A331741 Squares s such that A331733(s) = sigma(A225546(n)) is congruent to 2 modulo 4.

16, 144, 400, 784, 1936, 2704, 3600, 4624, 5776, 7056, 8464, 10000, 13456, 15376, 17424, 19600, 21904, 24336, 26896, 29584, 35344, 38416, 41616, 44944, 48400, 51984, 55696, 59536, 67600, 71824, 76176, 80656, 85264, 90000, 94864, 99856, 104976, 110224, 115600, 121104, 126736, 132496, 138384, 144400, 150544, 163216, 169744, 176400

Offset: 1

Author

Antti Karttunen, Feb 03 2020

Keywords

• nonn

Comments

Squares s for which A331733(s) is two times an odd number, i.e., squares s such that A007814(A331733(s)) == 1.

For each term k present, A006519(k) = 2^(2^e), with A000040(1+e) == 1 (mod 4). See A191218, A228058.

Equal to the sequence A225546(A191218(n)), for n >= 1, when its elements are sorted into ascending order.

Crossrefs

Cf. A006519, A007814, A010052, A067403, A191218, A225546, A228058, A331733, A331734.

Programs

Mathematica

Select[Range[100]^2, Mod[DivisorSigma[1, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@# + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]], 4] == 2 &] (* Michael De Vlieger, Feb 08 2020 *)

PARI

k=0; for(n=1,500,if(!(n%2)&&(1==A007814(A331733(n^2))),k++; write("b331741.txt", k, " ", n^2); print(n^2, " -> ", factor(n^2),", ")));

Formula

{n: A010052(n)*A007814(A331733(n)) == 1}.