A354502 Squarefree semiprimes p*q such that (p*q+1+p-q)/2 and (p*q+1+q-p)/2 are prime.

35, 65, 77, 133, 143, 161, 209, 215, 221, 235, 265, 335, 377, 391, 403, 413, 451, 517, 527, 551, 553, 565, 583, 623, 635, 667, 685, 707, 721, 731, 763, 779, 793, 817, 835, 851, 871, 893, 917, 923, 965, 1007, 1057, 1067, 1133, 1147, 1157, 1207, 1243, 1247, 1271, 1273, 1313, 1333, 1337, 1363, 1385

Offset

1,1

Comments

All terms are odd.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(3) = 77 = 7*11 is a term because p = 7, q = 11, (7*11+1+11-7)/2 = 41 and (7*11+1+7-11)/2 = 37 are prime.

Maple

filter:= proc(n) local p, q, t, k;
   if issqr(n) or numtheory:-bigomega(n) <> 2 then return false fi;
   p, q:= op(numtheory:-factorset(n));
   isprime((n+1+p-q)/2) and isprime((n+1+q-p)/2)
end proc:
select(filter, [seq(i, i=3..10000, 2)]);

Mathematica

Select[Range[1, 1400, 2], (f = FactorInteger[#])[[;; , 2]] == {1, 1} && PrimeQ[((p = f[[1, 1]])*(q = f[[2, 1]]) + 1 + p - q)/2] && PrimeQ[(p*q + 1 + q - p)/2] &] (* Amiram Eldar, Aug 16 2022 *)
sfspQ[n_]:=Module[{p, q}, {p, q}=FactorInteger[n][[;; , 1]]; AllTrue[{(n+1+p-q)/2, (n+1+q-p)/2}, PrimeQ]]; Select[Range[1500], SquareFreeQ[#]&&PrimeOmega[#]==2&&sfspQ[#]&] (* Harvey P. Dale, Mar 01 2024 *)

Crossrefs

Cf. A006881.

Sequence in context: A245274 A092256 A108172 * A176875 A292005 A338008

Adjacent sequences: A354499 A354500 A354501 * A354503 A354504 A354505

Keyword

nonn,less

Author

J. M. Bergot and Robert Israel, Aug 15 2022

Status

approved