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Numbers k such that sigma_1(k) does not divide sigma_2(k).
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%I #9 Jun 12 2024 16:34:13

%S 2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,21,22,23,24,26,27,28,29,30,31,

%T 32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,51,52,53,54,55,56,57,

%U 58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,76,77

%N Numbers k such that sigma_1(k) does not divide sigma_2(k).

%C Numbers k such that the antiharmonic mean of divisors of k is not an integer.

%C Antiharmonic mean of divisors of a number m = Product (p_i^e_i) is A001157(m)/A000203(m) = Product ((p_i^(e_i+1)+1)/(p_i+1)).

%C Numbers k such that A001157(k)/A000203(k) is not an integer.

%e a(12) = 15, sigma_2(15)/sigma_1(15)=260/24 = 65/6 (not integer).

%t Select[Range[100], Mod @@ DivisorSigma[{2, 1}, #] > 0 &] (* _Amiram Eldar_, Mar 22 2024 *)

%o (PARI) is(n) = {my(f = factor(n)); sigma(f, 2) % sigma(f);} \\ _Amiram Eldar_, Mar 22 2024

%Y Complement of A020487.

%Y Cf. A001157, A000203.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Mar 15 2009

%E More terms from _Amiram Eldar_, Mar 22 2024