A036263 Second differences of primes.

1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2, -2, 2, 2, 0, -4, 4, -2, -2, 4, -2, 2, 2, -4, -2, 2, -2, 2, 10, -10, 2, -4, 8, -8, 4, 0, -2, 2, 0, -4, 8, -8, 2, -2, 10, 0, -8, -2, 2, 2, -4, 8, -4, 0, 0, -4, 4, -2, -2, 8, 4, -10, -2, 2, 10, -8, 4, -8, 2, 2, 2, -2, 0, -2, 2, 2, -4, 4, 2, -8, 8, -8, 4, -2, 2, 2, -4, -2, 2, 8, -4

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

Examples

			a(3) = 5 + 11 - 2*7 = 16 - 14 = 2.
		

Links

• Edward Bernstein, Table of n, a(n) for n = 1..100000 (Terms 1 to 10000 from T. D. Noe)
• R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014. See Table 2.

Crossrefs

Cf. A001223, A036262, A051635, A006562, A051634, A147812, A147813, A182394, A079054, A064113 (places of zeros).

For records see A293154, A293155.

Programs

Haskell

a036263 n = a036263_list !! (n-1)
a036263_list = zipWith (-) (tail a001223_list) a001223_list
-- Reinhard Zumkeller, Oct 29 2011

Maple

A036263:=n->ithprime(n) + ithprime(n+2) - 2*ithprime(n+1); seq(A036263(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Mathematica

Table[Prime[n - 1] + Prime[n + 1] - 2*Prime[n], {n, 2, 105}]
Differences[Prime[Range[100]], 2] (* Harvey P. Dale, Oct 14 2012 *)

PARI

for(n=2,100,print1(prime(n+2)-2*prime(n+1)+prime(n)","))

Python

from sympy import prime
def A036263(n): return prime(n)-(prime(n+1)<<1)+prime(n+2) # Chai Wah Wu, Sep 28 2024

Formula

a(A064113(n)) = 0. - Reinhard Zumkeller, Jan 20 2012

a(n) = prime(n) + prime(n+2) - 2*prime(n+1). - Thomas Ordowski, Jul 21 2012

Conjecture: |a(1)| + |a(2)| + ... + |a(n)| ~ prime(n). - Thomas Ordowski, Jul 21 2012

a(n) = A001223(n+1) - A001223(n). - R. J. Mathar, Sep 19 2013

Sum_{i = 1..n - 1} a(i) = A046933(n), n >= 1. - Daniel Forgues, Apr 15 2014

Sum_{i = 2..n - 1} a(i) = prime(n + 1) - prime(n) - 2; Sum_{i = 2..n - 1} a(i) = 0 whenever prime(n) is a lesser of twin primes. - Hamdi Murat Yildirim, Jun 24 2014