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