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.

An Interesting Prime

Being born on the 3rd April 1949, my date of birth is often represented as 3 - 4 - 49. These numbers when concatenated form the prime number 3449. I was reminded of this number because of the factorisation of the number associated with my diurnal age today, 27592.$$27592 = 2^3 \times 3449 = 8 \times 3449$$So today my life can be divided into exactly eight equal parts, each of them 3449 days long which is about 9.44 years. The previous multiple$$7 \times 3449 = 24143$$occurred on May 10th 2015 when I was still working at the Shanghai Singapore International School. The next multiple$$9 \times 3449 = 31041$$ will fall on March 29th 2034, shortly before my 84th birthday (if I make it that far).

3449 forms the initial prime of a Cunningham chain of the first type with length exactly 3 and so: $$ 3449 \rightarrow 2 \times 3449 + 1 = 6889 \text{ (prime)} \\ 6889 \rightarrow 2 \times 6889 +1 = 13799 \text{ (prime)}$$Primes with this property form OEIS A059762. Another prime-related property of 3449 qualifies it for membership in OEIS A088483:

A088483: primes \( \textit{p} \) such that \(p^2+p-1\) and \(p^2+p+1\) are twin primes.

For 3449, the twin primes are \(11899049\) and \(11899051\). 

3449 is also a home prime with a homeliness of 3 because:$$ \begin{align} 611 &= 13 \times 47 \rightarrow 1347\\ 1347 &= 3 \times 449 \rightarrow 3449 \end{align}$$Not all primes are home primes of course. Take 613 as an example of a prime that is not a home prime because it cannot be formed by the concatenation of the prime factors of any number (the prime factors need to be concatenated in ascending order).

3449 is also a member of OEIS A153116:

A153116: primes \(p\) such that \(p^2 +12\) and \(p^2-12\) are also primes.

Here the two primes are \(11895589\) and \(11895613\). Additionally:$$ \text{period of}\frac{1}{3449}=\frac{3449-1}{8} = 431$$This property qualifies 3449 for membership of OEIS A056213:

A056213: primes \(p\) for which the period of reciprocal = \(\dfrac{p-1}{8}\).

3449 is a Sophie Germain prime because:$$2 \times 3449+1=6899 \text{ is prime}$$3449 also features in so-called "Golden Semiprimes" and this qualifies it for membership in OEIS A108544:

A108544: primes that are factors of distinct golden semiprimes (A108540).

I posted about these types of semiprimes in Semiprime Factor Ratios way back on the 26th August 2016. In that post I said that:

A golden semiprime is a number that factors to:

• \(p \times q\) where \(p\) and \(q\) are prime
• \( | \,p \, \phi - q \,| <1\) where \(\phi=\dfrac{\sqrt{5}+1}{2} \)

In the case of 3449, it is the \(q\) and \(p=2131\) and the golden semiprime is:$$7349819=2131 \times 3449$$The OEIS mentions 314 sequences in which 3449 makes an appearance and I've only dealt with a few of them here. However, I see 3449 as an important number in my life and didn't want its current occurrence to pass unnoticed.

New Telephone Number

It's always exciting to get a new telephone number because of the properties of the number that may turn up. Having arrived in Australia for a temporary stay, I needed a telephone number and the number that I was given was:$$0451591949 \rightarrow 451591949$$Now this number is prime but has additional "primeness" embedded in it because:$$ \text{Sum of digits is }47 \text{ and prime}\\47 \rightarrow 4 + 7 = 11 \\ 11 \rightarrow 1+1=2 \\ \text{ 2 (the digital root) is prime}$$Now 1949 and 1951 form a pair of twin primes but it turns out that:$$ 451591949 \text{ and } 4511591951 \\ \text{ also form a pair of twin primes}$$Furthermore:$$451591949 \text{ is a Sophie Germain prime} \\ 451591949 \times 2 + 1 = 903183899 \text{ a prime}$$\(451591949\) is also a Chen prime defined as a prime number  \(p\)  such that  \(p+2\)  is either a prime or a semiprime. Jing Run Chen, after which they are named, proved in 1966 that there are infinitely many such primes. Binbin Zhou has proved in 2009 that the Chen primes contain arbitrarily long arithmetic progressions.

