A306069 -id:A306069 - cp's OEIS frontend

A344731 Numbers k such that k divides A306069(k).

1, 275, 277, 3337, 3353, 3359, 39675, 39689, 472467, 797806459, 9501109507

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients A306069(k)/k are 1, 5, 5, 7, 7, 7, 9, 9, 11, 17, 19, ...

a(12) > 7.5*10^10, if it exists.

			a(1) = 1 since A306069(1) = 1 is divisible by 1.
a(2) = 275 since A306069(275) = 1375 = 5 * 275 is divisible by 275.

Cf. A286324, A306069.
The bi-unitary version of A050226.
Similar sequences: A064610, A344732, A344733.

f[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; s[1] = 1; s[n_] := s[n] = s[n - 1] + Times @@ f @@@ FactorInteger[n]; Select[Range[40000], Divisible[s[#], #] &]

A306071 Decimal expansion of Sum_{n>=1} (-1)^omega(n) phi(n)^2/n^4, where omega(n) is the number of distinct prime factors of n (A001221) and phi is Euler's totient function (A000010).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 19 2018

The constant A that appears in the asymptotic formulae for the sums of the bi-unitary divisor function (A306069) and the bi-unitary totient function (A306070).
The product in Suryanarayana's 1972 paper has a error that was corrected in his 1975 paper.
The probability that 2 randomly selected numbers will be unitary coprime (i.e. their largest common unitary divisor is 1). - Amiram Eldar, Aug 27 2019

			0.80733082163620503914...

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.

D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of bi-unitary divisors of an integer - II, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 508), preprint, arXiv:1310.7053 [math.NT] (2014) (see p. 21).

Cf. A000010, A001221, A286324, A306069, A306070.

cc = CoefficientList[Series[Log[1 - (p - 1)/(p^2*(p + 1))] /. p -> 1/x, {x, 0, 36}], x]; f = FindSequenceFunction[cc]; digits = 20; A = Exp[NSum[f[n + 1 // Floor]*(PrimeZetaP[n]), {n, 2, Infinity}, NSumTerms -> 16 digits, WorkingPrecision -> 16 digits]]; RealDigits[A, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
$MaxExtraPrecision = 1000; Do[Print[Zeta[2] * Exp[-N[Sum[q = Expand[(2*p^2 - 2*p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}], 120]]], {t, 300, 1000, 100}] (* Vaclav Kotesovec, May 29 2020 *)

prodeulerrat(1 - (p-1)/(p^2 * (p+1))) \\ Amiram Eldar, Mar 18 2021

Equals Product_{p prime} (1 - (p-1)/(p^2 * (p+1))).

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

a(1)-a(20) from Jean-François Alcover, Jun 19 2018
a(20)-a(24) from Jon E. Schoenfield, May 27 2019
More terms from Vaclav Kotesovec, May 29 2020

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

Original entry on oeis.org

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).

A306072 Decimal expansion of 2 * Sum_{p prime}(p^2-p-1)log(p)/(p^4+2p^3+1).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 19 2018

The constant B that appears in the asymptotic formula for the sum of the bi-unitary divisor function (A306069).

			0.405237031444223925085965099112185234104414172404198492623463629775379...

Cf. A085609, A306069, A335705, A335707.

cc = CoefficientList[Series[(p^2 - p - 1)/(p^4 + 2*p^3 + 1) /. p -> 1/x, {x, 0, 30}], x]; f = FindSequenceFunction[cc]; digits = 20; B = 2 NSum[f[n + 1 // Round]*(-PrimeZetaP'[n]), {n, 2, Infinity}, Method -> "AlternatingSigns", NSumTerms -> 10 digits, WorkingPrecision -> 5 digits]; RealDigits[B, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
ratfun = 2*(p^2 - p - 1)/(p^4 + 2*p^3 + 1); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}] (* Vaclav Kotesovec, Jun 17 2020 *)

a(1)-a(20) from Jean-François Alcover, Jun 19 2018

More digits from Vaclav Kotesovec, Jun 17 2020

A327573 Partial sums of the number of infinitary divisors function: a(n) = Sum_{k=1..n} id(k), where id is A037445.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 41, 43, 45, 49, 51, 55, 59, 63, 65, 73, 75, 79, 83, 87, 89, 97, 99, 103, 107, 111, 115, 119, 121, 125, 129, 137, 139, 147, 149, 153, 157, 161, 163, 167, 169, 173, 177, 181, 183, 191, 195, 203, 207, 211, 213, 221

Offset: 1

Amiram Eldar, Sep 17 2019

nonn

Differs from A306069 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. A037445, A327576.

Cf. A006218 (all divisors), A064608 (unitary), A306069 (bi-unitary), A145353 (exponential).

f[p_, e_] := 2^DigitCount[e, 2, 1]; id[1] = 1; id[n_] := Times @@ (f @@@ FactorInteger[n]); Accumulate[Array[id, 100]]

a(n) ~ 2 * c * n * log(n), where c = 0.366625... (A327576). [Corrected by Amiram Eldar, May 07 2021]

A344273 a(n) is the least k such that the average number of bi-unitary divisors of {1..k} is >= n.

Original entry on oeis.org

1, 6, 24, 80, 273, 960, 3336, 11480, 39648, 136952, 472416, 1630164, 5625480, 19412736, 66992016, 231184800, 797806152, 2753187210, 9501109380, 32787848746

Offset: 1

Amiram Eldar, May 13 2021

			a(2) = 6 since the average number of bi-unitary divisors of {1..6} is A306069(6)/6 = 13/6 > 2.

Cf. A286324, A306069, A306071.
The unitary version of A085829.
Similar sequences: A328331, A336304, A338891, A338943, A344272, A344274.

f[p_, e_] := If[OddQ[e], e + 1, e]; bdivnum[1] = 1; bdivnum[n_] := Times @@ (f @@@ FactorInteger[n]); seq={}; s = 0; k = 1; Do[While[s = s + bdivnum[k]; s < k*n, k++]; AppendTo[seq, k]; k++, {n, 1, 10}]; seq

Lim_{n->oo} a(n+1)/a(n) = exp(1/A) = 3.4509501567..., where A is A306071.

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A344731 Numbers k such that k divides A306069(k).

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A306071 Decimal expansion of Sum_{n>=1} (-1)^omega(n) phi(n)^2/n^4, where omega(n) is the number of distinct prime factors of n (A001221) and phi is Euler's totient function (A000010).

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

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A306072 Decimal expansion of 2 * Sum_{p prime}(p^2-p-1)*log(p)/(p^4+2*p^3+1).

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A327573 Partial sums of the number of infinitary divisors function: a(n) = Sum_{k=1..n} id(k), where id is A037445.

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A344273 a(n) is the least k such that the average number of bi-unitary divisors of {1..k} is >= n.

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A306072 Decimal expansion of 2 * Sum_{p prime}(p^2-p-1)log(p)/(p^4+2p^3+1).