A275196 Odd numbers n such that sigma(n) does not divide sigma(n^3).

9, 25, 27, 49, 63, 75, 81, 99, 117, 121, 125, 135, 147, 153, 169, 171, 175, 207, 225, 243, 245, 261, 275, 289, 297, 325, 333, 343, 361, 363, 369, 375, 387, 405, 425, 441, 475, 477, 507, 513, 525, 529, 531, 539, 549, 567, 575, 603, 605, 625, 637, 639, 675, 693, 711, 725, 729

Offset: 1

Author

Altug Alkan, Jul 20 2016

Keywords

• nonn

Comments

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.

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.

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

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, ...

Examples

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A175926.

Programs

Mathematica

Select[2Range[400] - 1, Not[Divisible[DivisorSigma[1, #^3], DivisorSigma[1, #]]] &] (* Alonso del Arte, Jul 20 2016 *)

PARI

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