A129374 -id:A129374 - cp's OEIS frontend

A107311 Decimal expansion of the solution to zeta(x) = 2.

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

Offset: 1

Ralf Stephan, May 20 2005

From Artur Jasinski, Dec 21 2024: (Start)
Borwein et al. (2007) proved (Theorem 3.1) that the real parts of the zeros of the partials sums of the Riemman zeta functions are not greater than this constant.
Conjecture 1: the real parts of the zeros of the prime zeta function are not greater than this constant.
Conjecture 2: the real parts of the zeros of the anyone subset of the prime zeta function are not greater than this constant. (End)

			zeta(1.72864723899818361813510301...) = 2.

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

Peter Borwein, Greg Fee, Ron Ferguson, and Alexa van der Waal, Zeros of Partial Sums of the Riemann Zeta Function, Experiment. Math. 16(1) (2007), pp. 21-40. See p. 25.
Einar Hille, A problem in "factorisatio numerorum", Acta Arithmetica, 2(1);134-144, 1936.
Hsien-Kuei Hwang, Distribution of the number of factors in random ordered factorizations of integers, Journal of Number Theory 81:1 (2000), pp. 61-92.
M. Klazar and F. Luca, On the maximal order of numbers in the "factorisatio numerorum" problem, arXiv:math/0505352 [math.NT], 2005-2006.

Cf. A129374, A247667.

x /. FindRoot[ Zeta[x] == 2, {x, 2}, WorkingPrecision -> 102] // RealDigits // First (* Jean-François Alcover, Mar 19 2013 *)

PARI
```
solve(X=1.5,2,zeta(X)-2)
```

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

A197953 a(n) = 1 + Sum_{d|n, d>1} d * a(n/d).

Original entry on oeis.org

1, 3, 4, 11, 6, 24, 8, 43, 22, 38, 12, 128, 14, 52, 54, 171, 18, 186, 20, 206, 74, 80, 24, 640, 56, 94, 130, 284, 30, 494, 32, 683, 114, 122, 118, 1226, 38, 136, 134, 1038, 42, 682, 44, 440, 432, 164, 48, 3072, 106, 488, 174, 518, 54, 1374, 182, 1436, 194

Offset: 1

Paul D. Hanna, Oct 19 2011

nonn

Logarithmic derivative of A129374, where g.f. G(x) of A129374 satisfies: G(x) = 1/(1-x) * G(x^2)*G(x^3)*G(x^4)*...*G(x^n)*...

			L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 24*x^6/6 +...
where
L(x) = -log(1-x) + L(x^2) + L(x^3) + L(x^4) + L(x^5) +...+ L(x^n) +...
also, exp(L(x)) is the g.f. of A129374:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 15*x^6 + 20*x^7 +...

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paul D. Hanna)

Cf. A067824, A129374, A307607.

PARI
```
{a(n)=sumdiv(n,d,d*if(d==1,1,a(n/d)))}
```

/* L.g.f. satisfies: L(x) = -log(1-x) + Sum_{n>1} L(x^n) */
{a(n)=local(L=x,X=x+x*O(x^n));for(i=1,n,L=-log(1-X)+sum(m=2,n,subst(L,x,x^m+x*O(x^n))));n*polcoeff(L,n)}

L.g.f. satisfies: L(x) = -log(1-x) + Sum_{n>1} L(x^n), where L(x) = Sum_{n>=1} a(n)*x^n/n.

A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 6, 19, 30, 96, 152, 461, 775, 1883, 3271, 8751, 14370, 34004, 59491, 140450, 239746, 541817, 932681, 2089189, 3606641, 7719178, 13398411, 28848808, 49603982, 103047935, 179154858, 370200348, 639269735, 1295389370, 2241994088, 4511677298, 7798101800, 15408901600

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Euler transform of A163767.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A000219, A001970, A006171, A129373, A129374, A163767, A174465, A280487.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      A(d$2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 34; CoefficientList[Series[Product[1/(1 - x^k)^Times@@(Binomial[# + k - 1, k - 1]&/@FactorInteger[k][[All, 2]]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Times@@(Binomial[# + d - 1, d - 1]&/@FactorInteger[d][[All, 2]]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A163767(k).

A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 52, 88, 152, 252, 416, 664, 1076, 1684, 2636, 4060, 6248, 9444, 14292, 21312, 31748, 46796, 68804, 100200, 145784, 210240, 302520, 432428, 616716, 873972, 1236136, 1738560, 2439936, 3407924, 4749160, 6589156, 9123976, 12582620, 17316052, 23745756

Offset: 0

Seiichi Manyama, Nov 04 2018

nonn

Convolution of A129373 and A129374. - Vaclav Kotesovec, Nov 05 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A074206, A129373, A129374.

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

a(n) ~ exp((1+r) * ((2^(1+r) - 1) * Gamma(1+r) * Zeta(1+r))^(1/(1+r)) * n^(r/(1+r)) / (r * 2^(r/(1+r)) * (-Zeta'(r))^(1/(1+r)))) * (-2*(2^(1+r) - 1) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(10*(1+r))) / (2^(7/25) * 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 05 2018

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 4, -1, 9, 3, 11, -4, 17, -2, 11, -24, 31, -3, 39, -35, 70, -14, 47, -107, 112, -27, 122, -163, 198, -90, 93, -409, 282, -108, 329, -487, 601, -160, 324, -1076, 835, -165, 907, -1298, 1478, -429, 565, -2973, 1745, -427, 1999, -3149, 3587, -528

Offset: 0

Seiichi Manyama, Nov 05 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Convolution inverse of A321326.

Cf. A067856, A073709, A129374.

b[n_] := If[n == 1, 1, Product[{p, e} = pe; If[2 == p, e--, If[e > 1, p = 0, p = -1]]; p^e, {pe, FactorInteger[n]}]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
a = etr[b];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 01 2019 *)

Euler transform of A067856.
G.f.: Product_{k>0} 1/(1 - x^k)^A067856(k).
Product_{k>0} A(x^k) = Product_{k>=0} 1/(1 - x^(2^k))^(2^k). (Cf. A073709.)

A308271 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)A(x^3)A(x^5)* ... A(x^prime(k)) ...

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 48, 64, 93, 122, 169, 224, 303, 395, 532, 686, 907, 1168, 1523, 1943, 2521, 3193, 4094, 5170, 6573, 8245, 10434, 13015, 16344, 20321, 25363, 31379, 39003, 48039, 59384, 72914, 89720, 109722, 134528, 163929, 200149, 243199, 295831, 358280

Offset: 0

Ilya Gutkovskiy, May 17 2019

nonn

Euler transform of A008480.

Cf. A000040, A008480, A129374, A308272.

g:= proc(n) option remember; (l-> add(i, i=l)!/
      mul(i!, i=l))(map(i-> i[2], ifactors(n)[2]))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      add(d*g(d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2019

terms = 47; A[] = 1; Do[A[x] = 1/(1 - x) Product[A[x^Prime[k]], {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^A008480(k).

cp's OEIS Frontend

A107311 Decimal expansion of the solution to zeta(x) = 2.

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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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A197953 a(n) = 1 + Sum_{d|n, d>1} d * a(n/d).

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PARI

PARI

Formula

A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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Maple

Mathematica

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A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*... .

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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 ...

A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

A308271 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)A(x^3)A(x^5)* ... A(x^prime(k)) ...