A128607 Perfect (or pure) powers pp such that sigma(pp) is also a perfect (pure) power.

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

Offset: 1

Author

Walter Kehowski, Mar 20 2007

Keywords

• nonn

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

Crossrefs

Cf. A000203, A001597, A128608.

Programs

Magma

[1] cat [n : n in [2..4*10^6] | IsPower(n) and IsPower(SumOfDivisors(n))]; // Vincenzo Librandi, Feb 15 2016

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-François Alcover, Apr 12 2019, after Robert Israel *)

PARI

isok(n) = (n==1) || (ispower(n) && ispower(sigma(n))); \\ Michel Marcus, Feb 14 2016

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