A318249 -id:A318249 - cp's OEIS frontend

A002746 Sum of logarithmic numbers.

1, 4, 13, 50, 203, 1154, 6627, 49356, 403293, 3858376, 33929377, 460614670, 5168544119, 64518640406, 946910125319, 16124114481720, 221243980745433, 4261440137319852, 68524390012831189, 1477309421907315082

Offset: 1

N. J. A. Sloane

nonn

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..450
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, A002744, A318249.

Table[Sum[Binomial[n,k] * DivisorSigma[0,k] * (k-1)!, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 16 2019 *)

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

a(n) = Sum_{k=1..n} 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, 12, 30, 380, 858, 1722 ... - Amiram Eldar, May 13 2020

Corrected and extended by Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

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

A338805 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1-x^j)^(-u/j).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 18, 28, 12, 1, 48, 170, 100, 20, 1, 480, 988, 870, 260, 30, 1, 1440, 7896, 7588, 3150, 560, 42, 1, 20160, 60492, 73808, 37408, 9100, 1064, 56, 1, 120960, 555264, 764524, 460656, 140448, 22428, 1848, 72, 1, 1451520, 5819904, 8448120, 5952700, 2162160, 436296, 49140, 3000, 90, 1

Offset: 1

Seiichi Manyama, Nov 10 2020

Also the Bell transform of A318249.

If we use sigma(n,1) in Vladeta Jovovic's formulas in A008298 then one gets the D'Arcais numbers, if we use sigma(n,0) then this sequence arises. # Peter Luschny, Jun 01 2022

			exp(Sum_{n>0} u*d(n)*x^n/n) = 1 + u*x + (2*u+u^2)*x^2/2! + (4*u+6*u^2+u^3)*x^3/3! + ... .
Triangle begins:
      1;
      2,     1;
      4,     6,     1;
     18,    28,    12,     1;
     48,   170,   100,    20,    1;
    480,   988,   870,   260,   30,    1;
   1440,  7896,  7588,  3150,  560,   42,  1;
  20160, 60492, 73808, 37408, 9100, 1064, 56, 1;

Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.

Column k=1..3 give A318249, A338810, A338811.
Row sums give A028342.
Cf. A000005 (d(n)), A008298, A264428.

# The function BellMatrix is defined in A264428 (with column k = 0).
BellMatrix(n -> n!*NumberTheory:-SumOfDivisors(n+1, 0), 9);
# Alternative:
P := proc(n, x) option remember; if n = 0 then 1 else
(1/n)*x*add(NumberTheory:-SumOfDivisors(n-k, 0)*P(k, x), k=0..n-1) fi end:
Trow := n -> seq(n!*coeff(P(n, x), x, k), k = 1..n):
seq(Trow(n), n = 0..10); # Peter Luschny, Jun 01 2022

a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSigma[0, n]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1-x^j+x*O(x^n))^(-u/j)), n), k)}

a(n) = if(n<1, 0, (n-1)!*numdiv(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

E.g.f.: exp(Sum_{n>0} u*d(n)*x^n/n), where d(n) is the number of divisors of n.
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} d(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} d(i_j)/i_j.

A330352 Expansion of e.g.f. -Sum_{k>=1} log(1 - log(1 + x)^k) / k.

Original entry on oeis.org

1, 1, 0, 10, -68, 818, -9782, 130730, -1835752, 27408672, -438578616, 7697802264, -150743052528, 3293454634416, -78787556904864, 2014008113598432, -54001416897306240, 1504891127666322048, -43527807706621236480, 1311515508480252542208

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Cf. A000005, A002744, A008275, A028342, A089064, A318249, A330351, A330353, A330354, A330493.

nmax = 20; CoefficientList[Series[-Sum[Log[1 - Log[1 + x]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: Sum_{i>=1} Sum_{j>=1} log(1 + x)^(i*j) / (i*j).
E.g.f.: log(Product_{k>=1} 1 / (1 - log(1 + x)^k)^(1/k)).
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * tau(k), where tau = A000005.

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

A318250 a(n) = (n - 1)! * sigma_2(n), where sigma_2(n) = sum of squares of divisors of n (A001157).

Original entry on oeis.org

1, 5, 20, 126, 624, 6000, 36000, 428400, 3669120, 47174400, 442713600, 8382528000, 81430272000, 1556755200000, 22666355712000, 445916959488000, 6067609067520000, 161837779783680000, 2317659281473536000, 66418224823222272000, 1216451004088320000000, 31165474724742758400000

Offset: 1

Ilya Gutkovskiy, Aug 22 2018

nonn

Cf. A000219, A001157, A038048, A318249.

