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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
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.
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
KEYWORD
nonn,base
AUTHOR
STATUS
approved