cp's OEIS Frontend

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

%I A128607 #26 Sep 08 2022 08:45:30
%S A128607 1,81,343,400,32400,1705636,3648100,138156516,295496100,1055340196,
%T A128607 1476326929,1857437604,2263475776,2323432804,2592846400,2661528100,
%U A128607 7036525456,10994571025,17604513124,39415749156,61436066769,85482555876,90526367376,97577515876,98551417041
%N A128607 Perfect (or pure) powers pp such that sigma(pp) is also a perfect (pure) power.
%C A128607 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.
%H A128607 Robert Israel, <a href="/A128607/b128607.txt">Table of n, a(n) for n = 1..58</a>
%p A128607 N:= 10^13: # to get all terms <= N
%p A128607 pows:= {1, seq(seq(n^k, n = 2 .. floor(N^(1/k))), k = 2 .. floor(log[2](N)))}:
%p A128607 filter:= proc(n) local s, F;
%p A128607    s:= numtheory:-sigma(n);
%p A128607    F:= map(t -> t[2], ifactors(s)[2]);
%p A128607    igcd(op(F)) >= 2
%p A128607 end proc:
%p A128607 filter(1):= true:
%p A128607 sort(convert(select(filter, pows),list)); # _Robert Israel_, Feb 14 2016
%t A128607 M = 10^13;
%t A128607 pows = {1, Table[Table[n^k, {n, 2, Floor[M^(1/k)]}], {k, 2, Floor[Log[2, M] ]}]} // Flatten // Union;
%t A128607 okQ[n_] := Module[{s, F}, s = DivisorSigma[1, n]; F = FactorInteger[s][[All, 2]]; GCD @@ F >= 2];
%t A128607 okQ[1] = True;
%t A128607 Select[pows, okQ] (* _Jean-François Alcover_, Apr 12 2019, after _Robert Israel_ *)
%o A128607 (PARI) isok(n) = (n==1) || (ispower(n) && ispower(sigma(n))); \\ _Michel Marcus_, Feb 14 2016
%o A128607 (Magma) [1] cat [n : n in [2..4*10^6] | IsPower(n) and IsPower(SumOfDivisors(n))]; // _Vincenzo Librandi_, Feb 15 2016
%Y A128607 Cf. A000203, A001597, A128608.
%K A128607 nonn
%O A128607 1,2
%A A128607 _Walter Kehowski_, Mar 20 2007
%E A128607 Missing terms 1, 10994571025, 17604513124, 39415749156 added by _Zak Seidov_, Feb 14 2016
%E A128607 Missing terms 61436066769, 90526367376, 97577515876, 98551417041 added by _Robert Israel_, Feb 14 2016