A278223 - OEIS

A278223

Least number with the same prime signature as the n-th odd number: a(n) = A046523(2n-1).

1, 2, 2, 2, 4, 2, 2, 6, 2, 2, 6, 2, 4, 8, 2, 2, 6, 6, 2, 6, 2, 2, 12, 2, 4, 6, 2, 6, 6, 2, 2, 12, 6, 2, 6, 2, 2, 12, 6, 2, 16, 2, 6, 6, 2, 6, 6, 6, 2, 12, 2, 2, 30, 2, 2, 6, 2, 6, 12, 6, 4, 6, 8, 2, 6, 2, 6, 24, 2, 2, 6, 6, 6, 12, 2, 2, 12, 6, 2, 6, 6, 2, 30, 2, 4, 12, 2, 12, 6, 2, 2, 6, 6, 6, 24, 2, 2, 30, 2, 2, 6, 6, 6, 12, 6, 2, 6, 6, 6, 6, 6, 2, 36, 2, 2

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OFFSET

1,2

COMMENTS

This sequence works as a filter for sequences related to the prime factorization of odd numbers by matching to any sequence that is obtained as f(2*n - 1), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). The last line in Crossrefs section lists such sequences that were present in the database as of Nov 11 2016, although some of the matches might be spurious.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..32769

Index entries for sequences computed from exponents in factorization of n.

FORMULA

a(n) = A046523(2n - 1).

a(n) = A046523(A064216(n)).

From Antti Karttunen, May 31 2017: (Start)

a(n) = A278222(A244153(n)).

a(n) = A278531(A245611(n)).

(End)

MATHEMATICA

a[n_] := Times @@ (Prime[Range[Length[f = FactorInteger[2*n - 1]]]]^Sort[f[[;; , 2]], Greater]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

PROG

(Scheme)

(define (A278223 n) (A046523 (+ n n -1)))

(define (A278223 n) (A046523 (A064216 n)))

(Python)

from sympy import factorint

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(2*n - 1) # Indranil Ghosh, May 11 2017

(Python)

from math import prod

from sympy import prime, factorint

def A278223(n): return prod(prime(i+1)**e for i, e in enumerate(sorted(factorint((n<<1)-1).values(), reverse=True))) # Chai Wah Wu, Sep 16 2022

CROSSREFS

Odd bisection of A046523.

Cf. A064216, A244153, A245611, A278222, A278224, A278531.

Sequences that partition or seem to partition N into same or coarser equivalence classes: A099774, A100007, A193773, A101871, A158280, A158315, A158647, A285716.

Sequence in context: A255336 A292929 A049627 * A368556 A134058 A345530

Adjacent sequences: A278220 A278221 A278222 * A278224 A278225 A278226

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 16 2016

STATUS

approved