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A128607
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Perfect (or pure) powers pp such that sigma(pp) is also a perfect (pure) power.
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5
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1, 81, 343, 400, 32400, 1705636, 3648100, 138156516, 295496100, 1055340196, 1476326929, 1857437604, 2263475776, 2323432804, 2592846400, 2661528100, 7036525456, 10994571025, 17604513124, 39415749156, 61436066769, 85482555876, 90526367376, 97577515876, 98551417041
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OFFSET
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1,2
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COMMENTS
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Denote by egcd(n) the gcd of all the powers in the prime factorization of n. In our context, a square has egcd=2, a cube has egcd=3 and so on. The only elements n in the sequence for which egcd(n)>2 are 81 and 343. Are there any others? Conjecture I: egcd(A128607(n))=2 for all n>2. Let A128608(n)=sigma(A128607(n)). Note that A128607(11)=1857437604=(2^2)*(3^2)*(11^2)*(653^2) has A128608(11)=5168743489=(7^3)*(13^3)*(19^3). Any other cubes or higher egcd's in A128608? Conjecture II: egcd(A128608(n))=2 for all n ne 11.
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LINKS
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MAPLE
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N:= 10^13: # to get all terms <= N
pows:= {1, seq(seq(n^k, n = 2 .. floor(N^(1/k))), k = 2 .. floor(log[2](N)))}:
filter:= proc(n) local s, F;
s:= numtheory:-sigma(n);
F:= map(t -> t[2], ifactors(s)[2]);
igcd(op(F)) >= 2
end proc:
filter(1):= true:
sort(convert(select(filter, pows), list)); # Robert Israel, Feb 14 2016
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MATHEMATICA
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M = 10^13;
pows = {1, Table[Table[n^k, {n, 2, Floor[M^(1/k)]}], {k, 2, Floor[Log[2, M] ]}]} // Flatten // Union;
okQ[n_] := Module[{s, F}, s = DivisorSigma[1, n]; F = FactorInteger[s][[All, 2]]; GCD @@ F >= 2];
okQ[1] = True;
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PROG
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(PARI) isok(n) = (n==1) || (ispower(n) && ispower(sigma(n))); \\ Michel Marcus, Feb 14 2016
(Magma) [1] cat [n : n in [2..4*10^6] | IsPower(n) and IsPower(SumOfDivisors(n))]; // Vincenzo Librandi, Feb 15 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Missing terms 1, 10994571025, 17604513124, 39415749156 added by Zak Seidov, Feb 14 2016
Missing terms 61436066769, 90526367376, 97577515876, 98551417041 added by Robert Israel, Feb 14 2016
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STATUS
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approved
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