A208576 Multiplicative persistence of n in factorial base.

0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2

Offset: 0

Author

Charles R Greathouse IV, Feb 28 2012

Keywords

• nonn
• base

Comments

Diamond and Reidpath prove that a(2n) = 1 for n > 0, a(n) = 2 if n is contains an even digit but no 0's in its factorial base representation. If a(n) > 2 then 3 | n.

Further modular properties can be easily proved. For example, a(n) > 2 implies that n is 33, 45, 81, or 93 mod 120.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• M. R. Diamond and D. D. Reidpath, A counterexample to conjectures by Sloane and Erdos concerning the persistence of numbers, Journal of Recreational Mathematics 29:2 (1998), pp. 89-92.
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A007623, A031346, A208575, A208277.

Programs

PARI

pr(n)=my(k=1,s=1);while(n,s*=n%k++;n\=k);s
a(n)=my(t);while(n>1,t++;n=pr(n));t

Formula

a(0) = a(1) = 0; for n > 1, a(n) = 1 + a(A208575(n)). - Antti Karttunen, Nov 14 2018