A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 128, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 256, 65, 11, 67, 136

Offset: 1

Author

Amiram Eldar, Mar 09 2023

Keywords

• nonn
• frac

Examples

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Crossrefs

Cf. A036537, A037445, A049417, A077609, A063947, A138302, A361317 (denominators).

Similar sequences: A099377, A103339.

Programs

Mathematica

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; a[1] = 1; a[n_] := Numerator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

PARI

a(n) = {my(f = factor(n), b); numerator(n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1)))); }

Formula

a(n) = numerator(n*A037445(n)/A049417(n)).

a(n)/A361317(n) <= A099377(n)/A099378(n), with equality if and only if n is in A036537.

a(n)/A361317(n) >= A103339(n)/A103340(n), with equality if and only if n is in A138302.