 4.

Figure 4: plethora of the digit 4

For digit 6, 7 and 9 only one result is found in each case. See Figures 5, 6, 7 and 8.

Figure 5: plethora of the digit 5

Figure 6: plethora of the digit 6

Figure 7: plethora of the digit 7

Figure 8: plethora of the digit 9

Here is a permalink to the algorithm used to generate these numbers. Overall then, the numbers which produce a single digit that accounts for at least 50% of the final product of digits are:

1833, 1887, 7189, 14833, 15589, 16242, 16405, 27542, 36449, 38006, 38319, 43589, 87731

A very exclusive club indeed. Of course these number properties are base-dependent and so  fall into the realm of recreational mathematics but numberphiles are indifferent to such divisions and simply delight in the dance of the digits.

Of Substrings and Divisors

Today I turned 27375 days old and this number has an interesting property in that:$$ \begin{align} 27375 &=375 \times 73  \\ &=5 \times 73 \times 75 \end{align}$$Looking at the numbers on the RHS of the equations, it can be seen that 5, 73, 75 and 375 are all substrings of the string 27375, considering the numbers as collections of characters rather than digits. The numbers with this property form OEIS A059470:

A059470

Numbers that are the products of distinct substrings (>1) of themselves and do not end in 0.

These numbers are not numerous and up to 40000 they are:

125, 375, 735, 1197, 1296, 1352, 1593, 1734, 2346, 3125, 4224, 4872, 5775, 8448, 9072, 11715, 12768, 13455, 14476, 14673, 15625, 16128, 17136, 17493, 18432, 21168, 22176, 23184, 23391, 27216, 27375, 27648, 27864, 32256, 34272, 34398, 36288, 36864, 37296, 39375

The breakdown into divisors/substrings is as follows with some numbers having more than one representation (permalink):

125 equals the product of [25, 5]

375 equals the product of [75, 5]

735 equals the product of [35, 3, 7]

1197 equals the product of [9, 19, 7]

1296 equals the product of [9, 2, 12, 6]

1352 equals the product of [2, 52, 13]

1593 equals the product of [9, 3, 59]

1734 equals the product of [17, 34, 3]

2346 equals the product of [3, 34, 23]

3125 equals the product of [25, 125]

4224 equals the product of [24, 2, 4, 22]

4872 equals the product of [87, 7, 8]

4872 equals the product of [2, 4, 87, 7]

5775 equals the product of [75, 77]

8448 equals the product of [48, 4, 44]

9072 equals the product of [72, 9, 2, 7]

11715 equals the product of [11, 15, 71]

12768 equals the product of [2, 7, 12, 76]

13455 equals the product of [13, 3, 345]

14476 equals the product of [7, 44, 47]

14673 equals the product of [3, 73, 67]

15625 equals the product of [625, 25]

16128 equals the product of [6, 8, 12, 28]

17136 equals the product of [3, 6, 7, 136]

17493 equals the product of [17, 3, 49, 7]

18432 equals the product of [32, 4, 18, 8]

21168 equals the product of [21, 6, 168]

22176 equals the product of [176, 21, 6]

23184 equals the product of [3, 4, 84, 23]

23391 equals the product of [339, 3, 23]

27216 equals the product of [6, 21, 216]

27216 equals the product of [2, 7, 72, 27]

27375 equals the product of [375, 73]

27375 equals the product of [5, 73, 75]

27648 equals the product of [64, 2, 8, 27]

27864 equals the product of [2, 6, 86, 27]

32256 equals the product of [32, 3, 6, 56]

34272 equals the product of [3, 42, 272]

34272 equals the product of [2, 34, 7, 72]

34272 equals the product of [2, 34, 3, 4, 42]

34398 equals the product of [9, 98, 39]

36288 equals the product of [2, 3, 36, 6, 28]

36864 equals the product of [64, 3, 4, 6, 8]

37296 equals the product of [2, 37, 7, 72]

37296 equals the product of [2, 7, 296, 9]

37296 equals the product of [3, 6, 7, 296]

39375 equals the product of [3, 5, 7, 375]

I adapted the code for generating the substrings from this source (see Figure 1).

While these numbers are not frequent, there are two coming up in the relatively near future (27648 and 27864) before there is a big gap to the next number, 32256.

