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

%I A278921 #34 Dec 19 2016 08:12:45
%S A278921 10,15,65,221,493,671,1147,1219,3439,5069,12209,14893,20737,24503,
%T A278921 30083,49813,61937,77507,91277,97297,100337,102719,109283,109783,
%U A278921 113521,132427,144301,178991,204851,244523,245041,246559,257149,258749,312167,339497,397219,433091,434617,461893,465763
%N A278921 Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1 and (q-p)^(q+1) + 1.
%C A278921 q is always a Pythagorean prime (A002144).
%C A278921 Semiprimes of the form p*q where p < q such that q divides p^(q+1) + k and (q-p)^(q+1) + k:
%C A278921 k = 1: (this sequence);
%C A278921 k = 2: 6, 33, 119, 247, 451, ...
%C A278921 k = 3: 14, 35, 91, 341, ...
%C A278921 k = 4: 39, 145, 371, ...
%C A278921 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
%H A278921 Charles R Greathouse IV, <a href="/A278921/b278921.txt">Table of n, a(n) for n = 1..10000</a>
%t A278921 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 *)
%o A278921 (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
%Y A278921 Cf. A001358, A002144, A006881, A279024.
%K A278921 nonn
%O A278921 1,1
%A A278921 _Juri-Stepan Gerasimov_, Dec 01 2016