A307160 -id:A307160 - cp's OEIS frontend

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

Amiram Eldar, Mar 27 2019

nonn

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.

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.

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

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]]

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).

A370904 Partial alternating sums of the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

1, -2, 2, -3, 3, -9, -1, -16, -6, -24, -12, -32, -18, -42, -18, -45, -27, -57, -37, -67, -35, -71, -47, -107, -81, -123, -83, -123, -93, -165, -133, -196, -148, -202, -154, -204, -166, -226, -170, -260, -218, -314, -270, -330, -270, -342, -294, -402, -352, -430

Offset: 1

Amiram Eldar, Mar 05 2024

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.

Cf. A188999, A307159, A307160.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[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 @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(# + 1) * bsigma[#] &, 100]]

bsigma(n) = {my(f = factor(n)); prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * bsigma(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A188999(k).

a(n) = -(11/53) * c * n^2 + O(n * log(n)^3), where c = A307160 (Tóth, 2017).

A327574 Decimal expansion of the constant that appears in the asymptotic formula for average order of the infinitary divisors sum function (A049417).

Original entry on oeis.org

7, 3, 0, 7, 1, 8, 2, 4, 2, 1, 2, 7, 3, 8, 4, 2, 2, 5, 8, 3, 8, 9, 7, 5, 4, 6, 8, 1, 7, 3, 5, 3, 0, 1, 6, 1, 9, 5, 7, 2, 5, 6, 4, 3, 3, 8, 6, 1, 7, 2, 7, 8, 6, 9, 7, 0, 7, 3, 3, 6, 7, 6, 2, 3, 0, 1, 0, 7, 9, 8, 8, 3, 3, 2, 8, 0, 0, 5, 3, 4, 6, 3, 7, 0, 2, 9, 9

Offset: 0

Amiram Eldar, Sep 17 2019

The asymptotic mean of the infinitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A049417(k)/k = 1.461436... is twice this constant. - Amiram Eldar, Jun 13 2020

			0.730718242127384225838975468173530161957256433861727...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Cf. A049417, A050376, A327566.

Cf. A013661 (corresponding constant for all divisors), A275480 (exponential), A306633 (unitary), A307160 (bi-unitary).

$MaxExtraPrecision = 1000; m = 1000; em = 10; f[x_] := Sum[Log[1 + x^(2^e)/(1 + 1/x^(2^e))], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals Limit_{k->oo} A327566(k)/k^2.

Equals (1/2) * Product_{P} (1 + 1/(P*(P+1))), where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

cp's OEIS Frontend

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

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A370904 Partial alternating sums of the sum of the bi-unitary divisors function (A188999).

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A327574 Decimal expansion of the constant that appears in the asymptotic formula for average order of the infinitary divisors sum function (A049417).

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