A306071 -id:A306071 - 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)

A350390 a(n) is the largest exponentially odd divisor of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 24, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 32, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Dec 28 2021

First differs from A331737 at n = 16.

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

Cf. A268335, A306071, A325837, A331737, A336643, A350389.

f[p_, e_] := If[OddQ[e], p^e, p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] - !(f[i,2]%2)));} \\ Amiram Eldar, Sep 18 2023

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A350390(n): return n*core(n)//prod(primefactors(n)) # Chai Wah Wu, Dec 30 2021

Multiplicative with a(p^e) = p^e if e is odd and p^(e-1) otherwise.
a(n) = n/A336643(n).
a(n) = n if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ (1/2)*c*n^2, where c = Product_{p prime} 1-(p-1)/(p^2*(p+1)) = 0.8073308216... (A306071).
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(2*s-1)). - Amiram Eldar, Sep 18 2023

A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 27 2019

			1.602317102305418052349626315621161003776939495785572...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z3).
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 2.37 and 3.18, pp. 1346 and 1354).
Abdelhakim Smati and Jie Wu, On the exponential divisor function, Publications de l'Institut Mathématique, Vol. 61 (1997), pp. 21-32.
László Tóth, An order result for the exponential divisor function, Publ. Math. Debrecen, Vol. 71, No. 1-2 (2007), pp. 165-171, arXiv preprint,, arXiv:0708.3552 [math.NT], 2007.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.10, p. 30).
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, Journal de théorie des nombres de Bordeaux, Vol. 7, No. 1, (1995), pp. 133-141.

Cf. A000005, A047994, A049419, A145353, A361013, A361967, A379517, A379518.

Cf. A059956 (constant for unitary divisors), A306071 (bi-unitary), A327576 (infinitary).

$MaxExtraPrecision = 1500; m = 1500; em = 500; f[x_] := 1 + Log[1 + Sum[x^e * (DivisorSigma[0, e] - DivisorSigma[0, e - 1]), {e, 2, em}]]; c = Rest[ CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m] ]; RealDigits[ Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{k->oo} A145353(k)/k.
Equals Product_{p prime} (1 + Sum_{e >= 2} p^(-e) * (d(e) - d(e-1))), where d(e) is the number of divisors of e (A000005).
Equals Product_{p prime} (1 - 1/p) * (2 - (log(p-1) + QPolyGamma(0, 1, 1/p)) / log(p)). - Vaclav Kotesovec, Feb 27 2023
From Amiram Eldar, Dec 24 2024: (Start)
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} k/uphi(k) = lim_{m->oo} (1/m) * Sum_{k=1..m} A319677(k)/A319676(k), where uphi(k) is the unitary totient function (A047994).
Equals lim_{m->oo} (1/log(m)) * Sum_{k=1..m} 1/uphi(k) = lim_{m->oo} (1/log(m)) * A379517(m)/A379518(m).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A361967(k).
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k-1))).
Equals Product_{p prime} (1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1))). (End)

More digits from Vaclav Kotesovec, Jun 13 2021

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.

A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

Original entry on oeis.org

1, 2, 4, 7, 11, 14, 20, 27, 35, 41, 51, 59, 71, 80, 89, 104, 120, 132, 150, 164, 178, 193, 215, 232, 256, 274, 300, 321, 349, 364, 394, 425, 448, 472, 497, 526, 562, 589, 617, 648, 688, 709, 751, 786, 820, 853, 899, 935, 983, 1019, 1056, 1098, 1150, 1189

Offset: 1

Amiram Eldar, Jun 19 2018

nonn

The bi-unitary version of A002088 and A177754.

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

Cf. A002088, A177754, A116550, A306071.

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[bphi[n], {n, 1, 100}]] (* after Jean-François Alcover at A116550 *)
phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; Accumulate[Array[bphi, 100]] (* Amiram Eldar, Jun 30 2025 *)

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

a(n) = A*n^2/2 + O(n*log(n)^2), where A = A306071.

A327576 Decimal expansion of the constant that appears in the asymptotic formula for average order of the number of infinitary divisors function (A037445).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 17 2019

			0.366625276945381909556532720687001563033612155971009...

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. A037445, A050376, A327573.

Cf. A059956 (corresponding constant for unitary divisors), A306071 (bi-unitary).

m = 1000; em = 10; f[x_] := Sum[Log[1 - 1/(1 + 1/x^(2^e))^2], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; $MaxExtraPrecision = 1500; RealDigits[(1/2)*Exp[f[1/2] + f[1/3]]* Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - (1/2)^k - (1/3)^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals Limit_{n->oo} A327573(n)/(2 * n * log(n)). [Corrected by Amiram Eldar, May 07 2021]

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

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

A327575 Decimal expansion of the constant that appears in the asymptotic formula for average order of an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 17 2019

			0.328935838840335516355748487365220229577066523794694...

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. A091732, A050376, A327572.

Cf. A104141 (corresponding constant for phi), A065463 (unitary), A306071 (bi-unitary).

$MaxExtraPrecision = 1500; m = 1500; 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} A327572(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).

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.

A319593 Decimal expansion of the probability that an integer triple is pairwise unitary coprime.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 27 2019

Two numbers are unitary coprime if their largest common unitary divisor is 1.

			0.552306904157942811183227347309264708535455831404497...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 54.

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. 509), preprint, arXiv:1310.7053 [math.NT], 2013-2014 (see p. 22).

Cf. A065473, A077610, A306071.

$MaxExtraPrecision = 1000; nm = 1000; f[x_] := 1 - 4*x^2 + 7*x^3 - 9*x^4 + 8*x^5 - 2*x^6 - 3*x^7 + 2*x^8; c = LinearRecurrence[{-1, 3, -4, 5, -3, -1, 2}, {0, -8, 21, -68, 180, -503, 1428}, nm]; RealDigits[f[1/2] * f[1/3] * Zeta[2] * Zeta[3] * Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

zeta(2) * zeta(3) * prodeulerrat(1-4/p^2+7/p^3-9/p^4+8/p^5-2/p^6-3/p^7+2/p^8) \\ Amiram Eldar, Jun 29 2023

Equals zeta(2) * zeta(3) * Product_{p prime} (1 - 4/p^2 + 7/p^3 - 9/p^4 + 8/p^5 - 2/p^6 - 3/p^7 + 2/p^8).

cp's OEIS Frontend

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

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A350390 a(n) is the largest exponentially odd divisor of n.

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A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

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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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A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

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A327576 Decimal expansion of the constant that appears in the asymptotic formula for average order of the number of infinitary divisors function (A037445).

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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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