A246263 -id:A246263 - cp's OEIS frontend

A269861 Numbers n such that n and A048673(n) are of opposite parity.

4, 5, 7, 10, 12, 14, 15, 16, 17, 19, 21, 29, 30, 34, 36, 38, 40, 41, 42, 43, 44, 45, 48, 51, 52, 53, 55, 56, 57, 58, 61, 63, 64, 65, 67, 73, 77, 79, 82, 86, 87, 90, 91, 92, 100, 101, 102, 103, 106, 108, 110, 113, 114, 115, 120, 122, 123, 124, 125, 126, 127, 129, 130, 132, 134, 135, 136, 137, 140, 144, 146, 148, 149

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

Union of even terms of A246261 and odd terms of A246263.

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

Complement: A269860.
Left inverse: A269862.
Cf. A000035, A048673, A246261, A246263.
Cf. also A270431.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Select[Range@ 150, Xor[EvenQ@ f@ #, EvenQ@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

Other identities. For all n >= 1:

A269862(a(n)) = 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.

A292603 Doudna-tree reduced modulo 4: a(n) = A005940(1+n) mod 4.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 01 2017

nonn

			The first six levels of the binary tree (compare also to the illustrations given at A005940 and A292602):
                               1
                               |
                               2
                ............../ \..............
               3                               0
        ....../ \......                 ....../ \......
       1               2               1               0
      / \             / \             / \             / \
     /   \           /   \           /   \           /   \
    3     2         3     0         1     2         3     0
   / \   / \       / \   / \       / \   / \       / \   / \
  3   2 1   0     3   2 1   0     1   2 3   0     1   2 1   0

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A005940, A292602.
Cf. A004767 (gives the positions of 0's), A016813 (of 2's).
Cf. also A246261, A246263, A246271, A292271, A292274, A292375, A292377, A292381, A292383, A292384, A292583.

(define (A292603 n) (modulo (A005940 (+ 1 n)) 4))

a(n) = A010873(A005940(1+n)).
a(n) + 4*A292602(n) = A005940(1+n).
a(2n+1) = 2*a(n) mod 4.
a(A004767(n)) = 0.
a(A016813(n)) = 2.
a(2*A156552(A246261(n))) = 1.
a(2*A156552(A246263(n))) = 3.
a(n * 2^(1+A246271(A005940(1+n)))) = 1.

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

A269861 Numbers n such that n and A048673(n) are of opposite parity.

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

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A292603 Doudna-tree reduced modulo 4: a(n) = A005940(1+n) mod 4.

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

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