A176716 a(1) = 1; a(2*n) = prime(n)*a(n), a(2*n+1) = prime(n)*a(n) + a(n+1), where prime(n) is the n-th prime.

1, 2, 4, 6, 10, 20, 26, 42, 52, 110, 130, 260, 286, 442, 484, 798, 850, 1196, 1306, 3190, 3320, 4030, 4290, 9620, 9906, 11726, 12168, 19006, 19490, 22748, 23546, 42294, 43144, 50150, 51346, 72956, 74262, 87502, 90692, 226490, 229810, 242360, 246390, 318370, 322660, 356070, 365690, 856180, 866086

Offset

1,2

Comments

Conjecture: for n > 1, 4 divides a(n) iff n == 0 (mod 3), otherwise A007814(a(n)) = 1.

Conjecture is true (proof by mathematical induction). - Robert Israel, Feb 25 2020

Given the generating triangle of A176528, replace k's with prime(k); then take powers of the triangle resulting in a left-shifted vector considered as a sequence.

First few rows of the generating triangle M for this sequence:

1;

2;

2, 1;

0, 3;

0, 3, 1;

0, 0, 5;

0, 0, 5, 1;

0, 0, 0, 7;

0, 0, 0, 7, 1;

0, 0, 0, 0, 11;

0, 0, 0, 0, 11, 1;

0, 0, 0, 0, 0, 13;

0, 0, 0, 0, 0, 13, 1;

0, 0, 0, 0, 0, 0, 17;

...

Then sequence is Lim_{n->inf} M^n, the left-shifted vector considered as a sequence.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(10) = 110 = prime(5) * a(5) = 11 * 10.
a(7) = 26 = prime(3) * 4 + 6 = 5 * 4 + 6.
4 divides a(9) = 52 since 9 == 0 mod 3.
2 divides a(13) = 286 but not 4 since 13 == 1 mod 3.

Prog

(PARI) a(n) = if(n==1, 1, if (n % 2, a(n\2+1)+a(n\2)*prime(n\2), a(n/2)*prime(n/2))); \\ Michel Marcus, Dec 25 2020

Crossrefs

Cf. A007814, A176528, A000040.

Sequence in context: A034872 A032362 A282251 * A256056 A293281 A164143

Adjacent sequences: A176713 A176714 A176715 * A176717 A176718 A176719

Keyword

nonn

Author

Gary W. Adamson, Apr 24 2010

Extensions

Edited and more terms from Robert Israel, Feb 25 2020

Status

approved