A286582 -id:A286582 - cp's OEIS frontend

A286585 a(n) = A053735(A048673(n)).

1, 2, 1, 3, 2, 4, 2, 4, 3, 3, 3, 5, 1, 5, 2, 5, 2, 4, 2, 4, 2, 4, 3, 6, 5, 6, 3, 6, 4, 7, 3, 6, 3, 3, 3, 5, 3, 5, 5, 5, 4, 3, 4, 5, 4, 6, 1, 7, 5, 6, 4, 7, 2, 8, 4, 7, 4, 5, 3, 8, 4, 4, 4, 7, 4, 6, 2, 4, 5, 6, 3, 6, 4, 6, 5, 6, 4, 6, 4, 6, 7, 5, 3, 4, 5, 7, 6, 6, 5, 5, 4, 7, 3, 8, 1, 8, 5, 6, 3, 7, 6, 7, 2, 8

Offset: 1

Antti Karttunen, May 31 2017

nonn

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

Cf. A000079, A048673, A053735, A286582, A286583, A286584, A286586.

from sympy.ntheory.factor_ import digits
from sympy import factorint, nextprime
from operator import mul
def a053735(n): return sum(digits(n, 3)[1:])
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))//2
def a(n): return a053735(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017

(define (A286585 n) (A053735 (A048673 n)))

a(n) = A053735(A048673(n)).

For all n >= 0, a(A000079(n)) = n+1.

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

A286583 a(n) = A007814(A048673(n)).

Original entry on oeis.org

0, 1, 0, 0, 2, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 5, 2, 2, 0, 2, 0, 1, 0, 1, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 2, 0, 0, 0, 1, 4, 0, 1, 2, 1, 0, 1, 0, 0, 1, 1, 3, 1, 0, 2, 1, 2, 1, 0, 2, 0, 1, 3, 1, 0, 3, 3, 7, 1, 2, 0, 0, 0, 3, 0, 0, 1, 4, 0, 0, 1, 0, 0, 4, 0, 5, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 6

Offset: 1

Antti Karttunen, May 31 2017

nonn

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

Cf. A007814, A048673, A286582, A286584, A286585.

Cf. A246261 (positions of zeros), A246263 (of nonzeros).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A007814(n) = (valuation(n,2));
A286583(n) = A007814(A048673(n));

from sympy import factorint, nextprime, prod
def a007814(n): return 1 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2
def a(n): return a007814(a048673(n)) # Indranil Ghosh, Jun 12 2017

(define (A286583 n) (A007814 (A048673 n)))

a(n) = A007814(A048673(n)).

A286584 a(n) = A048673(n) mod 4.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2017

nonn

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

Cf. A010873, A048673, A286582, A286583, A286585.

Cf. A246261 (positions of odd terms), A246263 (of even terms).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A286584(n) = (A048673(n)%4);

from sympy import factorint, nextprime
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return a048673(n)%4 # Indranil Ghosh, Jun 12 2017

(define (A286584 n) (modulo (A048673 n) 4))

a(n) = A010873(A048673(n)) = A048673(n) mod 4.

A289623 a(n) = A055396(A048673(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 1, 1, 6, 5, 4, 9, 2, 7, 1, 13, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 16, 8, 1, 2, 10, 2, 30, 2, 3, 14, 3, 1, 23, 1, 17, 1, 1, 2, 4, 18, 1, 1, 4, 1, 1, 1, 35, 1, 15, 11, 1, 1, 1, 1, 3, 1, 1, 1, 1, 21, 1, 12, 1, 1, 1, 2, 1, 1, 1, 1, 1, 65, 3, 2, 1, 19, 20, 1, 1, 4, 56, 1, 32, 2, 1, 2, 1, 2, 1, 38, 6

Offset: 1

Antti Karttunen, Jul 16 2017

From the scatter plot it can be seen that the terms are grouped into two distinct populations by their magnitude, with significant gap between them.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A048673, A055396, A246263 (the positions of ones).

Cf. also A278224, A286582, A286583, A286584, A286586.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A289623(n) = A055396(A048673(n));

a(n) = A055396(A048673(n)).

cp's OEIS Frontend

A286585 a(n) = A053735(A048673(n)).

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

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A286583 a(n) = A007814(A048673(n)).

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A286584 a(n) = A048673(n) mod 4.

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A289623 a(n) = A055396(A048673(n)).

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