So overall I'm very happy with my new prime telephone number even though it will lapse and be discarded once I leave the country. For now though it's mine and prime!

Triple Strength Sphenic Numbers And More

Strong primes in cryptography are primes that are difficult to factor but in number theory a strong prime \(p_n\) has the property that:$$p_n>\frac{p_{n-1}+p_{n+1}}{2}$$If a sphenic number has three strong primes then it might be termed "triple strength" which is just a term that I came up with. These sorts of sphenic number constitute OEIS A363782:

A363782

Products of three distinct strong primes.

There are only 44 such sphenic numbers in the range up to 40,000 and they are (permalink):

5423, 6919, 7667, 11033, 11803, 12529, 13079, 13277, 14773, 16687, 18139, 18241, 18821, 18887, 20009, 20213, 21373, 22649, 23749, 24013, 25201, 25619, 25789, 26609, 27269, 27863, 28897, 29087, 30217, 30481, 30943, 32021, 32153, 32219, 33031, 33473, 34133, 35003, 35629, 35717, 36839, 37111, 38947, 39479

Take 5423 = 11 x 17 x 29 as an example. 11 is closer to 13 than it is to 7, 17 is closer to 19 than it is to 13 and 29 is closer to 31 than it is to 23.

By contrast, sphenic numbers in which all three primes are weak are more numerous. A weak prime \(p_n\) has the property that:$$p_n<\frac{p_{n-1}+p_{n+1}}{2}$$These are not listed in the OEIS but there are 161 in the range up to 40,000 and they are (permalink):

1729, 2093, 2821, 3059, 3913, 4123, 4277, 4991, 5551, 5681, 5719, 6251, 6643, 6923, 7553, 7567, 7657, 8099, 8113, 9269, 9331, 9373, 9709, 9821, 9919, 10199, 10283, 10621, 11039, 11609, 11753, 11837, 11921, 12649, 12857, 13237, 13363, 13547, 13699, 13741, 14053, 14147, 14329, 14497, 15029, 15067, 15197, 15841, 16471, 16583, 17329, 17423, 17549, 17563, 18011, 18031, 18109, 18193, 18239, 18361, 18487, 18791, 18941, 19313, 20069, 20083, 20501, 20539, 20839, 21091, 21203, 21827, 21931, 21973, 21983, 22211, 22351, 22379, 23653, 24017, 24073, 24311, 24521, 24583, 24661, 24817, 24983, 25327, 25441, 25669, 25753, 26273, 26467, 26611, 26657, 26663, 26789, 26887, 26923, 27307, 27683, 27911, 28427, 28483, 28847, 29141, 29281, 29419, 30163, 30457, 30659, 30667, 30797, 30989, 31003, 31073, 31171, 31759, 31901, 32039, 32053, 32123, 32357, 32591, 32669, 32767, 32809, 33449, 33511, 33787, 33887, 34013, 34099, 34333, 34853, 35399, 35441, 35861, 35867, 35929, 36043, 36239, 36271, 36491, 36869, 37177, 37219, 37271, 37297, 37513, 37639, 38003, 38311, 38399, 38801, 38893, 38969, 39169, 39277, 39403, 39431

It should be noted that the famous taxi cab number, 1729, is the first member of this sequence and it factorises as follows:$$1729=7 \times 13 \times 19$$With 1729, we see that 7 is weak because it's closer to 5 than 11, 13 is weak because it's closer to 11 than 17 and 19 is weak because it's closer to 17 than it is to 23.

