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A208277
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Smallest number of multiplicative persistence n in factorial base.
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3
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,base,more
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AUTHOR
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STATUS
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approved
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