A002746 -id:A002746 - cp's OEIS frontend

A346545 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(x)/k).

1, 1, 5, 26, 175, 1384, 12933, 135050, 1582901, 20380208, 286577757, 4352682256, 71247772121, 1244923243966, 23166410620637, 456940648889070, 9521696033968393, 208851154175983608, 4812156417656806393, 116112764199821653284, 2928658457243240595901, 77042063713731887400418

Offset: 0

Ilya Gutkovskiy, Sep 16 2021

nonn

Exponential transform of A002746.

Cf. A000005, A002746, A028342, A346546, A346547, A346548.

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002746[n_] := Sum[Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002746[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

E.g.f.: exp( exp(x) * Sum_{k>=1} d(k) * x^k / k ).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002746(k) * a(n-k).

A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.

Original entry on oeis.org

1, 3, 11, 57, 359, 2793, 25871, 273297, 3268199, 44132313, 659178431, 10710083937, 189256343639, 3636935896233, 75228664345391, 1657133255788977, 38770903634692679, 964609458391250553, 25470259163197390751, 709595190213796188417

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000005, A000629, A002746, A008277, A028342, A308554, A318249, A330352, A330353, A330354, A330445.

nmax = 20; CoefficientList[Series[-Sum[Log[1 - (Exp[x] - 1)^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: Sum_{i>=1} Sum_{j>=1} (exp(x) - 1)^(i*j) / (i*j).
E.g.f.: log(Product_{k>=1} 1 / (1 - (exp(x) - 1)^k)^(1/k)).
G.f.: Sum_{k>=1} (k - 1)! * tau(k) * x^k / Product_{j=1..k} (1 - j*x), where tau = A000005.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * tau(k).
a(n) ~ n! * (log(n) + 2*gamma - log(2) - log(log(2))) / (n * (log(2))^n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 14 2019

A002744 Sum of logarithmic numbers.

Original entry on oeis.org

1, 0, 1, 10, -17, 406, -1437, 20476, -44907, 1068404, -5112483, 230851094, -1942311373, 31916614874, -27260241361, 3826126294680, -37957167335671, 2169009251237640, -25847377785179111, 858747698098918338, -5611513985867158697, 154094365406716365118

Offset: 1

N. J. A. Sloane

sign

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..451
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
J. M. Gandhi, Logarithmic Numbers and the Functions d(n) and sigma(n), The American Mathematical Monthly, Vol. 73, No. 9 (1966), pp. 959-964, alternative link.
Index entries for sequences related to logarithmic numbers

Cf. A000005, A002746, A318249.

a[n_] := n! * Sum[(-1)^k * DivisorSigma[0, n - k]/k!/(n - k), {k, 0, n - 1}]; Array[a, 22] (* Amiram Eldar, May 13 2020 *)

a(n) = sum(k=1, n, (-1)^(n-k)*numdiv(k)*(k-1)!*binomial(n, k)); \\ Michel Marcus, May 13 2020

a(n) = Sum_{k=1..n} (-1)^(n-k)*A000005(k)*(k-1)!*binomial(n, k). - Vladeta Jovovic, Feb 09 2003
E.g.f.: -exp(-x) * log(Product_{k>=1} (1 - x^k)^(1/k)). - Ilya Gutkovskiy, Dec 11 2019
a(p) == -2 (mod p) for prime p. The pseudoprimes of this congruence are 4, 6, 20, 42, 1806, ... - Amiram Eldar, May 13 2020

Corrected and extended by Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

A260323 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

Original entry on oeis.org

1, 3, 2, 8, 6, 6, 24, 24, 24, 24, 89, 80, 60, 120, 120, 415, 450, 480, 360, 720, 720, 2372, 2142, 2730, 840, 2520, 5040, 5040, 16072, 17696, 10416, 21840, 6720, 20160, 40320, 40320, 125673, 112464, 151704, 184464, 15120, 60480, 181440, 362880, 362880

Offset: 1

N. J. A. Sloane, Jul 23 2015

			Triangle begins:
1,
3,2,
8,6,6,
24,24,24,24,
89,80,60,120,120,
415,450,480,360,720,720,
2372,2142,2730,840,2520,5040,5040,
...

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]

Rows, column sums give A002104, A002742, A002745, A002746.

Cf. A260322-A260325.

A260323 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( 1/(r-j*n)!/j,j=1..(r)/n) ;
        %*r! ;
    end if;
end proc:
for r from 1 to 20 do
    for n from 1 to r do
        printf("%a,",A260323(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

T[n_, k_] := If[n == 0, 1, If[k > n+1, 0, Sum[1/(n - j*k)!/j, {j, 1, n/k}]]]*n!;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2023, after R. J. Mathar *)

A356589 a(n) = n! * Sum_{k=1..n} sigma_k(k)/(k * (n-k)!).

Original entry on oeis.org

1, 7, 74, 1896, 83829, 6169915, 634444586, 89796130088, 16407420884385, 3792452363345383, 1076168167972120354, 368657061467873013440, 149787334364400115372677, 71262783791831946810277899, 39228224120114488162020163762

Offset: 1

Seiichi Manyama, Aug 14 2022

nonn

Cf. A002745, A002746, A356437, A356590.

a[n_] := n! * Sum[DivisorSigma[k, k]/(k*(n - k)!), {k, 1, n}]; Array[a, 15] (* Amiram Eldar, Aug 14 2022 *)

a(n) = n!*sum(k=1, n, sigma(k, k)/(k*(n-k)!));

my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, log(1-(k*x)^k)/k)))

E.g.f.: -exp(x) * Sum_{k>0} log(1 - (k*x)^k)/k.

a(n) ~ n! * n^(n-1). - Vaclav Kotesovec, Aug 17 2022

A356600 a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).

Original entry on oeis.org

1, 7, 38, 240, 1509, 12115, 96326, 929432, 9421089, 108909943, 1249105054, 17862483320, 241674418101, 3676733397363, 59149265744302, 1058605924855568, 18041587282787489, 363409114370324295, 6970858463185187062, 153017341796727034336, 3360005220780469981157

Offset: 1

Seiichi Manyama, Aug 15 2022

nonn

The average value of a(n) is zeta(3) * exp(1) * n * n!. - Vaclav Kotesovec, Aug 17 2022

Cf. A002745, A002746, A356589.

Cf. A356298.

Table[n! * Sum[DivisorSigma[2, k]/(k * (n-k)!), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 17 2022 *)

a(n) = n!*sum(k=1, n, sigma(k, 2)/(k*(n-k)!));

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, x^k/(k*(1-x^k)^2))))

my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, k*log(1-x^k))))

E.g.f.: exp(x) * Sum_{k>0} x^k/(k * (1 - x^k)^2).

E.g.f.: -exp(x) * Sum_{k>0} k * log(1 - x^k).

cp's OEIS Frontend

A346545 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(x)/k).

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A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.

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A002744 Sum of logarithmic numbers.

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A260323 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

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A356589 a(n) = n! * Sum_{k=1..n} sigma_k(k)/(k * (n-k)!).

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A356600 a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).

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