A131385 -id:A131385 - cp's OEIS frontend

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 362880, 362880, 725760, 2177280, 8709120, 43545600, 261273600, 1828915200, 14631321600, 131681894400, 263363788800, 526727577600, 2106910310400, 12641461862400, 101131694899200

Offset: 1

Hieronymus Fischer, Jul 11 2007

			a(12)=dp_10(1)*dp_10(2)*dp_10(3)*...*dp_10(11)*dp_10(12)=1*2*3*4*5*6* 7*8*9*1*(1*1)*(1*2).
a(345)=3*4*5*3^45*4^5*(3-1)!^100*(4-1)!^10*(5-1)!^1*9!^64.
a(1000)=9!^300. a(1111)=9!^321.

Hieronymus Fischer, Table of n, a(n) for n = 1..102

Cf. A131385, A010786, A131387, A007953, A007954, A037131.

with transforms;
f:=proc(n) option remember; if n = 0 then 1 else f(n-1)*digprod0(n); fi; end;[seq(f(n),n=0..40)]; # N. J. A. Sloane, Oct 12 2013

The following formulas are given for general bases p>1:

a(n)=product{1<=k<=n, dp_p(k)} where dp_p(k) = product of the nonzero digits of k in base p.

a(n)=(n mod p)!*product{00}(floor(n/p^j)mod p)^(1+(n mod p^j))*((floor(n/p^j)mod p)-1)!^(p^j).

Recurrence: a(n+k*p^m)=a(n)*k^n*a(k*p^m) for 0<=k

a(n)=n!, for 0<=n

a(k*p^m)=k*(p-1)!^(k*m*p^(m-1))*(k-1)!^(p^m) for 0<=k

a(n)=(p-1)!^((m*p^(m+1)-(m+1)*p^m+1)/(p-1)^2)=(p-1)!^(1+2*p+3*p^2+...+m*p^(m-1)) for n=1+p+p^2+...+p^m.

a(n)=(p-1)!^(k*(m*p^(m+1)-(m+1)*p^m+1)/(p-1)^2)*(k-1)!^(p*(p^m-1)/(p-1))*k^(k*(p^(m+1)-(m+1)*p+m)/(p-1)^2)*k!*k^m, for n=k*(1+p+p^2+...+p^m).

For p=10: a(10^n)=9!^(n*10^(n-1)).

Asymptotic behavior: a(10^n)=10^(0.5559763...*n*10^n). Hence it grows slower than the factorial A000142(10^n) for which we have (10^n)!=10^((n-0.43429448...)*10^n+n/2+0.3990899...+o(1/n)). Example: a(1000) has 1668 digits, whereas 1000! has 2568 digits.

New b-file from Hieronymus Fischer, Sep 10 2007

2 typos in the formula section removed by Hieronymus Fischer, Dec 05 2011

A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 14 2007

Given A131385(n) = Product_{k=1..n} floor((n+k)/k), then limit A131385(n+1)/A131385(n) = exp(c), where c = this constant. - Paul D. Hanna, Nov 26 2012

Closely related to A085361 (the exponent in the definition of A085291). - Yuriy Sibirmovsky, Sep 04 2016

			1.257746886944369630009899830495881528511540890508884868977540833522...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 62. [Jean-François Alcover, Mar 21 2013]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.
Mark W. Coffey, Series of zeta values, the Stieltjes constants and a sum S_gamma(n), arXiv:math-ph/0706.0345, 2007-2009, eq (38a).
Paul Erdős, S. W. Graham, Aleksandar Ivic and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Volume 138, Progress in Mathematics pp 337-355.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Cf. A000005, A002210, A027423, A075887, A131385, A244109, A001620, A085361, A345682.

SetDefaultRealField(RealField(100)); L:=RiemannZeta(); (&+[(-1)^(n+1)*Evaluate(L,n+1)/n: n in [1..10^3]]); // G. C. Greubel, Nov 15 2018

evalf(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Sum[ -Zeta'[1 + k], {k, 1, Infinity}] (* Vladimir Reshetnikov, Dec 28 2008 *)
Integrate[EulerGamma/x + PolyGamma[0, 1+x]/x, {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* or *) Integrate[x*Log[x]/((1-x)*Log[1-x]), {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* Jean-François Alcover, Feb 04 2013 *)
$MaxExtraPrecision = 200; NIntegrate[HarmonicNumber[t]/t, {t, 0, 1}, WorkingPrecision -> 105] (* Yuriy Sibirmovsky, Sep 04 2016 *)
digits = 120; RealDigits[NSum[(-1)^(n + 1)*Zeta[n + 1]/n, {n,1,Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

PARI
```
sumalt(s=1, (-1)^(s+1)/s*zeta(s+1) )
```

suminf(k=2, -zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

numerical_approx(sum((-1)^(k+1)*zeta(k+1)/k for k in [1..1000]), digits=100) # G. C. Greubel, Nov 15 2018