More Sequences Involving SOD and POD

The terms SOD and POD are used here to refer to Sum Of Digits and Product Of Digits. I've made a post titled SOD ET AL on June 29th 2021. Quite recently on March 10th 2024, I made another post titled Permutations Involving Sum and Product of Digits and like that post, this post involves a combination of SOD and POD.

On March 2nd 2024, I turned 27362 days old and the number 27362 has the following property as noted in my Airtable record:

27362 is a number \(n\) without the digit 0 with two distinct prime factors such that \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) both have two distinct prime factors. Here SOD stands for sum of digits and POD for product of digits. Note that this is different to the arithmetic and multiplicative digital roots of a number. Here the results for \(n\), \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) are: $$ \begin{align} 27362 = 2 \times 13681\\27382 = 2 \times 13691\\27866 = 2 \times 13933 \end{align}$$The members of this sequence from 27362 up to 40000 are (permalink):

27362, 27373, 27389, 27395, 27419, 27443, 27493, 27515, 27535, 27571, 27578, 27598, 27635, 27641, 27649, 27757, 27842, 27849, 27899, 27933, 27934, 28141, 28187, 28235, 28293, 28321, 28345, 28369, 28498, 28529, 28769, 28783, 28811, 28846, 28874, 28963, 29219, 29227, 29263, 29278, 29291, 29335, 29377, 29485, 29487, 29534, 29543, 29553, 29593, 29594, 29617, 29626, 29657, 29765, 29773, 29797, 29951, 31187, 31273, 31435, 31439, 31462, 31618, 31619, 31631, 31677, 31693, 31754, 31762, 31767, 31783, 31826, 31874, 31893, 32161, 32177, 32179, 32221, 32449, 32521, 32527, 32534, 32551, 32629, 32666, 32735, 32755, 32819, 32827, 32845, 32863, 32881, 33121, 33133, 33431, 33458, 33499, 33523, 33526, 33643, 33658, 33659, 33671, 33729, 33837, 33842, 33877, 33926, 33947, 33963, 33983, 34315, 34321, 34363, 34467, 34514, 34531, 34555, 34634, 34733, 34754, 34829, 34837, 34873, 34966, 34973, 34993, 35138, 35218, 35219, 35233, 35318, 35366, 35414, 35477, 35522, 35611, 35614, 35633, 35657, 35678, 35693, 35726, 35761, 35782, 35789, 35813, 35857, 35887, 35927, 36111, 36154, 36169, 36178, 36193, 36227, 36289, 36398, 36447, 36463, 36485, 36535, 36577, 36641, 36733, 36759, 36853, 36893, 36961, 37165, 37239, 37381, 37486, 37586, 37615, 37678, 37787, 37837, 37865, 37943, 37981, 38137, 38179, 38243, 38245, 38297, 38359, 38422, 38429, 38463, 38473, 38489, 38515, 38549, 38615, 38758, 38771, 38837, 38849, 38854, 38914, 38926, 38957, 38978, 38983, 38999, 39127, 39145, 39257, 39413, 39453, 39481, 39637, 39661, 39723, 39747, 39811, 39871, 39917, 39941, 39959

We can extend this idea to sphenic numbers and consider numbers \(n\) without the digit 0 with three distinct prime factors such that \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) both have three distinct prime factors. An example of such a number is 27544 where \(n\), \(n\) + SOD and \(n\) + POD factorise respectively as follows:$$ \begin{align} 27554 &= 2 \times 23 \times 599\\27577 &= 11 \times 23 \times 109\\28954 &= 2 \times 31 \times 467 \end{align}$$The numbers satisfying this condition from 27544 up to 40000 are:

