A125180 a(n) = 2*a(n-1) + prime(n) - prime(n-1), a(1)=2, where prime(n) denotes the n-th prime.

2, 5, 12, 26, 56, 114, 232, 466, 936, 1878, 3758, 7522, 15048, 30098, 60200, 120406, 240818, 481638, 963282, 1926568, 3853138, 7706282, 15412568, 30825142, 61650292, 123300588, 246601178, 493202360, 986404722, 1972809448, 3945618910

Offset: 1

Author

Gary W. Adamson, Nov 22 2006

Keywords

• nonn

Comments

Row sums of A125179.

Limit_{n->oo} a(n)/a(n-1) = 2.

Examples

			a(4)=26 because 4*prime(1)+2*prime(2)+prime(3)+prime(4) = 8+6+5+7 = 26.
		

Links

• James C. McMahon, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A125179, A000040, A000720, A110299.

Programs

Magma

[n eq 1 select 2 else 2*Self(n-1)+NthPrime(n)-NthPrime(n-1):n in [1..31]]; // Marius A. Burtea, Oct 17 2019

Maple

a[1]:=2: for n from 2 to 35 do a[n]:=2*a[n-1]+ithprime(n)-ithprime(n-1) od: seq(a[n],n=1..35);

Mathematica

a[1] = 2; a[n_] := 2*a[n - 1] + Prime[n] - Prime[n - 1]; Table[a[n], {n, 1, 31}] (* James C. McMahon, Dec 10 2023 *)

Formula

a(n) = prime(n) + Sum_{j=1..n-1} 2^(n-j-1)*prime(j), where prime(k) denotes the k-th prime.

a(n) = Sum_{i=0..prime(n)-1} 2^(n-1-pi(i)), where prime(n) = A000040(n) and pi(n) = A000720(n). - Ridouane Oudra, Oct 17 2019

a(1) = 2 and a(n) = prime(n) + Sum_{i=1..n-1} a(i) for n > 1. - Alexandre Herrera, Dec 10 2023

a(n) = A000040(n) + A110299(n-1), for n > 1. - Ridouane Oudra, Jul 27 2025

Extensions

Edited by N. J. A. Sloane, Dec 02 2006