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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.

%I A078465 #18 Jul 01 2025 21:40:31
%S A078465 1,1,1,2,2,4,5,8,12,16,26,36,55,81,118,177,257,384,564,833,1233,1813,
%T A078465 2685,3956,5845,8629,12731,18807,27746,40976,60481,89282,131816,
%U A078465 194562,287253,424018,625968,924077
%N 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.
%C A078465 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?)
%H A078465 T. D. Noe, <a href="/A078465/b078465.txt">Table of n, a(n) for n=1..500</a>
%e A078465 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.
%t A078465 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 *)
%o A078465 (Haskell)
%o A078465 import Data.List (genericIndex)
%o A078465 a078465 n = a078465_list `genericIndex` (n-1)
%o A078465 a078465_list = 1 : 1 : f 3 where
%o A078465    f x = (sum $ map (a078465 . (x -)) $
%o A078465          takeWhile (< x) a000040_list) : f (x + 1)
%o A078465 -- _Reinhard Zumkeller_, Jul 20 2012
%Y A078465 Cf. A078974 (the constant 1.47622...), A084256 (the constant 1/1.47622...)
%K A078465 easy,nice,nonn
%O A078465 1,4
%A A078465 _Miklos Kristof_, Jan 02 2003
%E A078465 Name corrected by _Sean A. Irvine_, Jul 01 2025