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

0,6

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

Formula

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

Prog

(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

Crossrefs

Cf. A007623, A031346, A208575, A208277.

Sequence in context: A318930 A235748 A204697 * A307897 A008651 A049107

Adjacent sequences: A208573 A208574 A208575 * A208577 A208578 A208579

Keyword

nonn,base

Author

Charles R Greathouse IV, Feb 28 2012

Status

approved