A228061 Numbers k such that k^2 = sigma(m) for some m.

1, 2, 6, 11, 12, 16, 18, 20, 24, 28, 30, 31, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 62, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 112, 114, 120, 124, 126, 128, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 172, 174

Offset: 1

Author

T. D. Noe, Sep 04 2013

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).
• Paul Pollack and Carl Pomerance, Square values of Euler's function, Bull. London Math. Soc., Vol. 46, No. 2 (2014), pp. 403-414; alternative link. See section 5.

Crossrefs

Cf. A000203, A038688 (squares of these numbers).

Cf. A221284 (similar numbers for the phi function).

Programs

Mathematica

nn = 40000; t = Select[Union[DivisorSigma[1, Range[nn]]], IntegerQ[Sqrt[#]] &]; t = Sqrt[t]; t = Select[t, # < Sqrt[nn] &]
With[{nn=50000},Union[Select[Sqrt[DivisorSigma[1,Range[nn]]],IntegerQ[ #] && #<=Sqrt[nn]&]]] (* Harvey P. Dale, Jul 12 2021 *)

PARI

lista(kmax) = for(k = 1, kmax, if(invsigmaNum(k^2) > 0, print1(k, ", "))); \\ Amiram Eldar, Aug 12 2024, using Max Alekseyev's invphi.gp

Formula

a(n) = sqrt(A038688(n)).