A285716 -id:A285716 - cp's OEIS frontend

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

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

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.

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences computed from exponents in factorization of n.

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.

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 *)

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

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

(define (A278223 n) (A046523 (+ n n -1)))
(define (A278223 n) (A046523 (A064216 n)))

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)

A291763 Binary encoding of 2-digits in ternary representation of A245612(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 3, 0, 1, 8, 1, 6, 5, 4, 1, 2, 1, 10, 23, 16, 1, 2, 13, 10, 9, 2, 7, 0, 1, 0, 3, 2, 1, 70, 21, 24, 45, 4, 33, 52, 1, 36, 3, 20, 25, 10, 21, 0, 17, 18, 5, 0, 13, 16, 3, 12, 1, 8, 1, 4, 5, 2, 5, 0, 1, 32, 139, 74, 41, 208, 49, 0, 89, 108, 11, 130, 65, 18, 103, 4, 1, 8, 73, 4, 5, 112, 41, 16, 49, 72, 19, 38, 41, 20, 1, 8, 33, 0, 35, 86, 9, 38

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A245612, A285712, A285714, A285715, A285716, A289814, A291759, A291760.

a(n) = A289814(A245612(n)).

A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 3, 1, 1, 2, 1, 2, 3, 2, 2, 2, 3, 1, 2, 1, 2, 4, 1, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 1, 2, 2, 2, 4, 1, 1, 3, 1, 1, 2, 2, 2, 3, 2

Offset: 1

Andrew S. Plewe, Feb 20 2004

Omega(n) of the odd integers follows a pattern similar to A001222, with 4 maxima instead of 2 - i.e. between 2^n and (2^(n+1) - 1) there are two numbers with exactly n factors (2^n and 2^(n-1) * 3) while the odd integers have 4 maxima (3^n, 3^(n-1) * 5, 3^(n-1) * 7, 5^2*3^(n-2)) between 3^n and 3^(n+1) - 1.

			Omega(1) = 0, Omega(9) = 2 (3 * 3 = 9), Omega (243) = 5 (3 * 3 * 3 * 3 * 3 = 243), Omega(51) = 2 (3 * 17 = 51).
For n = 92, A001222(2*92 - 1) = A001222(183) = 2 as 183 = 3*61, thus a(92) = 2. - _Antti Karttunen_, May 31 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000

One more than A285716 (after the initial term).
Cf. A001222, A000120, A092523, A099774, A099990, A244153, A278223, A286582.
Cf. A006254 (positions of ones).

a[n_] := PrimeOmega[2*n - 1]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

a(n) = bigomega(2*n-1) \\ Michel Marcus, Jul 26 2013, edited to reflect the changed starting offset by Antti Karttunen, May 31 2017

a(n) = Omega(2n-1). [Odd bisection of A001222.]
From Antti Karttunen, May 31 2017: (Start)
For n >= 1, a(n) = A000120(A244153(n)).
For n >= 2, a(n) = 1+A285716(n).
(End)

Starting offset changed to 1 and the definition modified respectively. Also values of the initial term and of term a(92) (= 2, previously a(91) = 1) corrected by Antti Karttunen, May 31 2017

A285714 a(1) = 0; for n > 1, a(n) = 1 + a(A285712(n)).

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 5, 3, 6, 7, 4, 8, 3, 3, 9, 10, 5, 4, 11, 6, 12, 13, 4, 14, 4, 7, 15, 5, 8, 16, 17, 5, 6, 18, 9, 19, 20, 4, 5, 21, 4, 22, 7, 10, 23, 6, 11, 8, 24, 6, 25, 26, 5, 27, 28, 12, 29, 9, 7, 7, 5, 13, 4, 30, 14, 31, 8, 5, 32, 33, 15, 6, 10, 5, 34, 35, 8, 11, 36, 16, 9, 37, 6, 38, 6, 9, 39, 5, 17, 40, 41, 18, 12, 7, 6, 42, 43, 7, 44, 45, 19, 10, 13

Offset: 1

Antti Karttunen, Apr 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A029837, A245611, A285712, A285713, A285715, A285716.

;; First definition uses memoization-macro definec:
(definec (A285714 n) (if (<= n 1) 0 (+ 1 (A285714 (A285712 n)))))
(define (A285714 n) (A029837 (+ 1 (A245611 n))))

a(1) = 0; for n > 1, a(n) = 1 + a(A285712(n)).
a(n) = A029837(1+A245611(n)).
a(n) = A285715(n) + A285716(n).

A285715 a(n) = A000120(A245611(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 2, 1, 9, 10, 4, 3, 11, 5, 12, 13, 2, 14, 3, 6, 15, 4, 7, 16, 17, 3, 5, 18, 8, 19, 20, 2, 4, 21, 1, 22, 6, 9, 23, 5, 10, 7, 24, 4, 25, 26, 3, 27, 28, 11, 29, 8, 5, 6, 4, 12, 2, 30, 13, 31, 7, 2, 32, 33, 14, 5, 9, 3, 34, 35, 6, 10, 36, 15, 8, 37, 4, 38, 5, 7, 39, 3, 16, 40, 41, 17, 11, 6, 3, 42, 43, 5, 44, 45, 18, 9, 12, 8, 7

Offset: 1

Antti Karttunen, Apr 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000120, A245611, A285712, A285714, A285716.

Cf. A007051 (positions of 0 and 1's).

;; First implementation uses memoization-macro definec:
(definec (A285715 n) (if (<= n 2) (- n 1) (+ (if (= 2 (modulo n 3)) 0 1) (A285715 (A285712 n)))))
(define (A285715 n) (A000120 (A245611 n)))

a(1) = 0, a(2) = 1, for n > 2, a(n) = a(A285712(n)) + [n <> 2 mod 3]. (Where [] is Iverson bracket, giving here 1 if n is of the form 3k or 3k+1, and 0 if it is of the form 3k+2.)

a(n) = A000120(A245611(n)).

cp's OEIS Frontend

A278223 Least number with the same prime signature as the n-th odd number: a(n) = A046523(2n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python

Scheme

Formula

A291763 Binary encoding of 2-digits in ternary representation of A245612(n).

Views

Author

Keywords

Links

Crossrefs

Formula

A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A285714 a(1) = 0; for n > 1, a(n) = 1 + a(A285712(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A285715 a(n) = A000120(A245611(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula