A348969 -id:A348969 - cp's OEIS frontend

A348510 a(n) = A099377(n) - n, where A099377(n) is the numerator of the harmonic mean of the divisors of n.

0, 2, 0, 8, 0, -4, 0, 24, 18, 10, 0, 6, 0, -7, -10, 64, 0, 18, 0, 0, 0, 0, 0, -8, 50, 26, 0, -25, 0, -20, 0, 32, -22, 34, 0, 288, 0, 0, 0, -8, 0, -35, 0, -22, 0, -23, 0, 72, 0, 50, -34, 104, 0, -36, 0, 0, 0, 58, 0, -30, 0, -31, 126, 384, 0, -55, 0, 0, -46, -35, 0, 216, 0, 74, 150, 38, 0, -52, 0, 320, 324, 82, 0, -75

Offset: 1

Antti Karttunen, Oct 31 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A099377, A250094 (positions of zeros), A348968, A348969.

a[n_] := Numerator[DivisorSigma[0, n]/DivisorSigma[-1, n]] - n; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

A099377(n) = { my(d=divisors(n)); numerator(#d/sum(k=1, #d, 1/d[k])); }; \\  From A099377
A348510(n) = (A099377(n)-n);

A348968 a(n) = gcd(n, A099377(n)), where A099377(n) is the numerator of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 6, 13, 7, 5, 16, 17, 18, 19, 20, 21, 22, 23, 8, 25, 26, 27, 1, 29, 10, 31, 32, 11, 34, 35, 36, 37, 38, 39, 8, 41, 7, 43, 22, 45, 23, 47, 24, 49, 50, 17, 52, 53, 18, 55, 56, 57, 58, 59, 30, 61, 31, 63, 64, 65, 11, 67, 68, 23, 35, 71, 72, 73, 74, 75, 38, 77, 26, 79, 80, 81, 82, 83, 3

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

Cf. A057021, A099377, A348510, A348969.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A099377(n) = { my(d=divisors(n)); numerator(#d/sum(k=1, #d, 1/d[k])); }; \\ From A099377
A348968(n) = gcd(n, A099377(n));

a(n) = gcd(n, A099377(n)) = gcd(n, A348510(n)) = gcd(A099377(n), A348510(n)).
a(n) = n / A348969(n).
a(n) = A099377(n) / A057021(n). [Apparently, holds at least up to n = 2^25]

cp's OEIS Frontend

A348510 a(n) = A099377(n) - n, where A099377(n) is the numerator of the harmonic mean of the divisors of n.

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