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A299090 Number of "digits" in the binary representation of the multiset of prime factors of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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}.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

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.

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

Cf. A001511, A051903, A052409, A070939, A112798.

Sequence in context: A096309 A185102 A049419 * A046951 A159631 A050377

Adjacent sequences:  A299087 A299088 A299089 * A299091 A299092 A299093

KEYWORD

nonn,base

AUTHOR

Gus Wiseman, Feb 02 2018

EXTENSIONS

More terms from Antti Karttunen, Jul 29 2018

STATUS

approved

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Last modified October 14 01:36 EDT 2019. Contains 327994 sequences. (Running on oeis4.)