A226502 Let P(k) denote the k-th prime (P(1)=2, P(2)=3 ...); a(n) = P(n+1)P(n+3) - P(n)P(n+2).

11, 34, 36, 96, 60, 144, 160, 162, 360, 198, 320, 336, 352, 494, 460, 720, 378, 560, 718, 450, 972, 1020, 938, 1002, 816, 420, 864, 1752, 960, 2596, 810, 2204, 576, 2404, 1220, 1606, 1980, 1694, 1420, 2876, 744, 2694, 780, 3160, 2810, 3520, 3170, 1824, 1840, 1422, 3836

Offset: 1

Author

Ed Smiley, Jun 09 2013

Keywords

• nonn
• easy

Comments

Differences of the products of alternate primes.

Links

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

Crossrefs

First differences of A090076.

Programs

Mathematica

#[[2]]#[[4]]-#[[1]]#[[3]]&/@Partition[Prime[Range[60]],4,1] (* Harvey P. Dale, Jul 14 2025 *)

PARI

p=2;q=3;r=5;forprime(s=7,1e2,print1(q*s-p*r", ");p=q;q=r;r=s) \\ Charles R Greathouse IV, Jun 10 2013

Formula

a(n) >> n log n and this is probably sharp: on Dickson's conjecture there are infinitely many a(n) < kn log n for any k > 4. The constant 4 comes from 8 + 2 - 6 - 0 n the prime quadruplet (p+0, p+2, p+6, p+8). On Cramér's conjecture a(n) = O(n log^3 n). Unconditionally a(n) << n^1.525 log n. - Charles R Greathouse IV, Jun 10 2013