A298252 Even integers n such that n-3 is prime.

6, 8, 10, 14, 16, 20, 22, 26, 32, 34, 40, 44, 46, 50, 56, 62, 64, 70, 74, 76, 82, 86, 92, 100, 104, 106, 110, 112, 116, 130, 134, 140, 142, 152, 154, 160, 166, 170, 176, 182, 184, 194, 196, 200, 202, 214, 226, 230, 232, 236, 242, 244, 254, 260, 266, 272, 274, 280

Offset: 1

Author

David James Sycamore, Jan 15 2018

Keywords

• nonn
• easy
• less

Comments

Subsequence of A005843, same as A113935 with first term (5) excluded, since it is odd, not even. Index in A056240 of terms in A288313 (except for first two terms 2,4 of latter).

The terms in this sequence, combined with those in A297925 and A298366 form a partition of A005843(n); n>=3 (nonnegative numbers>=6). This is because any even integer n>=6 satisfies either(i) n-3 is prime, (ii) n-5 prime but n-3 composite, or (iii) n-5 and n-3 both composite.

a(n) is the smallest even number e > prime(n+1) such that e has a Goldbach partition containing prime(n+1). - Felix Fröhlich, Aug 18 2019

Examples

			a(1)=6 because 6-3=3; prime, and no smaller even number has this property; also a(1)=A113935(2)=6.  a(2)=8 because 8-3=5 is prime; also A113935(3)=8.
12 is not in the sequence because 12-3 = 9, composite.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A005843, A056240, A113935, A288313.

Programs

GAP

Filtered([1..300],n->IsEvenInt(n) and IsPrime(n-3)); # Muniru A Asiru, Mar 23 2018

Magma

[NthPrime(n+1) +3: n in [1..70]]; // G. C. Greubel, May 21 2019

Maple

N:=200
  for n from 6 to N by 2 do
if isprime(n-3) then print(n);
end if
end do

Mathematica

Select[2 Range@125, PrimeQ[# - 3] &] (* Robert G. Wilson v, Jan 15 2018 *)
Select[Prime[Range[100]]+3,EvenQ] (* Harvey P. Dale, Mar 07 2022 *)

PARI

a(n) = prime(n + 1) + 3 \\ David A. Corneth, Mar 23 2018

Sage

[nth_prime(n+1) +3 for n in (1..70)] # G. C. Greubel, May 21 2019

Formula

a(n) = A113935(n+1), n>=1.

A056240(a(n)) = A288313(n+2).

a(n) = prime(n + 1) + 3 = A113935(n + 1). - David A. Corneth, Mar 23 2018