A107311 -id:A107311 - cp's OEIS frontend

A253249 Number of nonempty chains in the divides relation on the divisors of n.

1, 3, 3, 7, 3, 11, 3, 15, 7, 11, 3, 31, 3, 11, 11, 31, 3, 31, 3, 31, 11, 11, 3, 79, 7, 11, 15, 31, 3, 51, 3, 63, 11, 11, 11, 103, 3, 11, 11, 79, 3, 51, 3, 31, 31, 11, 3, 191, 7, 31, 11, 31, 3, 79, 11, 79, 11, 11, 3, 175, 3, 11, 31, 127, 11, 51, 3, 31, 11, 51

Offset: 1

Geoffrey Critzer, Jun 04 2015

nonn

For prime p, a(p)=3.
a(2^k) = 2^(k+1)-1.
For integers of the form n = p_1*p_2*...*p_k we have a(n) = A007047(k).
The value of a(n) depends only on the exponents in the prime factorization of n.

			a(10) = 11 because we have: {1}, {2}, {5}, {10}, {1|2}, {1|5}, {1|10}, {2|10}, {5|10}, {1|2|10}, {1|5|10}.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A002033, A007047, A074206, A107311, A378219 (Dirichlet inverse).

with(numtheory):
b:= proc(n) option remember: 1+ `if`(n=1, 0,
       add(b(d), d=divisors(n) minus {n}))
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 04 2015

Table[Total[Table[Length[Select[Subsets[Divisors[n], {k}],Apply[And, Map[Apply[Divisible, #] &,Partition[Reverse[#], 2, 1]]] &]], {k, 1,PrimeOmega[n] + 1}]], {n, 1, 100}]

Dirichlet g.f.: zeta(s)^2*A(s) where A(s) is the Dirichlet g.f. for A074206. - Geoffrey Critzer, May 23 2018
Sum_{k=1..n} a(k) ~ -4*n^r / (r*Zeta'(r)), where r = A107311 = 1.728647238998183618135103... is the root of the equation zeta(r) = 2. - Vaclav Kotesovec, Jan 31 2019
a(n) = 4*A002033(n-1) - 1 for n > 1. - Geoffrey Critzer, Aug 19 2020

A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 15, 20, 35, 48, 76, 103, 166, 221, 333, 451, 671, 894, 1303, 1730, 2479, 3288, 4615, 6086, 8502, 11142, 15299, 20034, 27285, 35514, 47937, 62168, 83259, 107650, 142929, 184090, 243207, 312041, 409210, 523709, 683261, 871239, 1130703

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A074206, A129373, A129375.

{a(n)=local(A=1+x);for(i=2,n,A=1/(1-x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

G.f.: A(x) = Product_{n>=1} 1/(1 - x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ exp((1 + 1/r) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(10*(1+r))) / ((2*Pi)^(29/50) * sqrt(1+r) * n^((6 + 5*r)/(10*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

A050369 Number of ordered factorizations of n into 2 kinds of 2, 3 kinds of 3, ...

Original entry on oeis.org

1, 2, 3, 8, 5, 18, 7, 32, 18, 30, 11, 96, 13, 42, 45, 128, 17, 144, 19, 160, 63, 66, 23, 480, 50, 78, 108, 224, 29, 390, 31, 512, 99, 102, 105, 936, 37, 114, 117, 800, 41, 546, 43, 352, 360, 138, 47, 2304, 98, 400, 153, 416, 53, 1080, 165, 1120, 171, 174, 59, 2640

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

Dirichlet inverse of (A000027*A153881). - Mats Granvik, Jan 03 2009

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A074206.

a[1]=1; a[n_]:=a[n]=n*Sum[If[d==n,0,a[d]/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}] (* Vaclav Kotesovec, Feb 02 2019 *)

Dirichlet g.f.: 1/(2-zeta(s-1)).
a(n) = n*Sum_{d divides n, d1, a(1)=1. - Vladeta Jovovic, Feb 09 2002
Sum_{k=1..n} a(k) ~ -n^(1+r) / ((1+r)*Zeta'(r)), where r = A107311 = 1.728647238998183618135103010297... is the root of the equation Zeta(r) = 2. - Vaclav Kotesovec, Feb 02 2019
G.f. A(x) satisfies: A(x) = x + 2*A(x^2) + 3*A(x^3) + 4*A(x^4) + ... - Ilya Gutkovskiy, May 10 2019
For n > 0, a(n) = n * A074206(n). - Vaclav Kotesovec, Mar 18 2021

A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 9, 13, 19, 26, 34, 52, 67, 89, 123, 166, 214, 295, 380, 501, 660, 858, 1098, 1461, 1858, 2384, 3072, 3940, 4975, 6410, 8070, 10234, 12946, 16322, 20412, 25848, 32201, 40261, 50287, 62728, 77681, 96885, 119673, 148197, 183108, 225974

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A074206; A129374, A129375.

