A306072 -id:A306072 - cp's OEIS frontend

A286324 a(n) is the number of bi-unitary divisors of n.

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 6, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8

Offset: 1

Michel Marcus, May 07 2017

a(n) is the number of terms of the n-th row of A222266.

			From _Michael De Vlieger_, May 07 2017: (Start)
a(1) = 1 since 1 is the empty product; all divisors of 1 (i.e., 1) have a greatest common unitary divisor that is 1. 1 is a unitary divisor of all numbers n.
a(p) = 2 since 1 and p have greatest common unitary divisor 1.
a(6) = 4 since the divisor pairs {1, 6} and {2, 3} have greatest common unitary divisor 1.
a(24) = 8 since {1, 24}, {2, 12}, {3, 8}, {4, 6} have greatest unitary divisors {1, {1, 3, 8, 24}}, {{1, 2}, {1, 3, 4, 12}}, {{1, 3}, {1, 8}}, {{1, 4}, {1, 2, 3, 6}}: 1 is the greatest common unitary divisor among all 4 pairs. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
D. Suryanarayana, The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions pp 273-282, Lecture Notes in Mathematics book series (LNM, volume 251).
Eric Weisstein's World of Mathematics, Biunitary Divisor.

Cf. A222266, A188999, A293185 (indices of records), A340232, A350390.

Cf. A000005, A034444 (unitary), A037445 (infinitary).

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, 1 &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 90}] (* Michael De Vlieger, May 07 2017 *)
f[p_, e_] := If[OddQ[e], e + 1, e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120] (* Amiram Eldar, Dec 19 2018 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = #biudivs(n);

a(n)={my(f=factor(n)[,2]); prod(i=1, #f, my(e=f[i]); e + e % 2)} \\ Andrew Howroyd, Aug 05 2018

for(n=1, 100, print1(direuler(p=2, n, (X^3 - X^2 + X + 1) / ((X-1)^2 * (X+1)))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = e + (e mod 2). - Andrew Howroyd, Aug 05 2018
a(A340232(n)) = 2*n. - Bernard Schott, Mar 12 2023
a(n) = A000005(A350390(n)) (the number of divisors of the largest exponentially odd number dividing n). - Amiram Eldar, Sep 01 2023
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Let f(s) = Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (p-1)/((p+1)*p^2)) = A306071 = 0.80733082163620503914865427993003113402584582508155664401800520770441381...,
f'(1) = f(1) * Sum_{p prime} 2*(p^2 - p - 1) * log(p) /(p^4 + 2*p^3 + 1) = f(1) * 0.40523703144422392508596509911218523410441417240419849262346362977537989... = f(1) * A306072
and gamma is the Euler-Mascheroni constant A001620. (End)

A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 41, 45, 47, 51, 53, 57, 61, 65, 67, 75, 77, 81, 85, 89, 91, 99, 101, 107, 111, 115, 119, 123, 125, 129, 133, 141, 143, 151, 153, 157, 161, 165, 167, 175, 177, 181, 185, 189, 191, 199, 203, 211, 215, 219, 221

Offset: 1

Amiram Eldar, Jun 19 2018

nonn

The bi-unitary version of A006218 and A064608.

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

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

Cf. A001620, A006218, A064608, A286324, A306071, A306072.

fun[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; bdivnum[n_] := If[n==1,1,Times @@ (fun @@@ FactorInteger[n])]; Accumulate@ Array[bdivnum, {60}]

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = sum(k=1, n, #biudivs(k)); \\ Michel Marcus, Jun 20 2018

a(n) = A*n*(log(n) + 2*gamma - 1 + B) + O(n^(1/2)*exp(-A * log(n)^(3/5) * log(log(n))^(-1/5))), where gamma = A001620, A = A306071 and B = A306072.

A335705 Decimal expansion of Sum_{primes p} 2(p-3) log(p) / (p^3 + p - 2).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 18 2020

			0.079339315977982136748150057589344450315501605856610561211269...

Cf. A208133, A306072, A335706.

ratfun = 2*(p-3) / (p^3 + p - 2); 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}]

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.

Original entry on oeis.org

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

A345364 Decimal expansion of Sum_{p primes} p * (log(p))^2 / (p-1)^3.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 16 2021

			2.0914802823489018573384036648086053401546322612324184299409135322256726453113...

Eric Weisstein's World of Mathematics, Prime Factor, formula (15)-(16), (constant V).

Cf. A085609, A306072, A335705, A335706, A335707, A345308.

ratfun = p/(p - 1)^3; zetas = 0; ratab = Table[konfun = Together[Simplify[ratfun - c*(p^power/(p^power - 1)^2)]]; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*(-Zeta'[power]^2 / Zeta[power]^2 + Zeta''[power] / Zeta[power]) /. sol; ratfun = konfun /. sol, {power, 2, 30}]; Do[Print[N[Sum[Log[p]^2*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 100, 1000, 100}]

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A286324 a(n) is the number of bi-unitary divisors of n.

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A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

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A335705 Decimal expansion of Sum_{primes p} 2*(p-3) * log(p) / (p^3 + p - 2).

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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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A345364 Decimal expansion of Sum_{p primes} p * (log(p))^2 / (p-1)^3.

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A335705 Decimal expansion of Sum_{primes p} 2(p-3) log(p) / (p^3 + p - 2).