A078465 Primonacci numbers: a(n)=a(n-2)+a(n-3)+a(n-5)+a(n-7)+a(n-11)+...+a(n-p(k))+... until n <= p(k), where p(k) is the k-th prime. a(1)=a(2)=1.

1, 1, 1, 2, 2, 4, 5, 8, 12, 16, 26, 36, 55, 81, 118, 177, 257, 384, 564, 833, 1233, 1813, 2685, 3956, 5845, 8629, 12731, 18807, 27746, 40976, 60481, 89282, 131816, 194562, 287253, 424018, 625968, 924077

Offset: 1

Author

Miklos Kristof, Jan 02 2003

Keywords

• easy
• nice
• nonn

Comments

a(n)/a(n-1) -> 1.476229...=1/x, where x satisfies the Sum x^p(n)=1 equation, i.e. x^2+x^3+x^5+x^7+x^11+... =1. (What constant is it?)

Examples

			a(12) = 36 = a(12-2)+a(12-3)+a(12-5)+a(12-7)+a(12-11) = a(10)+a(9)+a(7)+a(5)+a(1) = 16+12+5+2+1 = 36.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..500

Crossrefs

Cf. A078974 (the constant 1.47622...), A084256 (the constant 1/1.47622...)

Programs

Haskell

import Data.List (genericIndex)
a078465 n = a078465_list `genericIndex` (n-1)
a078465_list = 1 : 1 : f 3 where
   f x = (sum $ map (a078465 . (x -)) $
         takeWhile (< x) a000040_list) : f (x + 1)
-- Reinhard Zumkeller, Jul 20 2012

Mathematica

a[1] = a[2] = 1; a[n_] := a[n] = Sum[a[n - Prime[k]], {k, 1, PrimePi[n]}]; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Mar 22 2011 *)

Extensions

Name corrected by Sean A. Irvine, Jul 01 2025