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.
e (* from A000043 *) = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, ...}; PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; Do[p = 2^(e[[n]] - 1)*(2^e[[n]] - 1); Print[p - PrevPrim[p]], {n, 1, 20}] #-NextPrime[#,-1]&/@PerfectNumber[Range[15]] (* The program generates the first 15 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Nov 22 2023 *)
a(3)=3 because 496 + 3 is a prime.
e (* from A000043 *) = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, ...}; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[p = 2^(e[[n]] - 1)*(2^e[[n]] - 1); Print[NextPrim[p] - p], {n, 1, 20}]
We have a(3) = 33550336 since 33550337 is prime and there is no other such perfect number less than a(3) and that exceeds a(2) = 28.
{e=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787];} /* exponents of Mersenne primes */
for(n=1,#e,p=(2^e[n]-1)*(2^(e[n]-1));if(ispseudoprime(p+1),print1(p,", ")));
for(n=0,10^5,ispseudoprime(2^(n-1)*(2^n-1)+1) && print1(n,", "))
12 is a term, because 8191 * 2^12 + 1 = 8191 * 4096 + 1 = 33550337 is prime. (also a term of A061644).
is(k) = isprime(8191 * 2^k + 1);
Comments