A278921 Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1 and (q-p)^(q+1) + 1.

10, 15, 65, 221, 493, 671, 1147, 1219, 3439, 5069, 12209, 14893, 20737, 24503, 30083, 49813, 61937, 77507, 91277, 97297, 100337, 102719, 109283, 109783, 113521, 132427, 144301, 178991, 204851, 244523, 245041, 246559, 257149, 258749, 312167, 339497, 397219, 433091, 434617, 461893, 465763

Offset: 1

Author

Juri-Stepan Gerasimov, Dec 01 2016

Keywords

• nonn

Comments

q is always a Pythagorean prime (A002144).

Semiprimes of the form p*q where p < q such that q divides p^(q+1) + k and (q-p)^(q+1) + k:

k = 1: (this sequence);

k = 2: 6, 33, 119, 247, 451, ...

k = 3: 14, 35, 91, 341, ...

k = 4: 39, 145, 371, ...

For every positive odd number q (whether prime or not), every integer p in 0..q, and every integer k, if q divides p^(q+1) + k, then it necessarily follows that q also divides (q-p)^(q+1) + k; thus, this sequence could be more simply defined as "Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1." - Jon E. Schoenfield, Dec 07 2016

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A002144, A006881, A279024.

Programs

Mathematica

Take[#, 41] &@ Union@ Flatten@ Table[Function[q, q Select[Prime@ Range@ n, Function[p, And[Divisible[p^(q + 1) + 1, q], Divisible[(q - p)^(q + 1) + 1, q]]]]]@ Prime@ n, {n, 600}] (* Michael De Vlieger, Dec 02 2016 *)

PARI

list(lim)=my(v=List()); forprime(q=5,lim\2, if(q%4>2, next); forprime(p=2,min(lim\q,q-2), if(Mod(p,q)^(q+1)==-1 && Mod(q-p,q)^(q+1)==-1, listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Dec 02 2016