27554, 27671, 27745, 27813, 27914, 27982, 28118, 28217, 28226, 28326, 28353, 28355, 28366, 28514, 28535, 28713, 28819, 28878, 28954, 29559, 29589, 29622, 29829, 29926, 29955, 29958, 31215, 31274, 31538, 31611, 31623, 31642, 31659, 31726, 31742, 31983, 32151, 32195, 32218, 32241, 32326, 32394, 32421, 32457, 32542, 32631, 32739, 32829, 32862, 32883, 32997, 33226, 33297, 33319, 33341, 33438, 33454, 33586, 33734, 33765, 33882, 33971, 34131, 34143, 34359, 34498, 34539, 34622, 34655, 34683, 34773, 34941, 34953, 34959, 34977, 35165, 35185, 35265, 35371, 35529, 35686, 35866, 35949, 36177, 36249, 36381, 36417, 36534, 36597, 36669, 36698, 36718, 36743, 36933, 37118, 37222, 37247, 37262, 37378, 37383, 37497, 37522, 37542, 38234, 38253, 38337, 38355, 38361, 38395, 38566, 38674, 39219, 39238, 39242, 39263, 39277, 39282, 39369, 39462, 39515, 39538, 39542, 39621, 39639

These sequences do not appear in the OEIS and I certainly won't be submitting them (pearls before swine) but they are interesting examples of sequences arising from a combination of SOD and POD.

Permutations Involving Sum and Product of Digits

One sequence that I was surprised NOT to find in the OEIS was one that involves adding the sum and product of a number's digits to the number itself and then comparing the two results. If the results are different but one is a permutation of the other, then the original number is a member of the sequence. Obviously numbers containing the digit 0 will not qualify as the product of the digits will always be 0. This first number to qualify is 36 where we have:

• sum of digits = 3 + 6 = 9
• number + sum of digits = 45
• product of digits = 3 x 6 = 18
• number + product of digits = 54
• 45 and 54 are permutations of the digits 4 and 5

In the range up to 40,000, there are 80 such numbers and they are (permalink):

36, 156, 438, 1145, 3228, 3348, 3414, 3711, 4314, 4689, 5769, 5949, 6219, 7311, 8343, 9216, 11245, 11257, 11439, 11523, 11558, 11619, 12145, 12512, 12821, 13266, 13512, 14346, 14512, 15123, 15212, 15312, 15412, 15512, 15612, 15712, 15812, 16119, 16236, 16344, 16512, 17512, 18221, 18484, 18512, 18551, 18844, 21145, 21512, 21699, 21821, 22314, 23214, 23238, 24216, 24574, 25112, 25474, 27237, 27369, 27999, 28121, 28233, 29331, 31266, 31512, 31896, 32214, 32238, 33597, 34299, 34461, 34554, 34632, 34776, 35112, 35445, 36216, 37341, 38232

For most of these numbers, the two results of adding the sum and the product of the digits to the number produce permutations with digits that are not identical to the original number. However, there are three numbers where this is indeed the case and these numbers are 5769, 14346 and 27369 (permalink):

• 5769 --> 5796 and 7659
• 14346 --> 14364 and 14634
• 27369 --> 27396 and 29637

These numbers are listed in the OEIS and form the initial members of OEIS A246421:

A246421

Numbers \(n\) such that (\(n\) + digit sum of \(n\)) and (\(n\) + digit product of \(n\)) are nontrivial permutations of the digits of \(n\).

All the digit sums and the digit products are multiples of 9. The first members of the sequence are as follows:

5769, 14346, 27369, 41346, 52569, 56925, 94725, 122346, 126135, 129213, 143658, 152469, 154269, 155169, 157914, 162135, 192213, 212346, 216135, 219213, 221346, 236124, 238959, 245925, 261135, 263124, 291213, 326124, 328536, 344925, 361647, 362124, 367425, 368892, 392436, 413658

I would surmise that such a series is finite because as the numbers get larger the size of the product of digits when added to the original number generates numbers with far more digits. There are two associated OEIS sequences to OEIS A246421 and they are:

A246420

Numbers \(m\) such that (\(m\) + digit sum of \(m\)) is a permutation of the decimal digits of \(m\).

Listed below are some of numbers coming up for me in terms of my diurnal age (with 27369 marking the starting point):

