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A338170 a(n) is the number of divisors d of n such that tau(d) divides sigma(d).

%I A338170 #28 May 28 2025 16:35:59
%S A338170 1,1,2,1,2,3,2,1,2,2,2,3,2,3,4,1,2,3,2,3,4,3,2,3,2,2,3,3,2,6,2,1,4,2,
%T A338170 4,3,2,3,4,3,2,7,2,4,5,3,2,3,3,2,4,2,2,5,4,4,4,2,2,8,2,3,4,1,4,7,2,3,
%U A338170 4,6,2,3,2,2,4,3,4,6,2,3,3,2,2,7,4,3,4,4,2,7,4,4,4,3,4,4,2,4,5,3,2,6,2,2,8
%N A338170 a(n) is the number of divisors d of n such that tau(d) divides sigma(d).
%C A338170 a(n) is the number of arithmetic divisors d of n.
%C A338170 a(n) = tau(n) = A000005(n) for numbers n from A334420.
%C A338170 See A338171 and A338172 for sum and product such divisors.
%C A338170 a(n) = 1 iff n = 2^k (A000079). - _Bernard Schott_, Dec 06 2020
%H A338170 Antti Karttunen, <a href="/A338170/b338170.txt">Table of n, a(n) for n = 1..65537</a>
%F A338170 a(n) = Sum_{d|n} c(d), where c(n) is the arithmetic characteristic of n (A245656).
%F A338170 a(p) = 2 for odd primes p (A065091).
%e A338170 a(6) = 3 because there are 3 arithmetic divisors of 6 (1, 3 and 6):
%e A338170 sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3.
%t A338170 a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* _Amiram Eldar_, Oct 15 2020 *)
%o A338170 (Magma) [#[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];
%o A338170 (PARI) a(n) = sumdiv(n, d, !(sigma(d) % numdiv(d))); \\ _Michel Marcus_, Oct 15 2020
%Y A338170 Inverse Möbius transform of A245656.
%Y A338170 Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
%Y A338170 Cf. A334420, A334421.
%Y A338170 Cf. A337326 (smallest numbers m with n such divisors).
%K A338170 nonn
%O A338170 1,3
%A A338170 _Jaroslav Krizek_, Oct 14 2020
%E A338170 Data section extended up to 105 terms by _Antti Karttunen_, Dec 12 2021