A307159 -id:A307159 - cp's OEIS frontend

A307160 Decimal expansion of the constant c in the asymptotic formula for the partial sums of the bi-unitary divisors sum function, A307159(k) ~ c*k^2.

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

Offset: 0

Amiram Eldar, Mar 27 2019

The asymptotic mean of the bi-unitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A188999(k)/k = 2*c = 1.505677... - Amiram Eldar, Jun 10 2020

			0.75283874100229431154333095155304127651952546756522...

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.

László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1, section 4.13.

Cf. A188999, A306071, A306072, A307159.

$MaxExtraPrecision = 1000; nm=1000; c = Rest[CoefficientList[Series[Log[1 - 2*x^3 + x^4 + x^5 - x^6],{x,0,nm}],x] * Range[0, nm]]; RealDigits[(Zeta[2]*Zeta[3]/2) * Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

Equals (zeta(2)*zeta(3)/2)* Product_{p}(1 - 2/p^3 + 1/p^4 + 1/p^5 - 1/p^6).

More terms from Vaclav Kotesovec, May 29 2020

A307161 Numbers n such that A307159(n) = Sum_{k=1..n} bsigma(k) is divisible by n, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

1, 2, 17, 37, 50, 56, 391, 919, 1399, 2829, 6249, 13664, 28829, 62272, 67195, 585391, 5504271, 6798541, 10763933, 866660818, 3830393407, 11044287758, 23058607363, 83159875881, 206501883259, 297734985607, 1087473543732, 1184060078117, 2789730557061, 2821551579466, 3529184155643

Offset: 1

Amiram Eldar, Mar 27 2019

nonn

The bi-unitary version of A056550.
The corresponding quotients are 1, 2, 13, 28, 38, 43, ... (see the link for more values).
a(32) > 10^13. - Giovanni Resta, May 28 2019

Amiram Eldar and Giovanni Resta, Table of n, a(n), A307159(a(n))/a(n) for n=1..31

Cf. A056550, A064611, A188999, A307043, A307159.

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]); seq={};s = 0; Do[s = s + bsigma[n]; If[Divisible[s,n], AppendTo[seq,n]], {n, 1, 10^6}]; seq

a(23)-a(31) from Giovanni Resta, Apr 20 2019

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).

A327566 Partial sums of the infinitary divisors sum function: a(n) = Sum_{k=1..n} isigma(k), where isigma is A049417.

Original entry on oeis.org

1, 4, 8, 13, 19, 31, 39, 54, 64, 82, 94, 114, 128, 152, 176, 193, 211, 241, 261, 291, 323, 359, 383, 443, 469, 511, 551, 591, 621, 693, 725, 776, 824, 878, 926, 976, 1014, 1074, 1130, 1220, 1262, 1358, 1402, 1462, 1522, 1594, 1642, 1710, 1760, 1838, 1910, 1980

Offset: 1

Amiram Eldar, Sep 17 2019

nonn

Differs from A307159 at n >= 16.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
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 (isigma), A327574.

Cf. A024916 (all divisors), A064609 (unitary), A307042 (exponential), A307159 (bi-unitary).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); Accumulate[Array[isigma, 52]]

a(n) ~ c * n^2, where c = 0.730718... (A327574).

A379615 Numerators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 4, 19, 107, 39, 61, 259, 89, 93, 857, 887, 181, 1303, 331, 1345, 4091, 4175, 21127, 4301, 21757, 87973, 88813, 90073, 90577, 1192621, 1201981, 1211809, 1221637, 1234741, 1240201, 626243, 89909, 45247, 15169, 30533, 153601, 2941819, 2956639, 20807623, 20876783

Offset: 1

Amiram Eldar, Dec 27 2024

			Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 89/40, 93/40, 857/360, 887/360, 181/72, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
V. Sitaramaiah and M. V. Subbarao, Asymptotic formulae for sums of reciprocals of some multiplicative functions, J. Indian Math. Soc., Vol. 57 (1991), pp. 153-167.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.13, p. 34.

Cf. A188999, A307159, A370904, A379616 (denominators), A379617.

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/bsigma[n], {n, 1, 50}]]]

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

a(n) = numerator(Sum_{k=1..n} 1/A188999(k)).

a(n)/A379616(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) / n), where A and B are constants.

A379616 Denominators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 3, 12, 60, 20, 30, 120, 40, 40, 360, 360, 72, 504, 126, 504, 1512, 1512, 7560, 1512, 7560, 30240, 30240, 30240, 30240, 393120, 393120, 393120, 393120, 393120, 393120, 196560, 28080, 14040, 4680, 9360, 46800, 889200, 889200, 6224400, 6224400, 889200, 1778400

Offset: 1

Amiram Eldar, Dec 27 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
V. Sitaramaiah and M. V. Subbarao, Asymptotic formulae for sums of reciprocals of some multiplicative functions, J. Indian Math. Soc., Vol. 57 (1991), pp. 153-167.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.13, p. 34.

Cf. A188999, A307159, A370904, A379615 (numerators), A379618.

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/bsigma[n], {n, 1, 50}]]]

bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / bsigma(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A188999(k)).

A379617 Numerators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 2, 11, 43, 53, 4, 37, 103, 23, 65, 71, 337, 2539, 1217, 2539, 7337, 7757, 1501, 7883, 7631, 31469, 30629, 31889, 6277, 84625, 82753, 423593, 82753, 426869, 421409, 216847, 213727, 108911, 11899, 24253, 119081, 2317139, 760853, 773203, 6889667, 7037867, 13946059

Offset: 1

Amiram Eldar, Dec 27 2024

			Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 103/120, 23/24, 65/72, 71/72, 337/360, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.13, p. 34.

Cf. A188999, A307159, A370904, A379615, A379618 (denominators).

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/bsigma[n], {n, 1, 50}]]]

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

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

a(n)/A379618(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) * n^c), where c = log(9/10)/log(2) = -0.152003..., and A and B are constants.

A379618 Denominators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 3, 12, 60, 60, 5, 40, 120, 24, 72, 72, 360, 2520, 1260, 2520, 7560, 7560, 1512, 7560, 7560, 30240, 30240, 30240, 6048, 78624, 78624, 393120, 78624, 393120, 393120, 196560, 196560, 98280, 10920, 21840, 109200, 2074800, 691600, 691600, 6224400, 6224400, 12448800

Offset: 1

Amiram Eldar, Dec 27 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.13, p. 34.

Cf. A188999, A307159, A370904, A379616, A379617 (numerators).

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/bsigma[n], {n, 1, 50}]]]

bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / bsigma(k); print1(denominator(s), ", "))};

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

cp's OEIS Frontend

A307160 Decimal expansion of the constant c in the asymptotic formula for the partial sums of the bi-unitary divisors sum function, A307159(k) ~ c*k^2.

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A307161 Numbers n such that A307159(n) = Sum_{k=1..n} bsigma(k) is divisible by n, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

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

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A327566 Partial sums of the infinitary divisors sum function: a(n) = Sum_{k=1..n} isigma(k), where isigma is A049417.

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A379615 Numerators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

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A379616 Denominators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

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A379617 Numerators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

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A379618 Denominators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

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