A286820 a(n) = smallest positive multiple of n whose factorial base representation contains only 0's and 1's.

1, 2, 3, 8, 25, 6, 7, 8, 9, 30, 33, 24, 26, 126, 30, 32, 153, 126, 152, 120, 126, 726, 5888, 24, 25, 26, 27, 728, 145, 30, 31, 32, 33, 5066, 840, 144, 5883, 152, 5070, 120, 123, 126, 129, 5192, 720, 5888, 752, 144, 147, 150, 153, 728, 848, 864, 46200, 728

Offset: 1

Author

Rémy Sigrist, Jun 24 2017

Keywords

• nonn
• base

Comments

All terms belong to A059590.

a(n) = n iff n belongs to A059590.

The sequence is well defined: for any n > 0: according to the pigeonhole principle, among the n+1 first repunits in factorial base (A007489), there must be two distinct terms equal modulo n; their absolute difference is a positive multiple of n, and contains only 0's and 1's in factorial base.

This sequence is to factorial base what A004290 is to decimal base.

Examples

			The first terms are:
n   a(n)     a(n) in factorial base
--  ----     ----------------------
1   1        1
2   2        1,0
3   3        1,1
4   8        1,1,0
5   25       1,0,0,1
6   6        1,0,0
7   7        1,0,1
8   8        1,1,0
9   9        1,1,1
10  30       1,1,0,0
11  33       1,1,1,1
12  24       1,0,0,0
13  26       1,0,1,0
14  126      1,0,1,0,0
15  30       1,1,0,0
16  32       1,1,1,0
17  153      1,1,1,1,1
18  126      1,0,1,0,0
19  152      1,1,1,1,0
20  120      1,0,0,0,0
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..2000
• Wikipedia, Factorial number system
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A004290, A007489, A059590, A284750.

Programs

PARI

isA059590(n) = my (r=2); while (n, if (n%r > 1, return (0), n\=r; r++)); return (1)
a(n) = forstep (m=n, oo, n, if (isA059590(m), return (m)))