A379579 Numerators of the partial sums of the reciprocals of the powerfree part function (A055231).

1, 3, 11, 17, 91, 16, 117, 152, 187, 381, 4261, 13553, 178499, 90322, 30441, 35446, 607587, 1300259, 24875091, 25521737, 77027101, 38733998, 895731799, 932913944, 1044460379, 2097501253, 2320594123, 2352464533, 68444564327, 11443370128, 355822756173, 389249504528

Offset

1,2

References

D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math., Vol. 12, No. 2 (1981), pp. 175-190.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d'un nombre, Thèse de doctorat, Université Laval, Québec (2018).

Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.11, pp. 31-32.

Formula

a(n) = numerator(Sum_{k=1..n} 1/A055231(k)).

a(n)/A379580(n) = A * n^(1/2) + B * n^(1/3) + O(n^(1/5)), where A = A328013, and B = (zeta(2/3)/zeta(2)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = -2.59305556147555965163... .

Example

Fractions begin with 1, 3/2, 11/6, 17/6, 91/30, 16/5, 117/35, 152/35, 187/35, 381/70, 4261/770, 13553/2310, ...

Mathematica

f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/powfree[n], {n, 1, 50}]]]

Prog

(PARI) powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / powfree(k); print1(numerator(s), ", "))};

Crossrefs

Cf. A055231, A328013, A370900, A370901, A379580 (denominators), A379581.

Sequence in context: A145701 A072982 A124787 * A080306 A128346 A128031

Adjacent sequences: A379576 A379577 A379578 * A379580 A379581 A379582

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 26 2024

Status

approved