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A128607 Perfect (or pure) powers pp such that sigma(pp) is also a perfect (pure) power. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Robert Israel, Table of n, a(n) for n = 1..58

MAPLE

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

MATHEMATICA

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;

Select[pows, okQ] (* Jean-Fran├žois Alcover, Apr 12 2019, after Robert Israel *)

PROG

(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

CROSSREFS

Cf. A000203, A001597, A128608.

Sequence in context: A232284 A337044 A337045 * A180090 A217967 A237384

Adjacent sequences:  A128604 A128605 A128606 * A128608 A128609 A128610

KEYWORD

nonn

AUTHOR

Walter Kehowski, Mar 20 2007

EXTENSIONS

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

STATUS

approved

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Last modified July 1 06:50 EDT 2022. Contains 354952 sequences. (Running on oeis4.)