A201498 a(n) = (prime(n) - 1)*(prime(n+1) - 1)/2 + 3.

4, 7, 15, 33, 63, 99, 147, 201, 311, 423, 543, 723, 843, 969, 1199, 1511, 1743, 1983, 2313, 2523, 2811, 3201, 3611, 4227, 4803, 5103, 5409, 5727, 6051, 7059, 8193, 8843, 9387, 10215, 11103, 11703, 12639, 13449, 14279, 15311, 16023, 17103, 18243, 18819, 19407

Offset: 1

Author

Zak Seidov, Dec 02 2011

Keywords

• nonn

Comments

Consider strictly increasing sequence with the rule:

a(n) is the smallest pairwise sum s of all previous terms such that s > a(n-1).

We start with some pair of coprime positive integers b < c, a(1)=b, a(2)=c; from now on, to find a(n) we use the above-mentioned rule. We observe that, for any seeds b,c, after some term, a(n) = a(n-1) + 1.

E.g., for b=7, c=12, we get 7, 12,1 9, 26, 31, 33, 38, 40, 43, 45, 47, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, ...

We stop at the term a(L=36) = 85 after which a(n) = a(n-1) + 1.

In the general case of arbitrary coprime b < c, the length of the sequence is L = 3 + (b-1)(c-1)/2, and a(L) = b*c + 1.

In A201498, we present the dependence of L(n) for the particular case b=prime(n) and c=prime(n+1).

Programs

Mathematica

#/2+3&/@(Times@@@Partition[Prime[Range[50]]-1,2,1])  (* Harvey P. Dale, Jun 01 2015 *)

PARI

p=2;forprime(q=3,1e3,print1((p-1)*(q-1)/2+3", ");p=q) \\ Charles R Greathouse IV, Dec 05 2011

Formula

a(n) = A099407(n) + 3.