A228061 - OEIS

%I #19 Aug 12 2024 04:27:29

%S 1,2,6,11,12,16,18,20,24,28,30,31,32,36,40,42,44,48,52,54,56,60,62,64,

%T 66,68,70,72,76,78,80,84,88,90,96,98,100,102,104,108,112,114,120,124,

%U 126,128,132,136,140,144,150,152,154,156,160,162,164,168,172,174

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

%H Amiram Eldar, <a href="/A228061/b228061.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts#Inversion_of_Multiplicative_Functions">PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions</a> (invphi.gp).

%H Paul Pollack and Carl Pomerance, <a href="https://doi.org/10.1112/blms/bdt097">Square values of Euler's function</a>, Bull. London Math. Soc., Vol. 46, No. 2 (2014), pp. 403-414; <a href="http://www.math.dartmouth.edu/~carlp/squaretotients5.pdf">alternative link</a>. See section 5.

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

%t nn = 40000; t = Select[Union[DivisorSigma[1, Range[nn]]], IntegerQ[Sqrt[#]] &]; t = Sqrt[t]; t = Select[t, # < Sqrt[nn] &]

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

%o (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

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

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

%K nonn

%O 1,2

%A _T. D. Noe_, Sep 04 2013