cp's OEIS Frontend

A299090 Number of "digits" in the binary representation of the multiset of prime factors of n.

%I A299090 #27 Apr 11 2025 10:32:15
%S A299090 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,
%T A299090 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,
%U A299090 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
%N A299090 Number of "digits" in the binary representation of the multiset of prime factors of n.
%C A299090 a(n) is also the binary weight of the largest multiplicity in the multiset of prime factors of n.
%C A299090 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,
%C A299090 bin({1,1,1,1,2,2,3,3,3}) = {1}{2,3}{3},
%C A299090 bin({1,1,1,1,1,2,2,2,2}) = {1,2}{}{1},
%C A299090 bin({1,1,1,1,1,2,2,2,3}) = {1}{2}{1,2,3}.
%C A299090 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
%H A299090 Antti Karttunen, <a href="/A299090/b299090.txt">Table of n, a(n) for n = 1..65537</a>
%H A299090 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A299090 a(n) = A070939(A051903(n)), n>1.
%F A299090 If m is a set then bin(m) has only one "digit" m; so a(n) = 1 if n is squarefree.
%F A299090 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.
%F A299090 a(n) = A061395(A225546(n)). - _Peter Munn_, Feb 10 2020
%F A299090 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
%e A299090 36 has prime factors {2,2,3,3} with binary representation {2,3}{} so a(36) = 2.
%e A299090 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}{}{}.
%t A299090 Table[If[n===1,0,IntegerLength[Max@@FactorInteger[n][[All,2]],2]],{n,100}]
%o A299090 (PARI)
%o A299090 A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
%o A299090 A299090(n) = if(1==n,0,#binary(A051903(n))); \\ _Antti Karttunen_, Jul 29 2018
%o A299090 (Python)
%o A299090 from sympy import factorint
%o A299090 def A299090(n): return max(factorint(n).values(),default=0).bit_length() # _Chai Wah Wu_, Apr 11 2025
%Y A299090 Cf. A001511, A051903, A052409, A070939, A112798, A329050.
%Y A299090 Related to A061395 via A225546.
%K A299090 nonn,base
%O A299090 1,4
%A A299090 _Gus Wiseman_, Feb 02 2018
%E A299090 More terms from _Antti Karttunen_, Jul 29 2018