A324982 a(n) = numerator 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, 2, 5, 14, 7, 25, 9, 62, 23, 59, 13, 1819, 15, 109, 245, 3382, 19, 1987, 21, 2731, 465, 257, 25, 250747, 271, 355, 775, 22295, 31, 405385, 33, 28434, 1121, 599, 1253, 6726169, 39, 745, 1557, 642763, 43, 1556549, 45, 28657, 61031, 1085, 49, 765671783, 713
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) = 14.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Magma
[Numerator(&+[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]];
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Mathematica
Array[Numerator@ DivisorSum[#, Apply[Times, Divisors@ #]/DivisorSigma[0, #] &] &, 49] (* Michael De Vlieger, Mar 24 2019 *)
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PARI
a(n) = numerator(sumdiv(n, d, vecprod(divisors(d))/numdiv(d))); \\ Michel Marcus, Mar 23 2019
Formula
a(p) = p + 2 for p = odd primes.
Comments