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

1,2

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.

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.

Example

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, ...

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]

Prog

(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)))); }

Crossrefs

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

Similar sequences: A099377, A103339.

Sequence in context: A021962 A097672 A103339 * A361782 A353990 A092383

Adjacent sequences: A361313 A361314 A361315 * A361317 A361318 A361319

Keyword

nonn,frac

Author

Amiram Eldar, Mar 09 2023

Status

approved