A324505 a(n) = numerator of Sum_{d|n} (d/pod(d)) where pod(k) = the product of the divisors of k (A007955).
1, 2, 2, 5, 2, 19, 2, 21, 7, 31, 2, 529, 2, 43, 46, 169, 2, 1135, 2, 1441, 64, 67, 2, 52513, 11, 79, 64, 2801, 2, 117001, 2, 2705, 100, 103, 106, 1122553, 2, 115, 118, 238561, 2, 317521, 2, 6865, 6886, 139, 2, 20247937, 15, 8251, 154, 9569, 2, 557443, 166
Offset: 1
Examples
Sum_{d|n} (d/pod(d)) for n >= 1: 1, 2, 2, 5/2, 2, 19/6, 2, 21/8, 7/3, 31/10, 2, 529/144, 2, 43/14, 46/15, 169/64, ...
For n=4; Sum_{d|4} (d/pod(d)) = 1/pod(1) + 2/pod(2) + 4/pod(4) = 1/1 + 2/2 + 4/8 = 5/2; a(4) = 5.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..20000
Programs
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Magma
[Numerator(&+[d / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]];
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Mathematica
Table[Numerator[Sum[k/Product[j, {j, Divisors[k]}], {k, Divisors[n]}]], {n, 1, 60}] (* G. C. Greubel, Mar 04 2019 *) -
PARI
A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); A324505(n) = numerator(sumdiv(n,d,(d/A007955(d)))); \\ Antti Karttunen, Jan 27 2025
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Sage
[sum(k/product(j for j in k.divisors()) for k in n.divisors()).numerator() for n in (1..60)] # G. C. Greubel, Mar 04 2019
Formula
a(p) = 2 for p = primes.
Comments