A022884 -id:A022884 - cp's OEIS frontend

A022885 Primes p=prime(k) such that prime(k) + prime(k+3) = prime(k+1) + prime(k+2).

5, 7, 11, 13, 23, 37, 53, 73, 97, 101, 103, 109, 137, 157, 179, 191, 223, 251, 263, 307, 353, 373, 389, 409, 419, 433, 457, 479, 487, 541, 563, 571, 593, 683, 691, 701, 757, 809, 821, 853, 859, 877, 883, 911, 977, 1019, 1039, 1049, 1087, 1103

Offset: 1

Clark Kimberling

nonn

These are primes p for which the subsequent alternate prime gaps are equal, so (p(k+3)-p(k+2))/(p(k+1)-p(k)) = 1. It is conjectured that the most frequent alternate prime gaps ratio is one. - Andres Cicuttin, Nov 07 2016

			Starting from 5, the four consecutive primes are 5, 7, 11, 13; and they satisfy 5 + 13 = 7 + 11. So 5 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 Vincenzo Librandi)

Cf. A022884, A260179, A261470.

[NthPrime(n): n in [1..200] | (NthPrime(n)+NthPrime(n+3)) eq (NthPrime(n+1)+NthPrime(n+2))]; // Vincenzo Librandi, Nov 08 2016

Transpose[Select[Partition[Prime[Range[500]],4,1],First[#]+Last[#] == #[[2]]+#[[3]]&]][[1]] (* Harvey P. Dale, May 23 2011 *)

isok(p) = {my(k = primepi(p)); (p == prime(k)) && ((prime(k) + prime(k+3)) == (prime(k+1) + prime(k+2)));} \\ Michel Marcus, Jan 15 2014

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    p, q, r, s = [2, 3, 5, 7]
    while True:
        if p + s == q + r: yield p
        p, q, r, s = q, r, s, nextprime(s)
print(list(islice(agen(), 50))) # Michael S. Branicky, May 31 2024

a(n) = A000040(A022884(n)). - Amiram Eldar, May 06 2020

Name edited by Michel Marcus, Jan 15 2014

A261470 a(n) = prime(n+3) - prime(n+2) - prime(n+1) + prime(n).

Original entry on oeis.org

1, 2, 0, 0, 0, 0, 4, -2, 0, 2, -4, 0, 4, 2, -4, 0, 2, -4, 2, 2, 0, 4, -2, -6, 0, 0, 0, 12, 0, -8, -2, 4, 0, -4, 4, -2, 0, 2, -4, 4, 0, -6, 0, 8, 10, -8, -10, 0, 4, -2, 4, 4, -4, 0, -4, 0, 2, -4, 6, 12, -6, -12, 0, 12, 2, -4, -4, -6, 4, 4, 0, -2, -2, 0, 4, -2, 0

Offset: 1

Altug Alkan, Aug 20 2015

			a(5) = 19 - 17 - 13 + 11 = 0.

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

Cf. A022884, A022885, A093667.

Cf. A001223.

a261470 n = a261470_list !! (n-1)
a261470_list = zipWith (-) (drop 2 a001223_list) a001223_list
-- Reinhard Zumkeller, Aug 22 2015

Table[Prime[i+3] - Prime[i+2] - Prime[i+1] + Prime[i], {i,100}] (* G. C. Greubel, Aug 20 2015 *)

first(m)=vector(m,i,prime(i+3)+prime(i)-prime(i+1)-prime(i+2)) \\ Anders Hellström, Aug 20 2015

a(n) = A001223(n+2) - A001223(n). - Reinhard Zumkeller, Aug 22 2015

A351124 a(n) is the least k > 0 such that the set { prime(n), ..., prime(n+k-1) } can be partitioned into two disjoint sets with equal sum, or -1 if no such k exists (prime(n) denotes the n-th prime number).

Original entry on oeis.org

3, 6, 4, 4, 4, 4, 8, 10, 4, 8, 8, 4, 10, 14, 6, 4, 6, 6, 8, 8, 4, 8, 12, 10, 4, 4, 4, 8, 4, 8, 6, 10, 4, 6, 8, 18, 4, 6, 8, 6, 4, 12, 4, 8, 10, 6, 10, 4, 8, 6, 8, 12, 10, 4, 6, 4, 8, 8, 10, 8, 12, 8, 4, 12, 6, 6, 8, 8, 14, 8, 4, 8, 10, 4, 10, 6, 4, 10, 8, 4, 4

Offset: 1

Rémy Sigrist, Feb 02 2022

nonn

Conjecture: all terms are positive.

			The first terms, alongside an appropriate partition {P, Q}, are:
  n   a(n)  P                     Q
  --  ----  --------------------  --------------------
   1     3  {2, 3}                {5}
   2     6  {3, 5, 7, 13}         {11, 17}
   3     4  {5, 13}               {7, 11}
   4     4  {7, 17}               {11, 13}
   5     4  {11, 19}              {13, 17}
   6     4  {13, 23}              {17, 19}
   7     8  {17, 29, 31, 43}      {19, 23, 37, 41}
   8    10  {19, 31, 41, 47, 53}  {23, 29, 37, 43, 59}
   9     4  {23, 37}              {29, 31}
  10     8  {29, 41, 47, 53}      {31, 37, 43, 59}

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A022884.

a(n) = { my (s=[0], k=0); forprime (p=prime(n), oo, s=setunion(apply (v -> v-p, s), apply (v -> v+p, s)); k++; if (setsearch(s, 0), return (k))) }

a(n) = 4 iff n belongs to A022884.

cp's OEIS Frontend

A022885 Primes p=prime(k) such that prime(k) + prime(k+3) = prime(k+1) + prime(k+2).

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A261470 a(n) = prime(n+3) - prime(n+2) - prime(n+1) + prime(n).

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A351124 a(n) is the least k > 0 such that the set { prime(n), ..., prime(n+k-1) } can be partitioned into two disjoint sets with equal sum, or -1 if no such k exists (prime(n) denotes the n-th prime number).

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Programs

PARI

Formula