Table[(n - 1)! DivisorSigma[2, n], {n, 1, 22}]
nmax = 22; Rest[CoefficientList[Series[Sum[x^k/(k (1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 22; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]

a(n) = (n-1)!*sigma(n,2); \\ Michel Marcus, Aug 22 2018

E.g.f.: Sum_{k>=1} x^k/(k*(1 - x^k)^2).
E.g.f.: -log(Product_{k>=1} (1 - x^k)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = o.g.f. of A000219.
a(p^k) = (p^(2*k+2) - 1)*(p^k - 1)!/(p^2 - 1), where p is a prime.

A338814 Expansion of e.g.f. log(Product_{k>0} (1 + x^k)^(1/k)).

Original entry on oeis.org

1, 0, 4, -6, 48, 0, 1440, -10080, 120960, 0, 7257600, -79833600, 958003200, 0, 348713164800, -3923023104000, 41845579776000, 0, 12804747411456000, -243290200817664000, 9731608032706560000, 0, 2248001455555215360000, -103408066955539906560000

Offset: 1

Seiichi Manyama, Nov 10 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 1..450

Column 1 of A338813.

Cf. A048272, A168243, A318249.

a[n_] := (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]; Array[a, 25] (* Amiram Eldar, Apr 28 2021 *)

N=40; x='x+O('x^N); Vec(serlaplace(log(prod(k=1, N, (1+x^k)^(1/k)))))

{a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)))}

a(n) = (n-1)! * A048272(n).

A352060 a(n) = (n - 1)! * omega(n), where omega(n) = number of distinct primes dividing n (A001221).

Original entry on oeis.org

0, 1, 2, 6, 24, 240, 720, 5040, 40320, 725760, 3628800, 79833600, 479001600, 12454041600, 174356582400, 1307674368000, 20922789888000, 711374856192000, 6402373705728000, 243290200817664000, 4865804016353280000, 102181884343418880000, 1124000727777607680000

Offset: 1

Seiichi Manyama, Mar 02 2022

nonn

Cf. A001221, A318249, A352012, A352058, A352059.

a[n_] := (n-1)! * PrimeNu[n]; Array[a, 25] (* Amiram Eldar, Mar 02 2022 *)

PARI
```
a(n) = (n-1)!*omega(n);
```

my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(-sum(k=1, N, isprime(k)*log(1-x^k)/k))))

E.g.f.: -Sum_{p prime} log(1-x^p)/p.

A354851 a(n) = (n-1)! * Sum_{d|n} d^(n/d).

Original entry on oeis.org

1, 3, 8, 54, 144, 2880, 5760, 206640, 1491840, 24675840, 43545600, 10298534400, 6706022400, 1195587993600, 33476463820800, 775450900224000, 376610217984000, 553805325545472000, 128047474114560000, 339876410542276608000, 6208765924866785280000

Offset: 1

Seiichi Manyama, Jun 08 2022

nonn

Cf. A055225, A318249, A354848.

a[n_] := (n - 1)! * DivisorSum[n, #^(n/#) &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)

PARI
```
a(n) = (n-1)!*sumdiv(n, d, d^(n/d));
```

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

a(n) = (n-1)! * A055225(n).
E.g.f.: -Sum_{k>0} log(1 - k * x^k)/k.
If p is prime, a(p) = (p-1)! + p!.

A353186 Expansion of e.g.f. 1/(1 - Sum_{k>=1} d(k) * x^k / k), where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 4, 22, 170, 1588, 18236, 240840, 3662424, 62456136, 1185150768, 24714979584, 562659843984, 13870798275072, 368324715871680, 10478253239415552, 317975367247809408, 10252138622419702656, 349999438215928660992, 12612365665457524786944, 478414908509124826439424

Offset: 0

Seiichi Manyama, Apr 29 2022

nonn

Cf. A000005, A028342, A129921, A305305, A318249, A340903.

d[k_] := d[k] = DivisorSigma[0, k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * d[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 21, 0] (* Amiram Eldar, Apr 30 2022 *)

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*numdiv(j)*binomial(i, j)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A318249(k) * binomial(n,k) * a(n-k).

cp's OEIS Frontend

A002746 Sum of logarithmic numbers.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A338805 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} (1-x^j)^(-u/j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A330352 Expansion of e.g.f. -Sum_{k>=1} log(1 - log(1 + x)^k) / k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A002744 Sum of logarithmic numbers.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A318250 a(n) = (n - 1)! * sigma_2(n), where sigma_2(n) = sum of squares of divisors of n (A001157).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A338814 Expansion of e.g.f. log(Product_{k>0} (1 + x^k)^(1/k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A352060 a(n) = (n - 1)! * omega(n), where omega(n) = number of distinct primes dividing n (A001221).

Views

Author

A338805 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1-x^j)^(-u/j).