27369, 27513, 27558, 27702, 27747, 27891, 27936, 28035, 28224, 28269, 28413, 28458, 28602, 28647, 28836, 29124, 29169, 29313, 29358, 29502, 29547, 29736, 29925, 30123, 30168, 30312, 30357, 30501, 30546, 30735, 30924, 30969, 31023, 31068, 31212, 31257, 31401, 31446, 31635, 31824, 31869, 32112, 32157, 32301, 32346, 32535, 32724, 32769, 32913, 32958, 33012, 33057, 33201, 33246, 33435, 33624, 33669, 33813, 33858, 34101, 34146, 34335, 34524, 34569, 34713, 34758, 34902, 34947, 35001, 35046, 35091, 35235, 35424, 35469, 35613, 35658, 35802, 35847, 36135, 36324, 36369, 36513, 36558, 36702, 36747, 36891, 36936, 37035, 37224, 37269, 37413, 37458, 37602, 37647, 37836, 38124, 38169, 38313, 38358, 38502, 38547, 38736, 38925, 39024, 39069, 39213, 39258, 39402, 39447, 39636, 39780, 39825

A243102

Numbers \(n\) such that the digits of (\(n\) + product of digits of \(n\)) are a nontrivial permutation of the digits of \(n\).

Listed below are some of numbers coming up for me in terms of my diurnal age (with 27369 marking the starting point):

27369, 28179, 28195, 29123, 29154, 29213, 29381, 29397, 29873, 31126, 31213, 31235, 31238, 31259, 31354, 31365, 31561, 31925, 32113, 32265, 32286, 32341, 32352, 32492, 32538, 32743, 32793, 33125, 33129, 33142, 33158, 33186, 33248, 33253, 33294, 33455, 33456, 33475, 33558, 33585, 33965, 33967, 34135, 34156, 34167, 34351, 34356, 34526, 34535, 34553, 34563, 34599, 34655, 34951, 35123, 35134, 35165, 35231, 35262, 35267, 35361, 35463, 35616, 35625, 35652, 35673, 35684, 35763, 35794, 35837, 35861, 35974, 36123, 36154, 36178, 36213, 36381, 36722, 36825, 36935, 37168, 37813, 37849, 38143, 38153, 39183, 39251

Four Fun Facts About Triangular Numbers

In my previous post (Happy Triangular Numbers) I made reference to a website Fascinating Triangular Numbers and this post I'd like to mention just four more of the "fun facts" mentioned there.

FUN FACT 1: 

The sum of two consecutive triangular numbers is a square number. This is easily proven as follow:$$ \begin{align} T_n+T_{n+1} &= \frac{n(n+1)}{2}+ \frac{(n+1)(n+2)}{2}\\ &= \frac{n+1}{2} \cdot (2n+2)\\ &= (n+1)^2 \end{align} $$FUN FACT 2:

The sum of the squares of two consecutive triangular numbers is also a triangular number. Again this is easily proven as follows:$$ \begin{align} \big (T_n \big )^2+ \big (T_{n+1} \big )^2&= \big (\frac{n(n+1)}{2} \big )^2+ \big ( \frac{(n+1)(n+2)}{2} \big )^2\\ &= \Big (\frac{n+1}{2} \Big )^2 \cdot \Big ( n^2+(n+2)^2 \Big )\\ &= \Big (\frac{n+1}{2} \Big )^2 \cdot \Big ( 2n^2+4n+4 \Big ) \\ &= \frac {(n^2 +2n+1) \cdot (n^2+2n+2)}{2} \\ &=T_{(n+1)^2} \end{align} $$FUN FACT 3:

There are infinitely many triangular numbers, which are also squares as given by the series 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056 etc. These can be termed as square triangular numbers. The \(n\)th Square Triangular number \(K_n\) can easily be obtained from the recursive formula: $$K_n = 34 \times K_{n-1} - K_{n-2} + 2$$So knowing the first two square triangular numbers i.e. \(K_1 = 1\) and \(K_2 = 36\) , all other successive Square Triangular numbers can be obtained. For example:$$ \begin{align} K_3 &= 34 \times K_2 - K_1 + 2 \\ &= 34 \times 36 -1 + 2 \\ &= 1225 \\ &= 35^2 \\ K_4 &= 34 \times K_3 - K_2 + 2 \\ &= 34 \times 1225 - 36 + 2 \\ &= 41616 \\ &=204^2 \end{align}$$FUN FACT 4:

There exist infinite triangular numbers that are simultaneously the sum, the difference and the product of two other triangular numbers. Here is a list of the initial such numbers:$$ \begin {align} 990 &= 1035 - 45 = 780 + 210 = 66 \times 15\\