{a(n)=local(A=1+x);for(i=2,n,A=(1+x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

G.f.: A(x) = Product_{n>=1} (1 + x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ exp((1 + 1/r) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(2 + 2*r)) / (2^(1/10) * sqrt(Pi) * sqrt(1+r) * n^((2+r)/(2 + 2*r))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

A247667 Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 22 2014

			0.55001000541315449183305812670222219646116822710271404...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 41.

Cf. A107311, A129373, A129374, A129375, A217598, A247668.

digits = 101; rho = x /. FindRoot[Zeta[x] == 2, {x, 2}, WorkingPrecision -> digits+5]; cm = -1/Zeta'[rho]; RealDigits[cm, 10, digits] // First

c_m = -1/zeta'(rho), where rho = 1.728647... is A107311, the real solution to zeta(rho) = 2.

Residue_{s = rho} 1/(2 - Zeta(s)). - Vaclav Kotesovec, Nov 04 2018

A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

Original entry on oeis.org

1, 1, 3, 13, 97, 621, 6571, 58633, 779073, 9317017, 138628531, 1977676581, 37384244833, 620735382853, 12434855135067, 245117537189281, 5651550278494081, 123266430844431153, 3128700944169196003

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A074206, A129373, A129374.

{a(n)=local(A=1+x);for(i=2,n,A=exp(x+x*O(x^n))*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); n!*polcoeff(A,n)}

E.g.f.: A(x) = exp( Sum_{n>=1} A074206(n)*x^n ) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ n! * exp(2/5 + (1 + 1/r) * (-Gamma(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-Gamma(1+r) / Zeta'(r))^(1/(2*(1+r))) / (sqrt(2*Pi*(1+r)) * n^((2+r)/(2*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

PARI program fixed by Vaclav Kotesovec, Feb 26 2014

A307604 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

Original entry on oeis.org

1, 1, 3, 6, 17, 28, 72, 122, 282, 493, 1027, 1790, 3673, 6300, 12205, 21117, 39782, 67989, 124937, 212189, 381705, 644625, 1136315, 1905352, 3312916, 5513005, 9443362, 15624026, 26445046, 43451200, 72751824, 118792691, 196966722, 319714816, 525316191, 847734183, 1381904765

Offset: 0

Ilya Gutkovskiy, Apr 18 2019

nonn

Euler transform of A050369.

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 17*x^4 + 28*x^5 + 72*x^6 + 122*x^7 + 282*x^8 + 493*x^9 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A050369, A074206, A129374, A307605, A307606, A307607.

terms = 36; A[] = 1; Do[A[x] = 1/(1 - x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*A074206(k)).

a(n) ~ (-Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(50*(2+r))) * exp(12/625 + ((2+r)/(1+r)) * (-Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(2+r)) * n^((1+r)/(2+r))) / (A^(144/625) * sqrt(2*Pi*(2+r)) * n^(1/2 + 1/(50*(2+r)))), where r = A107311 is the root of the equation zeta(r)=2 and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 18 2021

A217598 Decimal expansion of the coefficient of asymptotic expression of m(n), the number of multiplicative compositions of n.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 19 2013

From Amiram Eldar, Oct 16 2020: (Start)
Equals -1/(rho * zeta'(rho)), where rho is the root of zeta(rho) = 2 (A107311).
Equals lim_{k->oo} A173382(k)/k^rho. (End)

			0.318173652...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 293.

Cf. A074206, A107311 (rho), A173382.

rho = x /. FindRoot[Zeta[x] == 2, {x, 2}, WorkingPrecision -> 100]; RealDigits[-1/(rho*Zeta'[rho])] // First

a217598={my(rho=solve(x=1.1,2,zeta(x)-2));-1/(rho*zeta'(rho))} \\ Hugo Pfoertner, Oct 16 2020

A173382 Partial sums of A074206.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 10, 14, 16, 19, 20, 28, 29, 32, 35, 43, 44, 52, 53, 61, 64, 67, 68, 88, 90, 93, 97, 105, 106, 119, 120, 136, 139, 142, 145, 171, 172, 175, 178, 198, 199, 212, 213, 221, 229, 232, 233, 281, 283, 291, 294, 302, 303, 323, 326, 346, 349, 352, 353, 397, 398, 401, 409, 441, 444, 457

