A075473 -id:A075473 - cp's OEIS frontend

A076483 a(n) = n!*Sum_{k=1..n} (k-1)^k/k!.

0, 0, 1, 11, 125, 1649, 25519, 458569, 9433353, 219117905, 5677963451, 162457597961, 5087919552253, 173136159558361, 6361282619516343, 250987334850557369, 10584205713321808529, 475079402305823570849, 22614513693572549266291, 1137911105533216112417161

Offset: 0

Henry Bottomley, Oct 14 2002

nonn

Perhaps the largest possible number of ways of choosing (v1, v2, ..., vn), possibly with repetition, from {b1, b2, ..., bn} with b1 < b2 < ... < bn, such that v1 + v2 + ... + vn < b1 + b2 + ... + bn. Clearly the actual number of ways depends on the particular values of {b1, b2, ..., bn}, but {1, n, n^2, ..., n^(n-1)} produces this result for the number of sums strictly less than (n^n-1)/(n-1) = A023037(n).

			a(4) = 4!*(0^1/1! + 1^2/2! + 2^3/3! + 3^4/4!) = 0 + 12 + 32 + 81 = 125.

Seiichi Manyama, Table of n, a(n) for n = 0..386

Row sums of A076482.

Cf. A023037, A075473.

Table[n! Sum[(k-1)^k/k!, {k,n}], {n,0,17}] (* Stefano Spezia, Sep 11 2022 *)

a(n) = n!*sum(k=1, n, (k-1)^k/k!); \\ Seiichi Manyama, Jul 15 2023

Limit_{n->oo} a(n)/(e*a(n-1)) - n = -1/2.

Limit_{n->oo} a(n)/n^n = 1/(e-1).

A075478 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+27. Corresponds to selection of every 16th term from A074474.

Original entry on oeis.org

97, 74, 66, 14, 40, 17, 25, 14, 22, 27, 40, 14, 45, 27, 17, 14, 40, 38, 27, 14, 56, 17, 20, 14, 22, 27, 30, 14, 100, 30, 17, 14, 22, 33, 20, 14, 22, 17, 30, 14, 20, 30, 53, 14, 38, 20, 17, 14, 51, 25, 66, 14, 35, 17, 22, 14, 25, 20, 64, 14, 38, 40, 17, 14, 45, 25, 22, 14, 27

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3,11,19,27,35,43,51,55} provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g. {64k+19,192k+58,96k+29,288k+88,144k+44,72k+22,36k+11} submerge first below initial value at the 7th term,36k+11<64k+19.

			n=0: 64n+27=27, list={27, 82, 41, 46.23.70, ..}, i.e. the 97th term is the first that <27, the initial value.

Cf. A074473, A074474, A075476-A075482, A006370.

a(n)=A075473[64n+27], n=0, ..., 256

A075479 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+31. Corresponds to selection of every 16th term from A074474.

Original entry on oeis.org

92, 14, 35, 51, 17, 14, 25, 27, 22, 14, 64, 17, 22, 14, 61, 43, 131, 14, 27, 22, 17, 14, 33, 35, 22, 14, 53, 17, 20, 14, 43, 22, 22, 14, 45, 22, 17, 14, 35, 43, 20, 14, 25, 17, 25, 14, 20, 22, 27, 14, 38, 20, 17, 14, 27, 22, 30, 14, 25, 17, 33, 14, 40, 20, 69, 14, 115, 27, 17

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3,11,19,27,35,43,51,55} provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g. {64k+19,192k+58,96k+29,288k+88,144k+44,72k+22,36k+11} submerge first below initial value at the 7th term,36k+11<64k+19.

			n=1: 64n+31=95,list={95,286,143,430,215,646,323,970, 485,1456,728,364,182,91,274,...}, i.e. the 14th term=91 is the first that <95, the initial value, so a(1)=14.

Cf. A074473, A074474, A075476-A075482, A006370.

a(n)=A075473[64n+31], n=0, ..., 256

A075481 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+47. Corresponds to selection of every 16th term from A074474.

Original entry on oeis.org

89, 51, 14, 33, 22, 17, 14, 45, 27, 22, 14, 35, 17, 20, 14, 35, 22, 22, 14, 43, 22, 17, 14, 27, 128, 20, 14, 25, 17, 25, 14, 20, 22, 30, 14, 82, 20, 17, 14, 45, 22, 27, 14, 25, 17, 27, 14, 48, 20, 30, 14, 43, 30, 17, 14, 58, 61, 27, 14, 53, 17, 56, 14, 22, 30, 58, 14, 27, 53

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3,11,19,27,35,43,51,55} provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g. {64k+19,192k+58,96k+29,288k+88,144k+44,72k+22,36k+11} submerge first below initial value at the 7th term,36k+11<64k+19.

			n=2: 64n+47=175,list={175,526,263,790,395,1186,593,1780, 890,445,1336,668,334,167,502,251....}, i.e. the 14th term=167 is the first that <175, the initial value, so a(2)=14.

Cf. A074473, A074474, A075476-A075483, A006370.

a(n)=A075473[64n+47], n=0, ..., 256

cp's OEIS Frontend

A076483 a(n) = n!*Sum_{k=1..n} (k-1)^k/k!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A075478 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+27. Corresponds to selection of every 16th term from A074474.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A075479 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+31. Corresponds to selection of every 16th term from A074474.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A075481 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+47. Corresponds to selection of every 16th term from A074474.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula