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A299090
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Number of "digits" in the binary representation of the multiset of prime factors of n.
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13
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0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 2, 1
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OFFSET
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1,4
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COMMENTS
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a(n) is also the binary weight of the largest multiplicity in the multiset of prime factors of n.
Any finite multiset m has a unique binary representation as a finite word bin(m) = s_k..s_1 such that: (1) each "digit" s_i is a finite set, (2) the leading term s_k is nonempty, and (3) m = 1*s_1 + 2*s_2 + 4*s_3 + 8*s_4 + ... + 2^(k-1)*s_k where + is multiset union, 1*S = S as a multiset, and n*S = 1*S + (n-1)*S for n > 1. The word bin(m) can be thought of as a finite 2-adic set. For example,
bin({1,1,1,1,2,2,3,3,3}) = {1}{2,3}{3},
bin({1,1,1,1,1,2,2,2,2}) = {1,2}{}{1},
bin({1,1,1,1,1,2,2,2,3}) = {1}{2}{1,2,3}.
a(n) is the least k such that columns indexed k or greater in A329050 contain no divisors of n. - Peter Munn, Feb 10 2020
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LINKS
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FORMULA
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If m is a set then bin(m) has only one "digit" m; so a(n) = 1 if n is squarefree.
If m is of the form n*{x} then bin(m) is obtained by listing the binary digits of n and replacing 0 -> {}, 1 -> {x}; so a(p^n) = binary weight of n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (1 - 1/zeta(2^k)) = 1.47221057635756400916... . - Amiram Eldar, Jan 05 2024
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EXAMPLE
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36 has prime factors {2,2,3,3} with binary representation {2,3}{} so a(36) = 2.
Binary representations of the prime multisets of each positive integer begin: {}, {2}, {3}, {2}{}, {5}, {2,3}, {7}, {2}{2}, {3}{}, {2,5}, {11}, {2}{3}, {13}, {2,7}, {3,5}, {2}{}{}.
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MATHEMATICA
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Table[If[n===1, 0, IntegerLength[Max@@FactorInteger[n][[All, 2]], 2]], {n, 100}]
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PROG
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(PARI)
A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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