login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Numbers k such that tau(k^2) + 2*sigma(k^2) and 2*tau(k^2) + sigma(k^2) are both prime.
1

%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