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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

12, 18, 24, 28, 36, 42, 48, 52, 58, 66, 72, 78, 84, 88, 94, 102, 108, 114, 118, 132, 136, 144, 156, 162, 168, 172, 178, 186, 198, 204, 216, 228, 234, 238, 246, 256, 262, 268, 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
Offset: 1

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Author

David James Sycamore, Jan 08 2018

Keywords

Comments

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).
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).
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.

Examples

			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.
18 is a term because 18 - 5 = 13 is prime, and 18 - 3 = 15 is composite.
16 is not a term because 16 - 5 = 11 and 16 - 3 = 13 are both prime.

Links

Crossrefs

Similar to A130038. Subsequence of A175222.
Cf. A049591, A107986.

Programs

  • GAP
    Filtered([8..500], k-> IsPrime(k-5) and not IsPrime(k-3) and (k mod 2)=0); # G. C. Greubel, May 21 2019
  • 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
  • Maple
    N:=100
for n from 8 to N by 2 do
if isprime(n-5) and not isprime(n-3) then print (n);
end if
end do
  • Mathematica
    Select[Range[6, 500, 2], And[PrimeQ[# - 5], ! PrimeQ[# - 3]] &] (* Michael De Vlieger, Jan 10 2018 *)
Select[Range[6, 500, 2], Boole[PrimeQ[# -{5, 3}]] == {1, 0} &] (* Harvey P. Dale, Jan 30 2024 *)
  • PARI
    isok(n) = !(n % 2) && isprime(n-5) && !isprime(n-3); \\ Michel Marcus, Jan 09 2018
  • 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

Formula

a(n) = A049591(n) + 5 = A107986(n+1) + 3 for all n >= 1.
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