A085289 -id:A085289 - cp's OEIS frontend

A085291 Decimal expansion of Alladi-Grinstead constant exp(c-1), where c is given in A085361.

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

Offset: 0

Eric W. Weisstein, Jun 25 2003

Named after the Indian-American mathematician Krishnaswami Alladi (b. 1955) and the American mathematician Charles Miller Grinstead (b. 1952). - Amiram Eldar, Jun 15 2021

			0.80939402054063913071793188059409131721595399242500030424202871504...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 120-122.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B22.

K. Alladi and C. Grinstead, On the decomposition of n! into prime powers, J. Number Theory, Vol. 9, No. 4 (1977), pp. 452-458.
Simon Plouffe, Alladi-Grinstead Constant.
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant.
Wikipedia, Multiplicative partitions of factorials.

Cf. A085288, A085289, A085290, A085361.

evalf(exp(sum((Zeta(n+1)-1)/n, n=1..infinity)-1), 120); # Vaclav Kotesovec, Dec 11 2015

$MaxExtraPrecision = 256; RealDigits[ Exp[ Sum[ N[(-1 + Zeta[1 + n])/n, 256], {n, 350}] - 1], 10, 111][[1]] (* Robert G. Wilson v, Nov 23 2005 *)

exp(suminf(n=1, (zeta(n+1)-1)/n) - 1) \\ Michel Marcus, May 19 2020

Equals exp(c-1), where c is Sum_{n>=1} (zeta(n+1) - 1)/n (cf. A085361).

Equals lim_{n->oo} (Product_{k=1..n} (k/n)*floor(n/k))^(1/n). - Benoit Cloitre, Jul 15 2022

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 24 2003

A085288 Number of sorted multiplicative partitions of n!.

Original entry on oeis.org

1, 1, 3, 3, 10, 10, 30, 75, 220, 220, 588, 588, 1568, 3696, 11616, 11616, 30492, 30492, 84700, 173250, 441000, 441000, 1262520, 2777544, 6957720, 16731660, 43506760, 43506760, 98658000, 98658000, 277101000, 541886400, 1322481600, 2715495552

Offset: 2

Eric W. Weisstein, Jun 23 2003

nonn

Number of decompositions of n! into factors of the form (p_k)^(e^k).

			4! = 3*8 = 2*3*4 = 2*2*2*3, so a(4)=3.
5! = 3*5*8 = 2*3*4*5 = 2*2*2*3*5, so a(5)=3.

Amiram Eldar, Table of n, a(n) for n = 2..6940
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Cf. A000142, A000688, A085289, A085290, A085291.

f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger[ n!]; Array[f, 34, 2] (* Robert G. Wilson v, Sep 22 2006 *)

a(n) = A000688(n!) = A000688(A000142(n)). - Vladeta Jovovic, Jun 27 2003

More terms from Vladeta Jovovic, Jun 27 2003

A085290 Max[p1^b1] over all sorted multiplicative partitions of n! of length n.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 16, 16, 16, 16, 16, 16, 16

Offset: 4

Eric W. Weisstein, Jun 23 2003

nonn

			6! = 2*2*2*2*5*9 = 2*2*3*3*4*5, the smallest terms of which are 2 and 2, so a(6)=Max[2,2]=2.

Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Cf. A085288, A085289, A085291.

Cf. A103332.

works(n, m) = local(f, s, l, p, x); f = factor(n!); s = 0; l = matsize(f)[1]; for (i = 1, l, p = f[i, 1]; x = 1; while (p^x < m, x++); s += f[i, 2]\x; if (f[i, 2] < x, return(0))); s >= n; a(n) = local(f, m); f = factor(n); m = 2; while (works(n, m), m++); m - 1 \\ David Wasserman, Jan 31 2005

More terms from David Wasserman, Jan 31 2005

A103332 Min[pn^bn] over all sorted multiplicative partitions of n! of length n.

Original entry on oeis.org

3, 5, 5, 7, 7, 8, 8, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 25, 25, 25, 25, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 49, 49, 49, 49, 53, 53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 61, 61, 64, 67, 67, 67, 67, 71, 71, 73, 73, 73

Offset: 4

David Wasserman, Jan 31 2005

a(n) >= A007917(n); a(n) is a power of some prime p <= n.

Cf. A085289, A085290.

works(n, m) = local(f, s, l, p, x); f = factor(n!); s = 0; l = matsize(f)[1]; if (f[l, 1] > m, return(0)); for (i = 1, l, p = f[i, 1]; x = 1; while (p^x <= m, x++); x--; s += f[i, 2]\x; if (f[i, 2]%x, s++)); s <= n; a(n) = local(f, m); f = factor(n); m = f[matsize(f)[1], 1]; while (!works(n, m), m++); m

cp's OEIS Frontend

A085291 Decimal expansion of Alladi-Grinstead constant exp(c-1), where c is given in A085361.

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A085288 Number of sorted multiplicative partitions of n!.

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A085290 Max[p1^b1] over all sorted multiplicative partitions of n! of length n.

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Author

Keywords

Examples

Links

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Programs

PARI

Extensions

A103332 Min[pn^bn] over all sorted multiplicative partitions of n! of length n.

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Author

Keywords

Comments

Crossrefs

Programs

PARI