Equals Sum_{s>=1} (-1)^(s+1)*zeta(s+1)/s.
Equals Sum_{k>=1} -zeta'(1 + k), where Zeta' is the derivative of the Riemann zeta function. - Vladimir Reshetnikov, Dec 28 2008
Equals Sum_{s>=1} log(1+1/s)/s. - Jean-François Alcover, Mar 26 2013
Equals Integral_{t=0..1} H(t)/t dt. Compare to A001620 = Integral_{t=0..1} H(t) dt. Where H(t) are generalized harmonic numbers. - Yuriy Sibirmovsky, Sep 04 2016
Equals lim_{n->oo} log(d(n!))*log(n)/n, where d(n) is the number of divisors of n (A000005) (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

Extended to 105 digits by Jean-François Alcover, Feb 04 2013

A131387 Product of the nonzero digital products of n for all the bases 1 to n (a 'total digital-product factorial').

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 12, 48, 48, 192, 2880, 2880, 25920, 552960, 3265920, 1935360, 116121600, 278691840, 9405849600, 26754416640, 94058496000, 3210529996800, 869100503040000, 423789959577600, 927040536576000, 135612787064832000

Offset: 1

Hieronymus Fischer, Jul 08 2007

a(n) = {p = 1; for (b=2, n, digs = digits(n, b); p *= prod(k=1, #digs, if (digs[k], digs[k], 1));); return (p);} \\ Michel Marcus, Jul 15 2013

a(n)=product{1<=p<=n, dp_p(n)} where dp_p(n) = product of the nonzero digits of n in base p.

Cf. A131385, A010786, A007954.

A075887 a(n) = 1 + n + n[n/2] + n[n/2][n/3] +... + n[n/2][n/3]...[n/n], where [x]=ceiling(x).

Original entry on oeis.org

1, 2, 5, 16, 45, 171, 421, 1968, 4553, 19225, 57261, 226854, 496309, 3136420, 6764563, 24850336, 84877201, 380461599, 805949533, 4411165990, 9288196621, 48275465722, 154143694937, 527401107276, 1100708161081, 8151403215501

Offset: 0

Paul D. Hanna, Oct 17 2002

a(n) ~ L^n where L = 3.517487255902369649399793699323864170685620..., with log(L) = Sum_{k=1..inf} log(k+1)/(k*(k+1)) = 1.2577468869443696300... (cf. A131688).

			a(5) = 171 = 1 +5[5/2] +5[5/2][5/3] +5[5/2][5/3][5/4] +5[5/2][5/3][5/4][5/5] = 1 + 5 + 5*3 + 5*3*2 + 5*3*2*2 + 5*3*2*2*1, here [x]=ceiling(x).

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A131385, A131688.

[1] cat [1 + (&+[(&*[Ceiling(n/k): k in [1..j]]): j in [1..n]]): n in [1..50]]; // G. C. Greubel, Oct 11 2018

Table[1 +Sum[Product[Ceiling[n/k], {k,1,j}], {j,1,n}], {n,0,50}] (* G. C. Greubel, Oct 11 2018 *)

{a(n) = 1 + sum(m=1,n,prod(k=1,m,ceil(n/k)))}
for(n=0,40,print1(a(n),", "))

a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} ceiling(n/k) for n>0 and a(0)=1.

A308820 a(n) = Product_{k=1..n} ceiling(n/k)!.

Original entry on oeis.org

1, 2, 12, 96, 2880, 34560, 5806080, 92897280, 25082265600, 2006581248000, 794606174208000, 19070548180992000, 208250386136432640000, 5831010811820113920000, 4198327784510482022400000, 3224315738504050193203200000, 14799609239733590386802688000000

Offset: 1

Ilya Gutkovskiy, Jun 26 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 1..178

Cf. A010786, A020696, A092143, A131385.

[(&*[Factorial(Ceiling(n/(n-j+1))): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Mar 08 2023

seq(mul(ceil(n/k)!, k=1..n), n=1..30); # Ridouane Oudra, Apr 10 2023

a[n_] := Product[Ceiling[n/k]!, {k, 1, n}]; Table[a[n], {n, 1, 17}]

a(n) = prod(k=1, n, ceil(n/k)!); \\ Michel Marcus, Jun 27 2019

def A308820(n): return product( factorial(ceil(n/(n-k+1))) for k in range(1,n+1))
[A308820(n) for n in range(1,21)] # G. C. Greubel, Mar 08 2023

a(n) = Product_{k=1..n-1} Product_{d|k} (d + 1).
a(n) = Product_{k=1..n-1} (k + 1)^floor((n-1)/k). - Ridouane Oudra, Apr 10 2023
a(n) = A131385(n)*A092143(n-1). - Ridouane Oudra, Sep 20 2024

A332233 Number of integer partitions lambda (of any k) satisfying n = max_{p:lambda} p*m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

Original entry on oeis.org

1, 1, 4, 10, 44, 84, 528, 864, 4944, 12720, 56832, 89856, 882432, 1209600, 6036480, 20017152, 98592768, 141834240, 1202135040, 1625702400, 12997877760, 35291013120, 124429271040, 191102976000, 2350327726080, 4064999178240, 15972386734080, 47163577466880