Primes that are the average of the previous prime and the next prime are called balanced primes. Sphenic numbers that are the product of three balanced primes are not numerous and there are none in the range up to 40,000. However, in the range up to one million, there are 82 of them. They are not listed in the OEIS and they are as follows (permalink):

41605, 45845, 55915, 68105, 69695, 98845, 135805, 149195, 157145, 160855, 165635, 173045, 182515, 194245, 201745, 206455, 222305, 227495, 250955, 258905, 271135, 277465, 292295, 292805, 297595, 314555, 322645, 324095, 337955, 362255, 393515, 400415, 441955, 462955, 464545, 465505, 476495, 479305, 486995, 490495, 505355, 512605, 512945, 525055, 564845, 575405, 593965, 606055, 625615, 634045, 640385, 640505, 688915, 709405, 723455, 740345, 743395, 762005, 766945, 769295, 773315, 779795, 779995, 785195, 798205, 819155, 839105, 845105, 858695, 865855, 876355, 877945, 881555, 931795, 941905, 954095, 960055, 963805, 963895, 971395, 989245, 999085

Take 41605 = 5 x 53 x 157 as an example. We have 5 as the average of 3 and 7, 53 as the average of 47 and 59 and 157 as the average of 151 and 163,

What about sphenic numbers that are the products of Sophie Germain primes? These turn out to be more numerous. There are 655 in the range up to 40000 out of a total of 7720. These numbers form OEIS A157346:

A157346

Products of 3 distinct Sophie Germain primes.

The initial members are (permalink):

30, 66, 110, 138, 165, 174, 230, 246, 290, 318, 345, 410, 435, 498, 506, 530, 534, 615, 638, 678, 759, 786, 795, 830, 890, 902, 957, 1038, 1074, 1130, 1146, 1166, 1245, 1265, 1310, 1334, 1335, 1353, 1398, 1434, 1506, 1595, 1686, 1695, 1730, 1749, 1758, 1790

Take 30 = 2 x 3 x 5 as an example. 2 x 2 + 1 = 5, 2 x 3 + 1 = 7 and 2 x 5 + 1 = 11.

What about sphenic numbers that are the product of primes that are the smaller of twin prime pairs? There are 277 such numbers in the range up to 40,000 and they are not listed in the OEIS. These numbers are as follows (permalink):

165, 255, 435, 561, 615, 885, 935, 957, 1065, 1353, 1479, 1515, 1595, 1605, 1947, 2055, 2091, 2235, 2255, 2343, 2465, 2685, 2865, 2955, 3009, 3245, 3333, 3405, 3485, 3531, 3567, 3585, 3621, 3905, 4035, 4215, 4521, 4665, 4917, 5015, 5133, 5151, 5205, 5423, 5457, 5555, 5885, 5907, 5945, 6035, 6177, 6285, 6303, 6465, 6501, 6915, 6987, 7257, 7491, 7535, 7599, 7667, 7815, 7887, 8195, 8535, 8555, 8585, 8733, 8787, 8877, 8985, 9095, 9129, 9255, 9273, 9309, 9615, 9741, 9845, 9885, 10047, 10263, 10295, 10505, 10835, 11033, 11451, 11577, 11645, 11919, 12095, 12135, 12189, 12315, 12405, 12423, 12485, 12567, 12665, 12855, 12963, 13079, 13145, 13161, 13215, 13277, 13719, 13827, 14223, 14331, 14555, 14645, 14795, 15213, 15215, 15285, 15455, 15465, 15515, 15573, 15735, 15861, 15915, 16235, 16365, 16617, 16745, 16851, 17105, 17139, 17193, 17265, 17697, 17877, 18327, 18435, 18777, 18821, 18887, 18939, 19085, 19155, 19295, 19335, 19515, 19749, 19767, 19785, 19865, 20009, 20213, 20315, 20361, 20705, 20793, 20945, 21153, 21369, 21405, 21513, 21605, 21747, 21765, 21935, 21981, 22017, 22215, 22305, 22649, 22791, 22865, 23045, 23403, 23493, 23511, 23705, 23885, 24105, 24231, 24249, 24285, 24447, 25005, 25355, 25455, 25619, 25815, 25955, 26373, 26435, 26571, 26609, 26697, 26805, 27057, 27093, 27291, 27695, 27863, 27921, 28065, 28085, 28155, 28281, 28565, 28655, 28965, 29019, 29073, 29087, 29181, 29235, 29397, 29495, 29795, 29955, 30189, 30405, 30545, 30549, 31215, 31295, 31305, 31467, 31565, 31665, 31683, 31737, 31935, 32021, 32115, 32219, 32421, 32691, 32915, 32945, 33087, 33473, 33555, 33609, 33627, 33807, 33935, 34005, 34023, 34133, 34563, 34617, 34635, 34655, 34869, 35003, 35013, 35085, 35255, 35615, 35715, 35717, 35855, 36003, 36245, 36453, 36635, 36695, 36839, 37497, 37983, 37985, 38127, 38235, 38253, 38865, 39005, 39155, 39185, 39855

