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

1,2

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.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1).

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)

Example

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

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 *)

Crossrefs

Cf. A051935, A056115.

Sequence in context: A154661 A275672 A087753 * A320300 A070762 A061208

Adjacent sequences: A233331 A233332 A233333 * A233335 A233336 A233337

Keyword

nonn,easy

Author

Michel Lagneau, Dec 18 2013

Status

approved