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A297925 Even numbers k such that k - 5 is prime but k - 3 is not prime.

%I A297925 #54 Mar 11 2025 12:30:00
%S A297925 12,18,24,28,36,42,48,52,58,66,72,78,84,88,94,102,108,114,118,132,136,
%T A297925 144,156,162,168,172,178,186,198,204,216,228,234,238,246,256,262,268,
%U A297925 276,282,288,298,312,318,322,336,342,354,358,364,372,378,384,388,394,402,406,414,426,438,444,448,454
%N A297925 Even numbers k such that k - 5 is prime but k - 3 is not prime.
%C A297925 Even numbers that are the sum of 5 and another prime, but not the sum of 3 and another prime. For n >= 1, a(n) - 5 = A049591(n), a(n) - 3 = A107986(n+1).
%C A297925 Let r(n) = a(n) - 5, Then r(n) is the greatest prime < a(n), and therefore A056240(a(n)) = 5*r(n). Furthermore, since r(n) + 2 must be composite, A056240(a(n)) = 5*A049591(n).
%C A297925 The terms in this sequence, combined with those in A298366 and A298252 form a partition of A005843(n);n>=3 (nonnegative even numbers>=6). This is because any even integer n>=6 satisfies either (i) n-3 is prime, (ii) n-5 is prime but n-3 is composite, or (iii) both n-5 and n-3 are composite.
%H A297925 G. C. Greubel, <a href="/A297925/b297925.txt">Table of n, a(n) for n = 1..10000</a>
%F A297925 a(n) = A049591(n) + 5 = A107986(n+1) + 3 for all n >= 1.
%e A297925 12 is a term because 12 - 5 = 7 is prime, and 12 - 3 = 9 is composite. Also A049591(1)+5=7+5=12 and A107986(2)+3=9+3=12.
%e A297925 18 is a term because 18 - 5 = 13 is prime, and 18 - 3 = 15 is composite.
%e A297925 16 is not a term because 16 - 5 = 11 and 16 - 3 = 13 are both prime.
%p A297925 N:=100
%p A297925 for n from 8 to N by 2 do
%p A297925 if isprime(n-5) and not isprime(n-3) then print (n);
%p A297925 end if
%p A297925 end do
%t A297925 Select[Range[6, 500, 2], And[PrimeQ[# - 5], ! PrimeQ[# - 3]] &] (* _Michael De Vlieger_, Jan 10 2018 *)
%t A297925 Select[Range[6, 500, 2], Boole[PrimeQ[# -{5, 3}]] == {1, 0} &] (* _Harvey P. Dale_, Jan 30 2024 *)
%o A297925 (PARI) isok(n) = !(n % 2) && isprime(n-5) && !isprime(n-3); \\ _Michel Marcus_, Jan 09 2018
%o A297925 (Magma) [n: n in [3..500] | IsPrime(n-5) and not IsPrime(n-3) and (n mod 2) eq 0]; // _G. C. Greubel_, May 21 2019
%o A297925 (Sage) [n for n in (3..500) if is_prime(n-5) and not is_prime(n-3) and (mod(n, 2)==0)] # _G. C. Greubel_, May 21 2019
%o A297925 (GAP) Filtered([8..500], k-> IsPrime(k-5) and not IsPrime(k-3) and (k mod 2)=0); # _G. C. Greubel_, May 21 2019
%Y A297925 Similar to A130038. Subsequence of A175222.
%Y A297925 Cf. A049591, A107986.
%K A297925 nonn
%O A297925 1,1
%A A297925 _David James Sycamore_, Jan 08 2018