A061836 a(n) = smallest k>0 such that k+n divides k!.

1, 5, 4, 3, 4, 5, 6, 5, 4, 6, 5, 7, 6, 7, 6, 5, 8, 7, 6, 5, 4, 7, 8, 7, 6, 5, 9, 8, 7, 7, 6, 9, 8, 7, 6, 5, 9, 8, 7, 6, 8, 7, 6, 11, 10, 9, 10, 9, 8, 7, 10, 9, 8, 7, 6, 5, 7, 13, 12, 11, 10, 9, 8, 7, 8, 7, 6, 13, 12, 11, 10, 9, 8, 7, 6, 9

Offset: 0

Author

Robert G. Wilson v, Jun 22 2001

Keywords

• nonn

Comments

Comments from M. F. Hasler, Feb 20 2020 (Start)

The index at which any n > 2 appears for the last time is given by A005096(n) = n! - n.

For m>2, a(n) > m for n > A005096(m).

The integer 1 appears only once as a(0), the integer 2 is the only positive integer which never appears. (End)

It would be nice to have an estimate for the growth of the upper envelope of this sequence - what is lim sup a(n)? The answer seems to be controlled by A333537. - N. J. A. Sloane, Apr 12 2020

Paul Zimmermann suggests that perhaps a(n) is O(log(n)^2). My estimate was n^(1/3), although that seems a bit low. - N. J. A. Sloane, Apr 09 2020

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
• N. J. A. Sloane, Table of n, a(n) for n = 0..100000

Crossrefs

Cf. A332584 for a "concatenation in base 10" variant.

See also A005096, A332558 (essentially identical to this one).

For records, see A333532 and A333533 (and A333537).

Programs

Mathematica

f[n_] := (k = 1; While[ !IntegerQ[ k! / (k + n) ], k++ ]; k); Table[ f[n], {n, 0, 75} ]

PARI

a(n) = my (f=1); for (k=1, oo, if ((f*=k)%(n+k)==0, return (k))) \\ Rémy Sigrist, Feb 17 2020

Extensions

"k>0" added to definition at the suggestion of Chai Wah Wu, Apr 09 2020. - N. J. A. Sloane, Apr 22 2020