1540 &= 1711 - 171 = 1485 + 55 = 55 \times 28\\

2850 &= 3003 - 153 = 2415 + 435 = 190 \times 15\\

4851 &= 5151 - 300 = 3081 + 1770 = 231 \times 21\\

8778 &= 10731 - 1953 = 7875 + 903 = 2926 \times 3\\

11781 &= 12246 - 465 = 11628 +153 = 561 \times 21\\

15400 &= 18721 - 3321 = 14365 + 1035 =1540 \times 10\\

26796 &= 27261 - 465 = 26565 + 231 = 406 \times 66\\

43956 &= 44551 - 595 = 41328 + 2628 = 666 \times 66 \end{align} $$Here are the initial triangular numbers along with their associated indices (permalink):

[(1, 1), (3, 2), (6, 3), (10, 4), (15, 5), (21, 6), (28, 7), (36, 8), (45, 9), (55, 10), (66, 11), (78, 12), (91, 13), (105, 14), (120, 15), (136, 16), (153, 17), (171, 18), (190, 19), (210, 20), (231, 21), (253, 22), (276, 23), (300, 24), (325, 25), (351, 26), (378, 27), (406, 28), (435, 29), (465, 30), (496, 31), (528, 32), (561, 33), (595, 34), (630, 35), (666, 36), (703, 37), (741, 38), (780, 39), (820, 40), (861, 41), (903, 42), (946, 43), (990, 44), (1035, 45), (1081, 46), (1128, 47), (1176, 48), (1225, 49), (1275, 50), (1326, 51), (1378, 52), (1431, 53), (1485, 54), (1540, 55), (1596, 56), (1653, 57), (1711, 58), (1770, 59), (1830, 60), (1891, 61), (1953, 62), (2016, 63), (2080, 64), (2145, 65), (2211, 66), (2278, 67), (2346, 68), (2415, 69), (2485, 70), (2556, 71), (2628, 72), (2701, 73), (2775, 74), (2850, 75), (2926, 76), (3003, 77), (3081, 78), (3160, 79), (3240, 80), (3321, 81), (3403, 82), (3486, 83), (3570, 84), (3655, 85), (3741, 86), (3828, 87), (3916, 88), (4005, 89), (4095, 90), (4186, 91), (4278, 92), (4371, 93), (4465, 94), (4560, 95), (4656, 96), (4753, 97), (4851, 98), (4950, 99), (5050, 100), (5151, 101), (5253, 102), (5356, 103), (5460, 104), (5565, 105), (5671, 106), (5778, 107), (5886, 108), (5995, 109), (6105, 110), (6216, 111), (6328, 112), (6441, 113), (6555, 114), (6670, 115), (6786, 116), (6903, 117), (7021, 118), (7140, 119), (7260, 120), (7381, 121), (7503, 122), (7626, 123), (7750, 124), (7875, 125), (8001, 126), (8128, 127), (8256, 128), (8385, 129), (8515, 130), (8646, 131), (8778, 132), (8911, 133), (9045, 134), (9180, 135), (9316, 136), (9453, 137), (9591, 138), (9730, 139), (9870, 140), (10011, 141), (10153, 142), (10296, 143), (10440, 144), (10585, 145), (10731, 146), (10878, 147), (11026, 148), (11175, 149), (11325, 150), (11476, 151), (11628, 152), (11781, 153), (11935, 154), (12090, 155), (12246, 156), (12403, 157), (12561, 158), (12720, 159), (12880, 160), (13041, 161), (13203, 162), (13366, 163), (13530, 164), (13695, 165), (13861, 166), (14028, 167), (14196, 168), (14365, 169), (14535, 170), (14706, 171), (14878, 172), (15051, 173), (15225, 174), (15400, 175), (15576, 176), (15753, 177), (15931, 178), (16110, 179), (16290, 180), (16471, 181), (16653, 182), (16836, 183), (17020, 184), (17205, 185), (17391, 186), (17578, 187), (17766, 188), (17955, 189), (18145, 190), (18336, 191), (18528, 192), (18721, 193), (18915, 194), (19110, 195), (19306, 196), (19503, 197), (19701, 198), (19900, 199), (20100, 200), (20301, 201), (20503, 202), (20706, 203), (20910, 204), (21115, 205), (21321, 206), (21528, 207), (21736, 208), (21945, 209), (22155, 210), (22366, 211), (22578, 212), (22791, 213), (23005, 214), (23220, 215), (23436, 216), (23653, 217), (23871, 218), (24090, 219), (24310, 220), (24531, 221), (24753, 222), (24976, 223), (25200, 224), (25425, 225), (25651, 226), (25878, 227), (26106, 228), (26335, 229), (26565, 230), (26796, 231), (27028, 232), (27261, 233), (27495, 234), (27730, 235), (27966, 236), (28203, 237), (28441, 238), (28680, 239), (28920, 240), (29161, 241), (29403, 242), (29646, 243), (29890, 244), (30135, 245), (30381, 246), (30628, 247), (30876, 248), (31125, 249), (31375, 250), (31626, 251), (31878, 252), (32131, 253), (32385, 254), (32640, 255), (32896, 256), (33153, 257), (33411, 258), (33670, 259), (33930, 260), (34191, 261), (34453, 262), (34716, 263), (34980, 264), (35245, 265), (35511, 266), (35778, 267), (36046, 268), (36315, 269), (36585, 270), (36856, 271), (37128, 272), (37401, 273), (37675, 274), (37950, 275), (38226, 276), (38503, 277), (38781, 278), (39060, 279), (39340, 280), (39621, 281), (39903, 282), (40186, 283), (40470, 284), (40755, 285), (41041, 286), (41328, 287), (41616, 288), (41905, 289), (42195, 290), (42486, 291), (42778, 292), (43071, 293), (43365, 294), (43660, 295), (43956, 296), (44253, 297), (44551, 298), (44850, 299), (45150, 300)]

