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A354502
Squarefree semiprimes p*q such that (p*q+1+p-q)/2 and (p*q+1+q-p)/2 are prime.
1
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
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
KEYWORD
nonn,less
AUTHOR
J. M. Bergot and Robert Israel, Aug 15 2022
STATUS
approved