A233334 a(1) = 1; for n > 1, a(n) is the smallest number > a(n-1) such that a(1) + a(2) + ... + a(n) is a composite number.

1, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Michel Lagneau, Dec 18 2013

Keywords

• nonn
• easy

Comments

{a(n)} = {1, 3, 4, 6, 7} union {9, 10, 11, 12, ...} and the sum s(n) = a(1) + a(2) + ... + a(n) is always composite because s(1) = 1, s(2) = 4, s(3) = 8, s(4) = 14 and for n = 5,6,7,... s(n) = (n-2)*(n+9)/2 = 21, 30, 40, 51, ... = A056115(n) for n >= 3.

Examples

			The third term is 4 because 1+3+4=8 is composite.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A051935, A056115.

Programs

Mathematica

p=1; lst={p}; Do[If[!PrimeQ[p+n], AppendTo[lst, n]; p=p+n], {n, 3, 70}]; lst
nxt[{c_,a_}]:=Module[{k=a+1},While[!CompositeQ[c+k],k++];{c+k,k}]; NestList[nxt,{1,1},70][[;;,2]] (* Harvey P. Dale, Dec 05 2023 *)

Formula

From Chai Wah Wu, Jan 28 2024: (Start)

a(n) = 2*a(n-1) - a(n-2) for n > 7.

G.f.: x*(-x^6 + x^5 - x^4 + x^3 - x^2 + x + 1)/(x - 1)^2. (End)