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

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

Author

Antti Karttunen, Oct 10 2016

Keywords

• nonn

Comments

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)

Examples

			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
		

Links

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

Crossrefs

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

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

Programs

Mathematica

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 *)

Python

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

Formula

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)).

Extensions

More linking formulas added by Antti Karttunen, Apr 16 2017