The results for FUN FACT 4 could be written in terms of these indices. For example:$$ \begin {align} 990 &= 1035 - 45 = 780 + 210 = 66 \times 15\\ \text{T}_{44} &= \text{T}_{45} -\text{T}_{9} = \text{T}_{39} + \text{T}_{20} = \text{T}_{11} \times \text{T}_{5} \end{align}$$

Product Sum Ratios

I'm surprised I've not made a post about this topic before. The topic concerns the result when the product of the digits of a number is divided by the sum of its digits. Of interest is:

• when is the result of this division an integer?
• what numbers are associated with records in the size of this integer?

Clearly 1 will start the ball rolling with a record of 1 but after that the next number is 36 with a product of 18 and a sum to 9, giving a record of 2. 66 is next with a product of 36 and a sum of 12, giving a record of 3. Figure 1 shows a list of the record numbers up 10,000 followed by a plot of these numbers (Figure 2). 

Figure 1: permalink

Figure 2: permalink

Extending out to 100,000, Figure 3 shows an extended table and Figure 4 shows a plot of the full range of record-breaking numbers.

Figure 3: permalink

Figure 4: permalink

I won't show a table of results up to one million but here are the 59 record-breaking numbers (permalink):

1, 36, 66, 88, 257, 268, 279, 369, 459, 578, 579, 678, 789, 999, 2589, 2688, 2799, 3699, 3789, 4599, 4689, 4789, 5788, 5889, 7889, 8888, 18999, 25889, 26789, 26888, 27788, 28899, 37899, 38889, 45999, 46899, 47799, 47889, 55899, 56889, 57789, 58999, 78999, 257899, 258889, 267999, 277899, 278889, 367899, 377889, 378888, 457899, 459999, 489999, 588999, 678999, 688899, 778899, 778999

The results for the final number 778999 are:$$ \frac{ 285768} {49} =5832$$Figure 5 shows a plot of these numbers.

Figure 5: permalink

These numbers define OEIS A240520 that clearly goes on forever:

A240520

Numbers that set a new integer record for the ratio between the product and the sum of their digits

Interestingly, if the search is extended for numbers up to two million, only one new number is added and that is 1899999 with a digit product of 472392 and a sum of 54, giving the integer 8748 as the result.

Extending the search to three million, we get four new numbers.

Figure 6: permalink

The plot of the full range of numbers is shown in Figure 7.

Figure 7: permalink

What we see in Figure 7 is that 1899999 is very much a singleton, sitting there is splendid isolation. This is even more evident when the search is extended to four million. Three new numbers appear (Figure 8) but they are close together (Figure 9). 

Figure 8: permalink

Figure 9: permalink