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A299090
Number of "digits" in the binary representation of the multiset of prime factors of n.
13
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
OFFSET
1,4
COMMENTS
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
FORMULA
a(n) = A070939(A051903(n)), n>1.
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.
a(n) = A061395(A225546(n)). - Peter Munn, Feb 10 2020
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
EXAMPLE
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}{}{}.
MATHEMATICA
Table[If[n===1, 0, IntegerLength[Max@@FactorInteger[n][[All, 2]], 2]], {n, 100}]
PROG
(PARI)
A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));
A299090(n) = if(1==n, 0, #binary(A051903(n))); \\ Antti Karttunen, Jul 29 2018
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Feb 02 2018
EXTENSIONS
More terms from Antti Karttunen, Jul 29 2018
STATUS
approved