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

Author

Amiram Eldar, Dec 26 2024

Keywords

• nonn
• easy
• frac

Examples

			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, ...
		

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.

Crossrefs

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

Programs

Mathematica

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

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), ", "))};

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... .