Take 165 = 3 x 5 x 11 as an example. 3 is the smaller of the twin prime pair (3, 5), 5 is the smaller of the twin prime pair (5, 7) and 11 is the smaller of the twin prime pair (11, 13). At first sight, there appears to be too many numbers because the lesser prime of a twin prime pair is always a strong prime if the prime is greater than 5. This is because in a twin prime pair \( (p, p + 2)\) with \(p > 5\), \(p\) is always a strong prime, since \(3\) must divide \(p − 2\), which cannot be prime. 

Once we take out any sphenic numbers containing 3 or 5, we are left with only 21 sphenic numbers that contain primes that are both strong and the lesser of twin primes. These numbers are (permalink):

5423, 7667, 11033, 13079, 13277, 18821, 18887, 20009, 20213, 22649, 25619, 26609, 27863, 29087, 32021, 32219, 33473, 34133, 35003, 35717, 36839

Take 5423 = 11 x 17 x 29 as an example. This example was given earlier as the first member of OEIS  A363782 where we saw that 11 is closer to 13 than it is to 7, 17 is closer to 19 than it is to 13 and 29 is closer to 31 than it is to 23. Now, additionally, it can be seen that 11 is the lesser of (11, 13), 17 is the lesser of (17, 19) and 29 is the lesser of (29, 31).

What about sphenic numbers that are the product of primes that are the larger of twin prime pairs? There are 166 in the range up to 40,000 and they are (permalink):

455, 665, 1085, 1235, 1505, 1729, 2015, 2135, 2555, 2795, 2821, 2945, 3605, 3815, 3913, 3965, 4085, 4123, 4745, 4865, 5285, 5551, 5719, 5795, 6335, 6643, 6665, 6695, 6755, 6935, 6965, 7085, 7657, 8015, 8113, 8435, 9035, 9331, 9373, 9455, 9485, 9709, 9785, 9815, 9905, 9919, 10355, 10621, 10955, 11315, 11765, 12215, 12545, 12649, 12935, 13115, 13205, 13237, 13699, 13741, 14345, 14497, 14735, 14885, 15067, 15155, 15665, 15695, 15841, 15965, 16205, 16471, 16895, 17195, 17329, 17563, 17615, 18031, 18109, 18305, 18335, 18361, 18395, 18487, 18905, 19985, 20083, 20345, 20839, 21035, 21545, 21665, 21755, 21931, 21973, 22145, 22265, 22351, 22505, 22685, 22895, 23135, 23405, 23435, 23653, 24073, 24583, 24661, 25327, 25441, 25669, 25745, 25753, 26467, 26885, 26923, 27365, 28055, 28145, 28385, 28483, 28805, 29015, 29419, 29735, 29885, 29915, 30065, 30095, 30163, 30457, 30845, 30905, 31003, 31171, 31415, 31759, 32053, 32465, 32767, 32809, 33155, 33245, 33995, 34099, 34333, 35495, 35735, 35929, 36043, 36155, 36785, 37115, 37205, 37297, 37355, 37595, 37639, 38255, 38311, 38915, 39065, 39277, 39403, 39785, 39995

Take 455 = 5 x 7 x 13 as an example. 5 is the larger of the twin prime pair (3, 5), 7 is the larger of the twin prime pair (5, 7) and 13 is the larger of the twin prime pair (11, 13).

Clearly we could go on and on but that's probably enough for now. The same approach could be applied to composite numbers that are the product of two distinct prime, four distinct primes etc. For example, numbers that are the product of four distinct strong primes form OEIS A363167:

A363167

Products of four distinct strong primes.

