OFFSET
0,2
COMMENTS
a(n) exists for all n, unlike (conjecturally) its decimal equivalent A003001. In particular, with k = a(n-1), a(n) <= k * k! + (k-1)! + ... + 2! + 1! < (a(n-1)+1)! for n > 1. Diamond & Reidpath ask if this upper bound can be improved.
a(5) <= 255429978433810461138446192454297813.
LINKS
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.
EXAMPLE
5 = 1*1!+2*2!, and so is 21 in factorial base; the product of its digits is 2*1 = 10_! and the product of its digits in factorial base is 0*1 = 0, so 5 has multiplicative persistence 2. Since it is the smallest, a(2) = 5.
633 = 51111_! -> 21_! -> 10_! -> 0_! is the least chain of length 3 and so a(3) = 633.
PROG
(PARI) pr(n)=my(k=1, s=1); while(n, s*=n%k++; n\=k); s
persist(n)=my(t); while(n>1, t++; n=pr(n)); t
a(n)=my(k=0); while(persist(k)!=n, k++); k \\ Charles R Greathouse IV, Jan 21 2013
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Charles R Greathouse IV, Feb 28 2012
STATUS
approved