A283975 -id:A283975 - cp's OEIS frontend

A284266 Odd bisection of A277700, binary weight of terms of A283975.

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

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000120, A007306, A277700, A283484, A283975, A284265.

A264977[n_]:= If[n<2, n, If[EvenQ[n], 2 A264977[n/2], BitXor[A264977[(n - 1)/2], A264977[(n + 1)/2]]]]; Table[DigitCount[A264977[2n + 1], 2, 1], {n, 0, 150}] (* Indranil Ghosh, Mar 28 2017 *)

a(n) = if(n<2, n, if(n%2, bitxor(a((n - 1)/2), a((n + 1)/2)), 2*a(n/2)));
b(n) = if(n<1, 0, b(n\2) + n%2);
for(n=0, 150, print1(b(a(2*n + 1)),", ")) \\ Indranil Ghosh, Mar 28 2017

(define (A284266 n) (A277700 (+ n n 1)))
(define (A284266 n) (A000120 (A283975 n)))

a(n) = A277700((2*n)+1).
a(n) = A000120(A283975(n)).
Other identities. For all n >= 0:
A007306(1+n) = a(n) + 2*A284265(n).

A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

Original entry on oeis.org

2, 6, 18, 30, 90, 270, 450, 210, 630, 6750, 20250, 9450, 15750, 47250, 22050, 2310, 6930, 330750, 3543750, 1653750, 4961250, 53156250, 24806250, 727650, 1212750, 57881250, 173643750, 18191250, 8489250, 25467750, 2668050, 30030, 90090, 40020750, 1910081250, 891371250, 9550406250, 455814843750, 212713593750

Offset: 0

Antti Karttunen, Oct 10 2016

nonn

From David A. Corneth, Oct 22 2016: (Start)
The exponents of the prime factorization of a(n) are first nondecreasing, then nonincreasing.
The exponent of 2 in the prime factorization of a(n) is 1. (End)

			A method to find terms of this sequence, explained by an example to find a(7). To find k = a(7), we find k such that A048675(k) = 2*7+1 = 15. 7 has the binary partitions: {[7, 0, 0], [5, 1, 0], [3, 2, 0], [1, 3, 0], [3, 0, 1], [1, 1, 1]}. To each of those, we prepend a 1. This gives the binary partitions of 15 starting with a 1. For example, for the first we get [1, 7, 0, 0]. We see that only [1, 5, 1, 0], [1, 3, 2, 0] and [1, 1, 1, 1] start nondecreasing, then nonincreasing, so we only check those. These numbers will be the exponents in a prime factorization. [1, 5, 1, 0] corresponds to prime(1)^1 * prime(2)^5 * prime(3)^1 * prime(4)^0 = 2430. We find that [1, 1, 1, 1] gives k = 210 for which A048675(k) = 15 so a(7) = 210. - _David A. Corneth_, Oct 22 2016

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A005811, A005940, A007306, A007949, A156552, A260443, A277189, A277323, A283484, A283975, A284267, A284268, A284563, A284564, A284573.

Cf. A277200 (same sequence sorted into ascending order).

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a[2 n + 1], {n, 0, 38}] (* Michael De Vlieger, Apr 05 2017 *)

from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a260443(n): return n + 1 if n<2 else a003961(a260443(n//2)) if n%2==0 else a260443((n - 1)//2)*a260443((n + 1)//2)
def a(n): return a260443(2*n + 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

a(n) = A260443((2*n)+1).
a(0) = 2; for n >= 1, a(n) = A260443(n) * A260443(n+1).
Other identities. For all n >= 0:
A007949(a(n)) = A005811(n). [See comments in A125184.]
A156552(a(n)) = A277189(n), a(n) = A005940(1+A277189(n)).
A048675(a(n)) = 2n + 1. - David A. Corneth, Oct 22 2016
A001222(a(n)) = A007306(1+n).
A056169(a(n)) = A284267(n).
A275812(a(n)) = A284268(n).
A248663(a(n)) = A283975(n).
A000188(a(n)) = A283484(n).
A247503(a(n)) = A284563(n).
A248101(a(n)) = A284564(n).
A046523(a(n)) = A284573(n).
a(n) = A277198(n) * A284008(n).
a(n) = A284576(n) * A284578(n) = A284577(n) * A000290(A284578(n)).

More linking formulas added by Antti Karttunen, Apr 16 2017

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A284266 Odd bisection of A277700, binary weight of terms of A283975.

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A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

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