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1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 3, 3, 2, 2, 1, 4, 3, 2, 2, 2, 2, 2, 3, 2, 2, 4, 9, 4, 5, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 9, 2, 2, 2, 5, 6, 2, 2, 2, 3, 2, 2, 2, 2, 6, 2, 6, 1, 2, 5, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 8, 5, 2, 2, 6, 2, 2, 9, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 6, 2, 3, 7, 3, 3, 3, 4, 2, 2, 3, 2, 8
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OFFSET
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1,4
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COMMENTS
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All terms are > 0 because from any k > 0, one certainly cannot reach 1 in less than A051903(k) iterations of the map x -> A003415(x).
One of the records occur at a(20457) = 38. The corresponding term of A351255 is A351255(20457) = A276086(A328116(20457)) = A276086(688352) = 442600020398400142264711707660915237 = 3 * 7^6 * 11^10 * 13^11 * 17^5 * 19. When starting iterating from this value with A003415, it first goes relatively smoothly in 11 steps to the first squarefree number encountered, 6201461846617177861789236821121654153, but after that, it still meanders for the additional 37 iterations (visiting mostly squarefree numbers, but also six numbers with max. exponent = 2, and one number with max. exponent = 3), before finally reaching zero.
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LINKS
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FORMULA
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PROG
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(PARI)
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n, s, 0); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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