Offset: 0

Alois P. Heinz, Feb 07 2020

nonn

a(0) = 1 by convention.

			a(2) = 4: 2, 11, 21, 211.
a(3) = 10: 3, 31, 32, 111, 311, 321, 2111, 3111, 3211, 32111.
a(4) = 44: 4, 22, 41, 42, 43, 221, 322, 411, 421, 422, 431, 432, 1111, 2211, 3221, 4111, 4211, 4221, 4311, 4321, 4322, 21111, 22111, 31111, 32211, 41111, 42111, 42211, 43111, 43211, 43221, 221111, 321111, 322111, 421111, 422111, 431111, 432111, 432211, 3221111, 4221111, 4321111, 4322111, 43221111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Integer Partition

Column sums of A134979.
Cf. A024916.
Cf. A131385 (partial sum).

b:= proc(n, i, m, t) option remember; `if`(n=0, m,
      `if`(i<1 or m=0 and n

$RecursionLimit = 2000;
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n==0, m, If[i<1 || m==0 && nJean-François Alcover, May 08 2020, after Maple *)

a(n) = Sum_{k=n..A024916(n)} A134979(k,n) for n > 0.

a(n) = A131385(n+1) - A131385(n), for n>0. - Ridouane Oudra, Oct 30 2023

A127482 Product of the nonzero digital products of all the prime numbers prime(1) to prime(n).

Original entry on oeis.org

2, 6, 30, 210, 210, 630, 4410, 39690, 238140, 4286520, 12859560, 270050760, 1080203040, 12962436480, 362948221440, 5444223321600, 244990049472000, 1469940296832000, 61737492466944000, 432162447268608000, 9075411392640768000, 571750917736368384000

Offset: 1

Alain Van Kerckhoven (alain(AT)avk.org), Sep 12 2007

			a(7) = dp_10(2)*dp_10(3)*dp_10(5)*dp_10(7)*dp_10(11)*dp_10(13)*dp_10(17) = 2*3*5*7*(1*1)*(1*3)*(1*7) = 4410.

Harvey P. Dale, Table of n, a(n) for n = 1..547

Cf. A000040, A007953, A007954, A051801, A101987, A131385, A131387, A131451.

a:= proc(n) option remember; `if`(n<1, 1, a(n-1)*mul(
     `if`(i=0, 1, i), i=convert(ithprime(n), base, 10)))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 11 2022

Rest[FoldList[Times,1,Times@@Cases[IntegerDigits[#],Except[0]]&/@ Prime[ Range[ 20]]]] (* Harvey P. Dale, Mar 19 2013 *)

f(n) = vecprod(select(x->(x>1), digits(prime(n)))); \\ A101987
a(n) = prod(k=1, n, f(k)); \\ Michel Marcus, Mar 11 2022

from math import prod
from sympy import sieve
def pod(s): return prod(int(d) for d in s if d != '0')
def a(n): return pod("".join(str(sieve[i+1]) for i in range(n)))
print([a(n) for n in range(1, 23)]) # Michael S. Branicky, Mar 11 2022

a(n) = Product_{k=1..n} dp_p(prime(k)) where prime(k)=A000040(k) and dp_p(m)=product of the nonzero digits of m in base p (p=10 for this sequence). - Hieronymus Fischer, Sep 29 2007
From Michel Marcus, Mar 11 2022: (Start)
a(n) = Product_{k=1..n} A051801(prime(k)).
a(n) = Product_{k=1..n} A101987(k). (End)

Corrected and extended by Hieronymus Fischer, Sep 29 2007

A145119 a(n) = Product_{k=1..n-1} (ceiling(n/k) - ceiling(n/k) mod 2).

Original entry on oeis.org

1, 2, 4, 16, 32, 96, 384, 1024, 2048, 10240, 30720, 73728, 294912, 688128, 1835008, 12582912, 25165824, 56623104, 283115520, 629145600, 1887436800, 11072962560, 26575110144, 57982058496, 231928233984, 753766760448, 1758789107712

Offset: 1

Reikku Kulon, Oct 02 2008

Bounded by A010786 and A131385.

Each term has many more factors of two than any other prime factor.

Cf. A010786, A131385, A131387.

a(n) = prod(k=1, n-1, ceil(n/k) - ceil(n/k) % 2); \\ Michel Marcus, Nov 17 2019

Showing 1-9 of 9 results.

cp's OEIS Frontend

A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).

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A131451 Product of the nonzero digital products of all the numbers 1 to n (a 'total digital-product factorial' in base 10).

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A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

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A131387 Product of the nonzero digital products of n for all the bases 1 to n (a 'total digital-product factorial').

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A075887 a(n) = 1 + n + n[n/2] + n[n/2][n/3] +... + n[n/2][n/3]...[n/n], where [x]=ceiling(x).

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A308820 a(n) = Product_{k=1..n} ceiling(n/k)!.

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A332233 Number of integer partitions lambda (of any k) satisfying n = max_{p:lambda} p*m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.