A147813 -id:A147813 - cp's OEIS frontend

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

N. J. A. Sloane

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

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.

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

For records see A293154, A293155.

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

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

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

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

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

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

A147812 Primes prime(n) such that prime(n+1) - prime(n) > prime(n+2) - prime(n+1).

Original entry on oeis.org

7, 13, 23, 31, 37, 53, 61, 67, 73, 89, 97, 103, 113, 131, 139, 157, 173, 181, 193, 211, 223, 233, 241, 263, 271, 277, 293, 307, 317, 337, 359, 373, 389, 409, 421, 433, 449, 457, 467, 479, 491, 509, 523

Offset: 1

Roger L. Bagula, Nov 13 2008

nonn

This was originally formulated as (-prime(n) + 2*prime(n+1) - prime(n+2))/((1 - prime(n) + prime(n+1))^(3/2)) > 0, which relates it to other sequences. This is equivalent since the denominator is always positive.

			The gap between 7 and the next prime, 11, is 4, which is greater than the next prime gap from 11 to 13, so 7 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A036263, A147813 (complement with respect to A000040).

import Data.List (findIndices)
a147812 n = a147812_list !! (n-1)
a147812_list = map (a000040 . (+ 1)) $ findIndices (< 0) a036263_list
-- Reinhard Zumkeller, Jan 20 2012

d2[n_] = Prime[n + 2] - 2*Prime[n + 1] + Prime[n]; d1[n_] = Prime[n + 1] - Prime[n]; k[n_] = -d2[n]/(1 + d1[n])^(3/2); Flatten[Table[If[k[n] > 0, Prime[n], {}], {n, 1, 100}]]
Select[Partition[Prime[Range[150]],3,1],#[[2]]-#[[1]]>#[[3]]-#[[2]]&][[All,1]] (* Harvey P. Dale, Mar 29 2022 *)

require 'mathn'
Prime.take(100).each_cons(3).select{ |a,b,c| b-a>c-b }.map(&:first)
-- Aaron Weiner, Dec 05 2013

Edited by Alonso del Arte and Joerg Arndt, Nov 01 2013

Simpler formula added by Aaron Weiner, Dec 05 2013

cp's OEIS Frontend

A036263 Second differences of primes.

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