This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A341658 #15 Jun 14 2024 19:00:18 %S A341658 29,43,53,59,61,67,83,107,157,173,193,227,277,283,317,347,563,653,733, %T A341658 787,877,907,997,1213,1237,1283,1307,1523,1867,2083,2693,2797,2803, %U A341658 3253,3413,3517,3643,3677,3733,3803,4253,4363,4547,4723,5387,5443,5483,5717 %N A341658 Primes p such that p^2 - 1 has 32 divisors. %C A341658 Conjecture: sequence is infinite. %C A341658 All terms are primes p such that p^2 - 1 is of the form 24*q*r = 2^3 * 3 * q * r (where q and r are distinct primes), with only three exceptions: 53, 107, and 193 (see Example section). %C A341658 For primes p > 3, p^2 - 1 = (p-1)*(p+1) will have p-1 and p+1 as consecutive even numbers (so one of them is divisible by 4, so their product is divisible by 8), and one of p-1 and p+1 will be divisible by 3. For each term other than 53, 107, and 193, the factors p-1 and p+1 are, in some order, numbers of the forms 2*q and 12*r or 4*q and 6*r. %e A341658 p = factorization %e A341658 n a(n) p^2 - 1 of (p^2 - 1) %e A341658 -- ---- ------- ------------------- %e A341658 1 29 840 2^3 * 3 * 5 * 7 %e A341658 2 43 1848 2^3 * 3 * 7 * 11 %e A341658 3 53 2808 2^3 * 3^3 * 13 %e A341658 4 59 3480 2^3 * 3 * 5 * 29 %e A341658 5 61 3720 2^3 * 3 * 5 * 31 %e A341658 6 67 4488 2^3 * 3 * 11 * 17 %e A341658 7 83 6888 2^3 * 3 * 7 * 41 %e A341658 8 107 11448 2^3 * 3^3 * 53 %e A341658 9 157 24648 2^3 * 3 * 13 * 79 %e A341658 10 173 29928 2^3 * 3 * 29 * 43 %e A341658 11 193 37248 2^7 * 3 * 97 %t A341658 Select[Range[6000], PrimeQ[#] && DivisorSigma[0, #^2 - 1] == 32 &] (* _Amiram Eldar_, Feb 26 2021 *) %o A341658 (PARI) isok(p) = isprime(p) && (numdiv(p^2-1) == 32); \\ _Michel Marcus_, Feb 26 2021 %Y A341658 Cf. A000005, A000040, A309906, A341655. %K A341658 nonn %O A341658 1,1 %A A341658 _Jon E. Schoenfield_, Feb 26 2021