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A334420 Numbers m such that sigma(d)/tau(d) is an integer for all divisors d of m.

%I A334420 #31 May 21 2025 18:45:31
%S A334420 1,3,5,7,11,13,15,17,19,21,23,29,31,33,35,37,39,41,43,47,49,51,53,55,
%T A334420 57,59,61,65,67,69,71,73,77,79,83,85,87,89,91,93,95,97,101,103,105,
%U A334420 107,109,111,113,115,119,123,127,129,131,133,137,139,141,143,145
%N A334420 Numbers m such that sigma(d)/tau(d) is an integer for all divisors d of m.
%C A334420 Sequences of numbers m from this sequence with k such divisors for 1 < k < 6:
%C A334420 k = 2: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... (A065091 - odd primes).
%C A334420 k = 3: 49, 169, 361, 961, 1369, 1849, 3721, 4489, 5329, 6241, 9409, ...
%C A334420 k = 4: 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, ...
%C A334420 k = 5: 923521, 13845841, 519885601, 1073283121, 1982119441, ...
%C A334420 See A334421 for sequence of the smallest numbers m with k such divisors.
%C A334420 All divisors of a member of the sequence are members of the sequence. - _Robert Israel_, May 01 2020
%C A334420 Numbers for which all divisors are in A003601. - _Michel Marcus_, May 02 2020
%H A334420 Robert Israel, <a href="/A334420/b334420.txt">Table of n, a(n) for n = 1..10000</a>
%F A334420 A324500(a(n)) = 1.
%e A334420 Number 15 with divisors 1, 3, 5 and 15 is a term because sigma(1)/tau(1) = 1/1 = 1, sigma(3)/tau(3) = 4/2 = 2, sigma(5)/tau(5) = 6/2 = 3, sigma(15)/tau(15) = 24/4 = 6.
%p A334420 filter:= n -> andmap(d -> numtheory:-sigma(d) mod numtheory:-tau(d)=0, numtheory:-divisors(n)):
%p A334420 select(filter, [$1..200]); # _Robert Israel_, May 01 2020
%t A334420 divQ[n_] := Divisible[DivisorSigma[1, n], DivisorSigma[0, n]]; Select[Range[150], AllTrue[Divisors[#], divQ] &] (* _Amiram Eldar_, Apr 29 2020 *)
%o A334420 (Magma) [m: m in [1..10^6] | &+[SumOfDivisors(d) mod NumberOfDivisors(d): d in Divisors(m)] eq 0];
%o A334420 (PARI) isok(m) = fordiv(m, d, if (sigma(d) % numdiv(d), return (0))); return(1); \\ _Michel Marcus_, Apr 29 2020
%Y A334420 Subsequence of A306639.
%Y A334420 Cf. A000005 (tau), A000203 (sigma), A003601, A324499, A324500, A334421.
%Y A334420 Includes A056911.
%K A334420 nonn
%O A334420 1,2
%A A334420 _Jaroslav Krizek_, Apr 29 2020