SUPER SUPER-PRIME NUMBERS

Super-prime numbers (also known as higher-order primes or prime-indexed primes or PIPs) are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence begins:

3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... (OEIS A006450).

The number associated with my diurnal age today, 26489, is one such super-prime because it is the 421st prime and the number 421 is prime. So what is a super super-prime? Well, this is a terminology of my own invention, but I think it aptly describes numbers such as 26489 that have the following properties:

A331031

The prime numbers that are prime-indexed primes and whose digit sum, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.

It took me a while to take that all in so let's take it step by step for the case of 26489:

• prime-indexed prime: we have seen that 26489 is the 421st prime and that 421 is prime
  
  
• digit sum is prime: the digits of 26489 total 29 and 29 is prime
  
  
• adjacent digit sum concatenation is prime: 
• 2+6 =8, 6+4=10, 4+8=12, 8+9=17
• these adjacent digit sums give 8101217 when concatenated
• 8101217 is prime
  
  
• adjacent digit difference concatenation is prime:
• |2-6|=4, |6-4|=2, |4-8|=4, |8-9|=1
• these adjacent digit difference give 4241 when concatenated
• 4241 is prime

Here is a permalink to SageMathCell if you want to test out the algorithm for generating the sequence of "super super-primes". Up to 20425103, there are 267 members. These are shown below:

41, 83, 401, 2063, 6863, 10909, 20063, 26489, 44621, 105229, 187067, 205507, 233267, 238547, 240047, 243301, 256307, 346763, 367021, 376003, 395581, 555707, 562181, 563467, 600203, 613243, 644843, 675263, 689789, 785801, 787601, 837667, 845381, 954263, 959389, 1070203, 1089463, 1379029, 1394389, 1550503, 1759489, 1777609, 1868567, 1948603, 1994143, 2002001, 2003321, 2034521, 2071481, 2104547, 2106389, 2184101, 2191529, 2217443, 2231407, 2298389, 2303681, 2312621, 2316203, 2334281, 2342309, 2362163, 2365201, 2387003, 2395747, 2416163, 2458747, 2473067, 2491007, 2501243, 2502767, 2505263, 2578403, 2610701, 2612521, 2629307, 2717129, 2742521, 2775781, 2824447, 2858747, 2877221, 2940521, 2960381, 3030409, 3058201, 3080729, 3161309, 3267067, 3339607, 3429667, 3489007, 3510509, 3528409, 3551881, 3598981, 3623401, 3643403, 3765589, 3775043, 3895981, 4143401, 4169621, 4277263, 4349089, 4364501, 4466443, 4576601, 4615601, 4645181, 4664263, 4928389, 4950409, 4979563, 5010407, 5043881, 5048921, 5049203, 5071103, 5110103, 5115203, 5135621, 5165707, 5297909, 5374307, 5533043, 5533681, 5535281, 5969309, 6000809, 6068443, 6146303, 6246029, 6260629, 6310243, 6345067, 6348781, 6405989, 6525643, 6535163, 6678109, 6740743, 6747421, 6856589, 7014881, 7161103, 7410889, 7415743, 7708009, 7813301, 8030963, 8108921, 8152447, 8207363, 8261381, 8512267, 8618567, 8669981, 8715181, 8720947, 8753707, 8787089, 8846429, 8854981, 8884621, 9061447, 9077521, 9277381, 9297907, 9302467, 9476647, 9792301, 9802343, 9913081, 9972343, 9974509, 9998701, 10013747, 10015903, 10045421, 10067809, 10070201, 10121143, 10180481, 10205207, 10223267, 10246729, 10330367, 10490863, 10500229, 10847621, 10893767, 10990121, 11113547, 11203301, 11228207, 11245547, 11265707, 11310647, 11608489, 11638903, 11725001, 11731963, 11878967, 12004309, 12054403, 12079121, 12382663, 12523909, 12579647, 12867409, 12987103, 13009303, 13162909, 13204847, 13248409, 13474789, 13609963, 13702301, 13836101, 13853263, 13918601, 14306203, 14400707, 14412407, 14504267, 14520403, 14637101, 14833543, 14918509, 15168529, 15230321, 15338801, 15439429, 15471889, 15616967, 15650321, 15944389, 16206103, 16420189, 16509343, 16578103, 16970143, 17046889, 17059843, 17309863, 17533189, 17653243, 18053389, 18223003, 18319243, 18329329, 18486581, 18505181, 18590563, 18665629, 18805667, 18970547, 19721201, 19948363, 19999303, 20132401, 20170421, 20273243, 20314867, 20390221, 20425103