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A280363
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a(n) = floor(log_p(n)) where p = A020639(n), i.e., the least prime factor of n.
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2
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0, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 4, 1, 4, 2, 4, 1, 4, 2, 4, 3, 4, 1, 4, 1, 5, 3, 5, 2, 5, 1, 5, 3, 5, 1, 5, 1, 5, 3, 5, 1, 5, 2, 5, 3, 5, 1, 5, 2, 5, 3, 5, 1, 5, 1, 5, 3, 6, 2, 6, 1, 6, 3, 6, 1, 6, 1, 6, 3, 6, 2, 6, 1, 6, 4, 6, 1, 6, 2, 6, 4, 6, 1, 6, 2, 6, 4, 6, 2, 6, 1, 6, 4, 6, 1, 6, 1, 6, 4, 6, 1, 6, 1, 6, 4, 6, 1, 6, 2, 6, 4, 6, 2, 6
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OFFSET
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1,4
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COMMENTS
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a(1) = 0 since 1 is the empty product.
a(p) = 1 since the exponent e of the largest power p^e of the prime divisor p is p^1 (i.e., p itself).
a(p^m) = m since the largest power p^e of the prime divisor p is p^m, (p^m itself), i.e., e = m.
a(n) is the greatest value of the power e of p^e across the prime divisors p of n such that p^e <= n.
Consider integers 1<=r<=n with all prime divisors p of r also dividing n. Let m be the smallest power n^m | r, and let e be the largest value of m across 1<=r<=n. This is A280274(n). This sequence underlies A280274: A280274(1) = 0, A280274(n) = 1 with n having omega(n) = 1. A280274(n) = a(n) for squarefree n. A280274(n) for all other n is ceiling(a(n)/k), with k being the multiplicity of p = A020639(n) in the prime decomposition of n.
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LINKS
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EXAMPLE
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a(10) = 3, because 2^3 = 8 and 5^1 = 5 are less than 10 = 2*5, and of the multiplicities of these numbers, 3 is the greatest.
a(12) = 3, because 2^3 = 8 and 3^2 = 9 are less than 12 = 2*2*3, and of the multiplicities of these numbers, 3 is the greatest.
a(16) = 4, because 2^4 = 16 = n, and is the largest power of the distinct prime divisor 2 of 16.
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MATHEMATICA
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Table[If[n == 1, 0, Floor[Log[FactorInteger[n][[1, 1]], n]]], {n, 120}]
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PROG
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(PARI) a(n) = if (n==1, 0, logint(n, vecmin(factor(n)[, 1]))); \\ Michel Marcus, Jan 01 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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