The 47 members up to one million are (permalink):

200651, 222343, 283679, 319957, 363341, 385033, 408221, 428417, 452353, 463573, 483923, 491249, 513689, 526031, 544357, 546601, 547723, 580261, 605693, 671143, 688721, 696377, 698819, 739211, 740333, 742951, 743699, 747881, 771661, 774367, 783343, 790801, 808027, 820369, 838013, 871607, 876293, 878713, 883949, 889559, 928609, 932437, 947903, 970717, 973709, 984533, 989791

Take 200651 = 11 x 17 x 29 x 37 as an example. 11 is closer to 13 than it is to 7, 17 is closer to 19 than it is to 13, 29 is closer to 31 than it is to 23 and 37 is closer to 41 than it is to 31. All the properties mentioned in this post are independent of the number base used.

Brazilian Primes

There's only one reference that I've made to Brazilian primes in this blog and that was in a post titled Fermat Primes and Brazilian Numbers on February 26th 2018 which is now over five years ago. I was reminded of them once again when a number associated with my diurnal age, 27061, was identified as a Brazilian prime. I wrote the following about Brazilian numbers in that earlier post:

These numbers are listed in OEIS A125134 and defined as:

Numbers \( n \) such that there is a natural number \( b \) with \( 1 < b < n-1 \) such that the representation of \( n \) in base \( b \) has all equal digits.

All even numbers \( \geq \) 8 are Brazilian numbers because: $$ \begin{align} 2p&=2(p-1)+2 \\&= 22 \end{align}$$in base \(p-1\) if \(p-1>2 \) and that is true if \(p \geq 4 \). 

The odd Brazilian numbers are listed in OEIS A257521 and are fairly common, with some being prime numbers:

7, 13, 15, 21, 27, 31, 33, 35, 39, 43, 45, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 127, 129, 133, 135, 141, 143, 145, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195, ...

As an example, take 27 in the above sequence which can be expressed as \(33_8 \). As for the even numbers, take a number like 28. It can be written as 2 x (14-1) + 2 and thus can be represented as \(22_{13} \).

Brazilian primes are simply Brazilian numbers that are prime. These constitute OEIS A085104:

A085104

Primes of the form \(1 + n + n^2 + n^3 + ... + n^k \) where \(n > 1\) and \( k > 1\).

For example, in the case of 27061 we have: $$ \begin{align} 27061& = 1+164+164^2 \\ &= 164^2+164+1\\&=111_{164} \end{align}$$These primes can be generated using this permalink. The initial members of the sequence are:

7, 13, 31, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023, 20593, 21757, 22621, 22651, 23563, 24181, 26083, 26407, 27061, 28057, 28393, 30103, 30941, 31153, 35533, 35911, 37057, 37831, 41413, 42643, 43891, 46441, 47743, 53593, 55933, 55987, 60271, 60763, 71023, 74257, 77563, 78121, 82657, 83233, 84391, 86143, 88741, 95791, 98911

The OEIS comments to this sequence are informative:

The number of terms \(k+1\) is always an odd prime, but this is not enough to guarantee a prime, for example 111 = 1 + 10 + 100 = 3 x 37.

The inverses of the Brazilian primes form a convergent series; the sum is slightly larger than 0.33.

It is not known whether there are infinitely many Brazilian primes. 

Brazilian primes can be written in the form:

$$ \dfrac{(n^p - 1)}{(n - 1}\\ \text{ where } p \text{ is an odd prime and } n > 1$$

The number of terms less than \(10^n\) are 1, 5, 14, 34, 83, 205, 542, 1445, 3880, 10831, 30699, 88285, ...

Brazilian primes fall into two classes:

• when \(n\) is prime, we get sequence OEIS A023195 except 3 which is not Brazilian,
• when \(n\) is composite, we get sequence OEIS A285017.

The conjecture that "No Sophie Germain prime is Brazilian (prime)"  is false because:$$ \begin{align} a(856) &= 28792661\\ &= 1 + 73 + 73^2 + 73^3 + 73^4 \\&= (11111)_{73} \end{align} $$and 28792661 is the 141385-th Sophie Germain prime.