%I #16 May 25 2020 10:01:03

%S 2,162,441,2704,4225,275194921

%N Numbers k such that A332221(k) = A156552(sigma(k)) is 2*{an odd square}.

%C Any even term of A332216 must occur also in this sequence.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e a(n) -> sigma(a(n)) -> A156552(sigma(a(n)))

%e 2 = 2^1 * 1^2 -> 3 = 3^1 -> 2 = 2^1 * 1^1,

%e 162 = 2^1 * 3^4 -> 363 = 3^1 * 11^2 -> 98 = 2^1 * 7^2,

%e 441 = 3^2 * 7^2 -> 741 = 3^1 * 13^1 * 19^1 -> 578 = 2^1 * 17^2,

%e 2704 = 2^4 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,

%e 4225 = 5^2 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,

%e and

%e 275194921 = 53^2 * 313^2 -> 281384229 = 3^1 * 7^1 * 181^2 * 409^1 -> 9671406556943421676716050 = 2^1 * 5^2 * 7^2 * 62829235873^2.

%t Select[Range@ 5000, And[IntegerQ[#], OddQ[#]] &@ Sqrt[#/2] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ DivisorSigma[1, #]]] &] (* _Michael De Vlieger_, Feb 12 2020 *)

%o (PARI)

%o \\ Needs also code from A156552:

%o istosq(n) = ((1==valuation(n,2))&&issquare(n/2));

%o for(n=1,2^25,if(istosq(A156552(sigma(n*n))),print1(n*n,", ")); if(istosq(A156552(sigma(2*n*n))),print1(2*n*n,", ")));

%Y Cf. A000203, A156552, A332216, A332217, A332221.

%Y Subsequence of A332217 ⊂ A067051 ⊂ A028982.

%K nonn,more

%O 1,1

%A _Antti Karttunen_, Feb 11 2020