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A124143
Perfect powers pp such that sigma(k) = pp for some positive integer k.
1
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
1,1
EXAMPLE
a(1) = 4 since sigma(3) = 4 = 2^2.
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 *)
PROG
(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
(PARI) is(n) = ispower(n) && invsigmaNum(n) > 0; \\ Amiram Eldar, Aug 02 2024, using Max Alekseyev's invphi.gp
CROSSREFS
Intersection of A001597 and A002191 \ {1}.
Cf. A065496.
Sequence in context: A162459 A129195 A180098 * A173617 A034041 A050442
KEYWORD
nonn
AUTHOR
Walter Kehowski, Dec 01 2006
STATUS
approved