A124143 Perfect powers pp such that sigma(k) = pp for some positive integer k.

4, 8, 32, 36, 121, 128, 144, 216, 256, 324, 400, 512, 576, 784, 900, 961, 1024, 1296, 1600, 1728, 1764, 1936, 2304, 2704, 2744, 2916, 3136, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5776, 5832, 6084, 6400, 7056, 7744, 7776, 8000, 8100, 8192, 9216, 9604

Offset: 1

Author

Walter Kehowski, Dec 01 2006

Keywords

• nonn

Examples

			a(1) = 4 since sigma(3) = 4 = 2^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Crossrefs

Intersection of A001597 and A002191 \ {1}.

Cf. A065496.

Programs

Magma

Set(Sort([SumOfDivisors(k): k in[1..10000], b in [2..15], a in [2..100] | SumOfDivisors(k) eq a^b])); // Jaroslav Krizek, Mar 10 2015

Magma

Set(Sort([SumOfDivisors(k): k in[A065496(n)]])); // Jaroslav Krizek, Mar 10 2015

Maple

with(numtheory); egcd := proc(n) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: P:={}: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; P:=P union {s}; fi od od; L; P;

Mathematica

powerQ[n_] := Block[{pf = FactorInteger@ n, min}, min = Min @@ Last /@ pf; min > 1 && AllTrue[Last /@ pf/min, IntegerQ]]; lim = 10000; Intersection[Select[Range@ lim, powerQ], DeleteDuplicates@ Sort[DivisorSigma[1, #] & /@ Range@ lim]] (* Michael De Vlieger, Mar 10 2015 *)

PARI

is(n) = ispower(n) && invsigmaNum(n) > 0; \\ Amiram Eldar, Aug 02 2024, using Max Alekseyev's invphi.gp