A239392 -id:A239392 - cp's OEIS frontend

A020483 Least prime p such that p+2n is also prime.

2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13, 11, 31, 7

Offset: 0

David W. Wilson

nonn

It is conjectured that a(n) always exists. a(n) has been computed for n < 5 * 10^11, with largest value a(248281210271) = 3307. - Jens Kruse Andersen, Nov 28 2004

If a(n) = a(n+1) = k, then 2*n + k and 2*(n+1) + k are twin primes. - Ya-Ping Lu, Sep 22 2020

			Given n = 2, we see that 2 + 2n = 6 = 2 * 3, but 3 + 2n = 7, which is prime, so a(2) = 3.
Given n = 3, we see that 2 + 2n = 8 = 2^3 and 3 + 2n = 9 = 3^2, but 5 + 2n = 11, which is prime, so a(3) = 5.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jens Kruse Andersen, Prime gaps (not necessarily consecutive), Yahoo! group "primenumbers", Nov 26 2004.
Jens Kruse Andersen, Mike Oakes, Ed Pegg Jr, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004.

Cf. A087711, A101042, A101043, A101044, A101045, A101046.
Cf. A101045, A239392 (record values).
Cf. A000040, A010051, A020484, A237055.
It is likely that A054906 is an identical sequence, although this seems to have not yet been proved. - N. J. A. Sloane, Feb 06 2017

P:=Filtered([1..10000],IsPrime);;
a:=List(List([0..110],n->Filtered(P,i->IsPrime(i+2*n))),Minimum); # Muniru A Asiru, Mar 26 2018

a020483 n = head [p | p <- a000040_list, a010051' (p + 2 * n) == 1]
-- Reinhard Zumkeller, Nov 29 2014

A020483 := proc(n)
    local p;
    p := 2;
    while true do
        if isprime(p+2*n) then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
seq(A020483(n),n=0..40); # R. J. Mathar, Sep 23 2016

Table[j = 1; found = False; While[!found, j++; found = PrimeQ[Prime[j] + 2i]]; Prime[j], {i, 200}]
leastPrimep2n[n_] := Block[{k = 1, p, q = 2 n}, While[p = Prime@k; !PrimeQ[p + q], k++]; p]; Array[leastPrimep2n, 102] (* Robert G. Wilson v, Mar 26 2008 *)

a(n)=forprime(p=2,,if(isprime(p+2*n), return(p))) \\ Charles R Greathouse IV, Mar 19 2014

If a(n) exists, a(n) < 2n, which of course is a great overestimate. - T. D. Noe, Jul 16 2002
a(n) = A087711(n) - n. - Zak Seidov, Nov 28 2007
a(n) = A020484(n) - 2n. - Zak Seidov, May 29 2014
a(n) = 2 if and only if n = 0. - Alonso del Arte, Mar 14 2018

a(0)=2 added by N. J. A. Sloane, Apr 25 2015

A370998 2*a(n) = m is the least even number m such that all sums m + prime(k), k=1..n are composite.

Original entry on oeis.org

1, 3, 11, 44, 44, 56, 56, 101, 101, 101, 359, 359, 359, 664, 664, 821, 821, 821, 866, 866, 866, 2623, 2623, 2623, 2623, 2944, 2944, 2944, 2944, 2944, 2944, 2944, 2944, 2944, 5171, 5171, 12839, 18833, 18833, 18833, 18833, 29947, 29947, 29947, 38002, 38002, 38002, 38002, 51551

Offset: 1

Hugo Pfoertner, Mar 09 2024

nonn

			a(1) = 1: prime(1) = 2; 2 + 2*a(1) = 4 is the first composite.
a(2) = 3: m = 6; since all sums prime(1) + 2*x are even, any x can be chosen. prime(2) = 3, 3 + 6 = 9, whereas 3 + 1*2 and 3 + 2*2 are prime.
a(3) = 11: m = 22; for any even m < 22 at least one of 3 + m or 5 + m would be prime, e.g., 3+2=5, 3+4=7, 5+6=11, 3+8=11, 5+12=17, 5+14=19, 3+16=19, 5+18=23, 3+20=23, but 3+22=25 and 5+22 are composite.

Robert Israel, Table of n, a(n) for n = 1..273

Cf. A239392 (records).

R:= 1: P:= [2]: r:= 1:
for n from 2 to 100 do
  P:= [op(P), ithprime(n)];
  for k from r while ormap(isprime,P +~ 2*k) do od:
  R:= R, k; r:= k;
od:
R; # Robert Israel, Mar 09 2025

from itertools import count
from sympy import prime, isprime
def A370998(n):
    ptuple = tuple(prime(k) for k in range(1,n+1))
    return next(filter(lambda m:not any(isprime(p+m) for p in ptuple),count(2,2)))>>1 # Chai Wah Wu, Mar 21 2024

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A020483 Least prime p such that p+2n is also prime.

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A370998 2*a(n) = m is the least even number m such that all sums m + prime(k), k=1..n are composite.

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A371069 Positions of records in A370998.

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