A324983 a(n) = denominator of Sum_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).
1, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 6, 2, 2, 4, 15, 2, 2, 2, 2, 4, 2, 2, 6, 6, 2, 4, 6, 2, 4, 2, 5, 4, 2, 4, 6, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 30, 6, 2, 4, 6, 2, 4, 4, 6, 4, 2, 2, 4, 2, 2, 4, 35, 4, 4, 2, 2, 4, 4, 2, 6, 2, 2, 12, 6, 4, 4, 2, 10, 20, 2, 2, 12, 4
Offset: 1
Examples
Sum_{d|n} (pod(d)/tau(d)) for n >= 1: 1, 2, 5/2, 14/3, 7/2, 25/2, 9/2, 62/3, 23/2, 59/2, ...
For n=4; Sum_{d|4} (pod(d)/tau(d)) = pod(1)/tau(1) + pod(2)/tau(2) + pod(4)/tau(4) = 1/1 + 2/2 + 8/3 = 14/3; a(4) = 3.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..20000
Programs
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Magma
[Denominator(&+[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
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Mathematica
Array[Denominator@ DivisorSum[#, Apply[Times, Divisors@ #]/DivisorSigma[0, #] &] &, 85] (* Michael De Vlieger, Mar 24 2019 *)
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PARI
a(n) = denominator(sumdiv(n, d, vecprod(divisors(d))/numdiv(d))); \\ Michel Marcus, Mar 23 2019
Formula
a(p) = 2 for p = odd primes.
Comments