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%I #14 Aug 27 2024 16:39:29
%S 1,2,3,4,6,11,12,17,18,24,33,60,69,94,131,138,173,187,198,200,214,226,
%T 263,282,290,311,347,360,400,426,428,495,498,502,521,583,606,622,653,
%U 675,771,822,850,902,911,1013,1020,1104,1127,1177,1195,1215,1243,1283,1366,1377,1402,1500,1714,1795
%N Numbers k such that tau(k^2) + 2*sigma(k^2) and 2*tau(k^2) + sigma(k^2) are both prime.
%C If tau(x) + 2*sigma(x) is prime, tau(x) must be odd so x must be a square.
%H Robert Israel, <a href="/A358788/b358788.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%p filter:= proc(n) uses numtheory; local s,t;
%p s:= sigma(n^2); t:= tau(n^2);
%p isprime(s+2*t) and isprime(2*s+t)
%p end proc:
%p select(filter, [$1..10000]);
%t Select[Range[1800], PrimeQ[(d = DivisorSigma[0, #^2]) + 2*(s = DivisorSigma[1, #^2])] && PrimeQ[2*d + s] &] (* _Amiram Eldar_, Dec 01 2022 *)
%t tsQ[n_]:=With[{t=DivisorSigma[0,n^2],s=DivisorSigma[1,n^2]},AllTrue[{t+2s,2t+s},PrimeQ]]; Select[Range[1800],tsQ] (* _Harvey P. Dale_, Aug 27 2024 *)
%Y Cf. A000005, A000203.
%K nonn
%O 1,2
%A _J. M. Bergot_ and _Robert Israel_, Nov 30 2022