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A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1). 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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)

LINKS

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

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)).

EXAMPLE

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

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 *)

PROG

(Scheme, two versions)

(define (A277324 n) (A260443 (+ 1 n n)))

(define (A277324 n) (if (zero? n) 2 (* (A260443 n) (A260443 (+ 1 n)))))

(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 xrange(101)] # Indranil Ghosh, Jun 21 2017

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).

Sequence in context: A197168 A288815 A277200 * A034881 A146345 A064842

Adjacent sequences:  A277321 A277322 A277323 * A277325 A277326 A277327

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 10 2016

EXTENSIONS

More linking formulas added by Antti Karttunen, Apr 16 2017

STATUS

approved

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Last modified July 16 23:49 EDT 2019. Contains 325092 sequences. (Running on oeis4.)