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

%I A298252 #30 Sep 08 2022 08:46:20
%S A298252 6,8,10,14,16,20,22,26,32,34,40,44,46,50,56,62,64,70,74,76,82,86,92,
%T A298252 100,104,106,110,112,116,130,134,140,142,152,154,160,166,170,176,182,
%U A298252 184,194,196,200,202,214,226,230,232,236,242,244,254,260,266,272,274,280
%N A298252 Even integers n such that n-3 is prime.
%C A298252 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).
%C A298252 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.
%C A298252 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
%H A298252 Muniru A Asiru, <a href="/A298252/b298252.txt">Table of n, a(n) for n = 1..5000</a>
%F A298252 a(n) = A113935(n+1), n>=1.
%F A298252 A056240(a(n)) = A288313(n+2).
%F A298252 a(n) = prime(n + 1) + 3 = A113935(n + 1). - _David A. Corneth_, Mar 23 2018
%e A298252 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.
%e A298252 12 is not in the sequence because 12-3 = 9, composite.
%p A298252 N:=200
%p A298252   for n from 6 to N by 2 do
%p A298252 if isprime(n-3) then print(n);
%p A298252 end if
%p A298252 end do
%t A298252 Select[2 Range@125, PrimeQ[# - 3] &] (* _Robert G. Wilson v_, Jan 15 2018 *)
%t A298252 Select[Prime[Range[100]]+3,EvenQ] (* _Harvey P. Dale_, Mar 07 2022 *)
%o A298252 (PARI) a(n) = prime(n + 1) + 3 \\ _David A. Corneth_, Mar 23 2018
%o A298252 (GAP) Filtered([1..300],n->IsEvenInt(n) and IsPrime(n-3)); # _Muniru A Asiru_, Mar 23 2018
%o A298252 (Magma) [NthPrime(n+1) +3: n in [1..70]]; // _G. C. Greubel_, May 21 2019
%o A298252 (Sage) [nth_prime(n+1) +3 for n in (1..70)] # _G. C. Greubel_, May 21 2019
%Y A298252 Cf. A005843, A056240, A113935, A288313.
%K A298252 nonn,easy,less
%O A298252 1,1
%A A298252 _David James Sycamore_, Jan 15 2018