A078974 -id:A078974 - cp's OEIS frontend

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

Miklos Kristof, Jan 02 2003

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?)

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

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

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

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 *)

Name corrected by Sean A. Irvine, Jul 01 2025

A359388 a(n) is the number of compositions of n into prime parts, with the 1st part equal to 2, the 2nd part less than or equal to 3, ..., and the k-th part less than or equal to prime(k), and so on.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 4, 5, 7, 11, 15, 24, 33, 50, 73, 105, 159, 229, 342, 501, 738, 1094, 1604, 2378, 3499, 5166, 7627, 11243, 16610, 24494, 36165, 53376, 78775, 116301, 171642, 253398, 374034, 552139, 815079, 1203166, 1776174, 2621938, 3870572, 5713798, 8434744

Offset: 0

Stefano Spezia, Dec 29 2022

nonn

			The 7 such compositions of n = 11 are:
[ 1]  (2, 2, 2, 2, 3);
[ 2]  (2, 2, 2, 3, 2);
[ 3]  (2, 2, 3, 2, 2);
[ 4]  (2, 3, 2, 2, 2);
[ 5]  (2, 2, 2, 5);
[ 6]  (2, 2, 5, 2);
[ 7]  (2, 3, 3, 3).

Alois P. Heinz, Table of n, a(n) for n = 0..4000
Index entries for sequences related to compositions

Cf. A000040, A004526, A023360, A078974, A326178, A359328.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-ithprime(j), i+1), j=1..min(i, numtheory[pi](n))))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 29 2022

a[n_]:=Coefficient[Expand[Sum[Product[Sum[x^Prime[i], {i, k}], {k,m}], {m, 0,Floor[n/2]}]],x,n]; Array[a,48,0]

G.f.: Sum_{m>=0} Product_{k=1..m} Sum_{i=1..k} x^prime(i).

a(n) ~ c*A078974^n, where c = 0.094587447... .

cp's OEIS Frontend

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.

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