A275196 - OEIS

%I #31 Jul 27 2016 09:59:52

%S 9,25,27,49,63,75,81,99,117,121,125,135,147,153,169,171,175,207,225,

%T 243,245,261,275,289,297,325,333,343,361,363,369,375,387,405,425,441,

%U 475,477,507,513,525,529,531,539,549,567,575,603,605,625,637,639,675,693,711,725,729

%N Odd numbers n such that sigma(n) does not divide sigma(n^3).

%C All terms are composite since sigma(p) = p + 1 and sigma(p^3) = p^3 + p^2 + p + 1 = (p + 1)(p^2 + 1) for p prime.

%C An odd number n with prime factorization Product_i p_i^(e_i) is in this sequence if and only if Product_i ((p_i^(3*e_i + 1) - 1)/(p_i^(e_i + 1) - 1)) is not an integer.

%C Nonsquare terms of this sequence are 27, 63, 75, 99, 117, 125, 135, 147, 153, 171, 175, 207, 243, 245, 261, 275, ...

%C Terms that are not perfect powers are 63, 75, 99, 117, 135, 147, 153, 171, 175, 207, 245, 261, 275, 297, 325, 333, 363, 369, 375, ...

%H Charles R Greathouse IV, <a href="/A275196/b275196.txt">Table of n, a(n) for n = 1..10000</a>

%e 63 is a term because sigma(63^3) = 437200 is not divisible by sigma(63) = 104.

%t Select[2Range[400] - 1, Not[Divisible[DivisorSigma[1, #^3], DivisorSigma[1, #]]] &] (* _Alonso del Arte_, Jul 20 2016 *)

%o (PARI) isok(n) = sigma(n^3) % sigma(n) != 0 && n % 2 == 1

%Y Cf. A000203, A175926.

%K nonn

%O 1,1

%A _Altug Alkan_, Jul 20 2016