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A278222 Filter-sequence for unary-binary encoding of prime factorization: least number with the same prime signature as A005940(n+1). 54
1, 2, 2, 4, 2, 6, 4, 8, 2, 6, 6, 12, 4, 12, 8, 16, 2, 6, 6, 12, 6, 30, 12, 24, 4, 12, 12, 36, 8, 24, 16, 32, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 4, 12, 12, 36, 12, 60, 36, 72, 8, 24, 24, 72, 16, 48, 32, 64, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 6, 30, 30, 60, 30, 210, 60, 120, 12, 60, 60, 180, 24, 120, 48, 96, 4, 12, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence can be used for filtering certain base-2 related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A005940(n+1)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Because the Doudna map n -> A005940(1+n) is an isomorphism from "unary-binary encoding of factorization" (see A156552) to the ordinary prime factorization domain, it follows that the equivalence classes of this sequence match with any such sequence b, where b(n) is computed from the lengths of 1-runs in the binary representation of n and the order of those 1-runs does not matter. Particularly, this holds for any sequence that is obtained as a "Run Length Transform", i.e., where b(n) = Product S(i), for some function S, where i runs through the lengths of runs of 1's in the binary expansion of n (see the index entry for some of the known ones).

However, this sequence itself is not a run length transform of any sequence (which can be seen for example from the fact that A046523 is not multiplicative).

Furthermore, this matches not only with sequences involving products of S(i), but with any sequence obtained with any commutative function applied cumulatively, like e.g., A000120 (binary weight, obtained in this case as Sum identity(i)), and A069010 (number of runs of 1's in binary representation of n, obtained as Sum signum(i)).

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to binary expansion of n

Index entries for sequences computed from exponents in factorization of n

Index entries for sequences computed with run length transform

FORMULA

a(n) = A046523(A005940(1+n)).

a(n) = A124859(A278159(n)).

a(n) = A278219(A006068(n)).

MATHEMATICA

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Array[If[# == 1, 1, Times @@ MapIndexed[ Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]]] &@ f[# - 1, 1, 1] &, 99] (* Michael De Vlieger, May 09 2017 *)

PROG

(Scheme) (define (A278222 n) (A046523 (A005940 (+ 1 n))))

(Python)

from sympy import prime, factorint

import math

def A(n): return n - 2**int(math.floor(math.log(n, 2)))

def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))

def a005940(n): return b(n - 1)

def P(n):

    f = factorint(n)

    return sorted([f[i] for i in f])

def a046523(n):

    x=1

    while True:

        if P(n) == P(x): return x

        else: x+=1

def a(n): return a046523(a005940(n + 1)) # Indranil Ghosh, May 05 2017

CROSSREFS

Cf. A005940, A156552, A006068, A124859, A278159.

Similar filter sequences: A278217, A278219 (other base-2 related variants), A069877 (base-10 related), A278226 (primorial base), A278234-A278236 (factorial base), A278243 (Stern polynomials), A278233 (factorization in ring GF(2)[X]), A046523 (factorization in Z).

Cf. also A286622 (rgs-version of this filter) and A286162, A286252, A286163, A286240, A286242, A286379, A286464, A286374, A286375, A286376, A286243, A286553 (various compound and other filters involving this sequence).

Sequences that partition N into same or coarser equivalence classes: too many to list all here (over a hundred). At least every sequence listed under index-entry "Run Length Transforms" is included, and also sequences like A000120 and A069010, plus their combinations like A136277.

Sequence in context: A072300 A210359 A286553 * A286588 A277705 A028913

Adjacent sequences:  A278219 A278220 A278221 * A278223 A278224 A278225

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 15 2016

STATUS

approved

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Last modified December 12 20:04 EST 2017. Contains 295954 sequences.