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A338172 a(n) is the product of those divisors d of n such that tau(d) divides sigma(d).

%I A338172 #22 Apr 10 2025 14:17:39
%S A338172 1,1,3,1,5,18,7,1,3,5,11,18,13,98,225,1,17,18,19,100,441,242,23,18,5,
%T A338172 13,81,98,29,40500,31,1,1089,17,1225,18,37,722,1521,100,41,1555848,43,
%U A338172 10648,10125,1058,47,18,343,5,2601,13,53,26244,3025,5488,3249,29
%N A338172 a(n) is the product of those divisors d of n such that tau(d) divides sigma(d).
%C A338172 a(n) is the product of arithmetic divisors d of n.
%C A338172 a(n) = pod(n) = A007955(n) for numbers n from A334420.
%F A338172 a(p) = p for odd primes p (A065091).
%e A338172 a(6) = 18 because there are 3 arithmetic divisors of 6 (1, 3 and 6): sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3. Product of this divisors is 18.
%t A338172 a[n_] := Times @@ Select[Divisors[n],  Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* _Amiram Eldar_, Oct 15 2020 *)
%o A338172 (Magma) [&*[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];
%o A338172 (PARI) a(n) = my(d=divisors(n)); prod(k=1, #d, if (sigma(d[k]) % numdiv(d[k]), 1, d[k])); \\ _Michel Marcus_, Oct 15 2020
%Y A338172 Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
%Y A338172 Cf. A334420, A334421.
%Y A338172 See A338170 and A338171 for number and sum of such divisors.
%K A338172 nonn
%O A338172 1,3
%A A338172 _Jaroslav Krizek_, Oct 14 2020