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A208576 Multiplicative persistence of n in factorial base. 4
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 (list; graph; refs; listen; history; text; internal format)
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 * A008651 A307897 A049107

Adjacent sequences:  A208573 A208574 A208575 * A208577 A208578 A208579

KEYWORD

nonn,base

AUTHOR

Charles R Greathouse IV, Feb 28 2012

STATUS

approved

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Last modified December 9 03:42 EST 2021. Contains 349625 sequences. (Running on oeis4.)