A307159 Partial sums of the bi-unitary divisors sum function: Sum_{k=1..n} bsigma(k), where bsigma is A188999.

1, 4, 8, 13, 19, 31, 39, 54, 64, 82, 94, 114, 128, 152, 176, 203, 221, 251, 271, 301, 333, 369, 393, 453, 479, 521, 561, 601, 631, 703, 735, 798, 846, 900, 948, 998, 1036, 1096, 1152, 1242, 1284, 1380, 1424, 1484, 1544, 1616, 1664, 1772, 1822, 1900, 1972, 2042

Offset: 1

Author

Amiram Eldar, Mar 27 2019

Keywords

• nonn

References

• D. Suryanarayana and M. V. Subbarao, Arithmetical functions associated with the biunitary k-ary divisors of an integer, Indian J. Math., Vol. 22 (1980), pp. 281-298.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1, section 4.13.

Crossrefs

Cf. A024916, A064609, A188999, A306069, A307042, A307160.

Programs

Mathematica

fun[p_,e_] := If[OddQ[e],(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); Accumulate[Array[bsigma, 60]]

Formula

a(n) ~ c * n^2, where c = (zeta(2)*zeta(3)/2) * Product_{p}(1 - 2/p^3 + 1/p^4 + 1/p^5 - 1/p^6) (A307160).