Offset: 0

Jonathan Vos Post, Feb 17 2010

Partial sums of number of ordered factorizations of n.

a(n) is also the number of ways to tile a strip of length at most log(n) with tiles having lengths {log(k) : k>=2}. - David Bevan, Jan 07 2025

			a(96) = 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 2 + 3 + 1 + 8 + 1 + 3 + 3 + 8 + 1 + 8 + 1 + 8 + 3 + 3 + 1 + 20 + 2 + 3 + 4 + 8 + 1 + 13 + 1 + 16 + 3 + 3 + 3 + 26 + 1 + 3 + 3 + 20 + 1 + 13 + 1 + 8 + 8 + 3 + 1 + 48 + 2 + 8 + 3 + 8 + 1 + 20 + 3 + 20 + 3 + 3 + 1 + 44 + 1 + 3 + 8 + 32 + 3 + 13 + 1 + 8 + 3 + 13 + 1 + 76 + 1 + 3 + 8 + 8 + 3 + 13 + 1 + 48 + 8 + 3 + 1 + 44 + 3 + 3 + 3 + 20 + 1 + 44 + 3 + 8 + 3 + 3 + 3 + 112.

Shikao Ikehara, On Kalmar's Problem in “Factorisatio Numerorum”, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 21 (1939) pp. 208-219.
Shikao Ikehara, On Kalmar's Problem in “Factorisatio Numerorum” II, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 23 (1941) pp. 767-774.
Kalmár, Laszlo. "Über die mittlere Anzahl der Produktdarstellungen der Zahlen.(Erste Mitteilung)'." Acta Litt. ac Scient. Szeged 5 (1931): 95-107.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
David Bevan and Julien Condé, Introducing irrational enumeration: analytic combinatorics for objects of irrational size, arXiv:2412.14682 [math.CO], 2024. See p. 11.
Ann Clifton, Eva Czabarka, Kevin Liu, Sarah Loeb, Utku Okur, Laszlo Szekely, and Kristina Wicke, Universal rooted phylogenetic tree shapes and universal tanglegrams, arXiv:2308.06580 [math.CO], 2023.
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A074206, A002033, A001055, A050324, A000670.

A025523 is an essentially identical sequence.

Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = 1 + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 0, 100}] (* Vaclav Kotesovec, Jan 31 2019 *)
Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; Join[{0}, Accumulate[a /@ Range[100]]] (* Vaclav Kotesovec, Jan 31 2019, after Jean-François Alcover, faster *)

a(n) = Sum_{i=0..n} A074206(i).

a(n) ~ -n^r / (r*Zeta'(r)), where r = A107311 = 1.728647238998183618135103... is the root of the equation Zeta(r) = 2. - Vaclav Kotesovec, Jan 31 2019

Terms corrected by N. J. A. Sloane, May 04 2016

A307605 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

Original entry on oeis.org

1, 1, 2, 5, 12, 20, 48, 81, 169, 305, 580, 1009, 1966, 3338, 6067, 10503, 18730, 31633, 55641, 93151, 160389, 267585, 452762, 747016, 1253644, 2049943, 3390786, 5516227, 9034745, 14572790, 23668066, 37918484, 61042425, 97231826, 155292944, 245774727, 389998116

Offset: 0

Ilya Gutkovskiy, Apr 18 2019

nonn

Weigh transform of A050369.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 20*x^5 + 48*x^6 + 81*x^7 + 169*x^8 + 305*x^9 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A050369, A074206, A129373, A307604, A307606.

terms = 36; A[] = 1; Do[A[x] = (1 + x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} (1 + x^k)^(k*A074206(k)).

a(n) ~ ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(4 + 2*r)) * exp((2+r)/(1+r) * ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(2+r)) * n^((1+r)/(2+r))) / (2^(1/50) * sqrt(Pi*(2+r)) * n^((3 + r)/(4 + 2*r))), where r = A107311 is the root of the equation zeta(r) = 2. - Vaclav Kotesovec, Mar 18 2021

cp's OEIS Frontend

A253249 Number of nonempty chains in the divides relation on the divisors of n.

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A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A050369 Number of ordered factorizations of n into 2 kinds of 2, 3 kinds of 3, ...

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A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A247667 Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.

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A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A307604 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

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A217598 Decimal expansion of the coefficient of asymptotic expression of m(n), the number of multiplicative compositions of n.

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A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A307604 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

A307605 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...