A358788 Numbers k such that tau(k^2) + 2*sigma(k^2) and 2*tau(k^2) + sigma(k^2) are both prime.

1, 2, 3, 4, 6, 11, 12, 17, 18, 24, 33, 60, 69, 94, 131, 138, 173, 187, 198, 200, 214, 226, 263, 282, 290, 311, 347, 360, 400, 426, 428, 495, 498, 502, 521, 583, 606, 622, 653, 675, 771, 822, 850, 902, 911, 1013, 1020, 1104, 1127, 1177, 1195, 1215, 1243, 1283, 1366, 1377, 1402, 1500, 1714, 1795

Offset

1,2

Comments

If tau(x) + 2*sigma(x) is prime, tau(x) must be odd so x must be a square.

Links

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

Example

a(3) = 3 is a term because tau(9) = 3 and sigma(9) = 13 so tau(9) + 2*sigma(9) = 29 and 2*tau(9) + sigma(9) = 19, and 29 and 19 are both prime.

Maple

filter:= proc(n) uses numtheory; local s, t;
  s:= sigma(n^2); t:= tau(n^2);
  isprime(s+2*t) and isprime(2*s+t)
end proc:
select(filter, [$1..10000]);

Mathematica

Select[Range[1800], PrimeQ[(d = DivisorSigma[0, #^2]) + 2*(s = DivisorSigma[1, #^2])] && PrimeQ[2*d + s] &] (* Amiram Eldar, Dec 01 2022 *)

Crossrefs

Cf. A000005, A000203.

Sequence in context: A255921 A111338 A029457 * A014859 A084166 A245569

Adjacent sequences: A358785 A358786 A358787 * A358789 A358790 A358791

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Nov 30 2022

Status

approved