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A297925 Even numbers k such that k - 5 is prime but k - 3 is not prime. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(PARI) isok(n) = !(n % 2) && isprime(n-5) && !isprime(n-3); \\ Michel Marcus, Jan 09 2018

(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

(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

(GAP) Filtered([8..500], k-> IsPrime(k-5) and not IsPrime(k-3) and (n mod 2)=0) # G. C. Greubel, May 21 2019

CROSSREFS

Similar to A130038. Subsequence of A175222.

Cf. A049591, A107986.

Sequence in context: A280014 A162151 A056773 * A341099 A175837 A136446

Adjacent sequences:  A297922 A297923 A297924 * A297926 A297927 A297928

KEYWORD

nonn

AUTHOR

David James Sycamore, Jan 08 2018

STATUS

approved

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)