cp's OEIS Frontend

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

%I A124143 #19 Aug 17 2024 15:04:58
%S A124143 4,8,32,36,121,128,144,216,256,324,400,512,576,784,900,961,1024,1296,
%T A124143 1600,1728,1764,1936,2304,2704,2744,2916,3136,3600,3844,4096,4356,
%U A124143 4624,4900,5184,5776,5832,6084,6400,7056,7744,7776,8000,8100,8192,9216,9604
%N A124143 Perfect powers pp such that sigma(k) = pp for some positive integer k.
%H A124143 Amiram Eldar, <a href="/A124143/b124143.txt">Table of n, a(n) for n = 1..10000</a>
%H A124143 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).
%e A124143 a(1) = 4 since sigma(3) = 4 = 2^2.
%p A124143 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;
%t A124143 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 *)
%o A124143 (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
%o A124143 (Magma) Set(Sort([SumOfDivisors(k): k in[A065496(n)]])); // _Jaroslav Krizek_, Mar 10 2015
%o A124143 (PARI) is(n) = ispower(n) && invsigmaNum(n) > 0; \\ _Amiram Eldar_, Aug 02 2024, using _Max Alekseyev_'s invphi.gp
%Y A124143 Intersection of A001597 and A002191 \ {1}.
%Y A124143 Cf. A065496.
%K A124143 nonn
%O A124143 1,1
%A A124143 _Walter Kehowski_, Dec 01 2006