A173037 Numbers k such that k-4, k-2, k+2 and k+4 are prime.

9, 15, 105, 195, 825, 1485, 1875, 2085, 3255, 3465, 5655, 9435, 13005, 15645, 15735, 16065, 18045, 18915, 19425, 21015, 22275, 25305, 31725, 34845, 43785, 51345, 55335, 62985, 67215, 69495, 72225, 77265, 79695, 81045, 82725, 88815, 97845

Offset

1,1

Comments

Average k of the four primes in two twin prime pairs (k-4, k-2) and (k+2, k+4) which are linked by the cousin prime pair (k-2, k+2).

All terms are odd composites; except for a(1) they are multiples of 5.

Subsequence of A087679, of A087680 and of A164385.

All terms except for a(1) are multiples of 15. - Zak Seidov, May 18 2014

One of (k-1, k, k+1) is always divisible by 7. - Fred Daniel Kline, Sep 24 2015

Terms other than a(1) must be equivalent to 1 mod 2, 0 mod 3, 0 mod 5, and 0,+/-1 mod 7. Taken together, this requires terms other than a(1) to have the form 210k+/-15 or 210k+105. However, not all numbers of that form belong to this sequence. - Keith Backman, Nov 09 2023

Links

Klaus Brockhaus, Table of n, a(n) for n = 1..28388 (terms < 10^9).

Formula

For n >= 2, a(n) = 15*A112540(n-1). - Michel Marcus, May 19 2014

From Jeppe Stig Nielsen, Feb 18 2020: (Start)

For n >= 2, a(n) = 30*A014561(n-1) + 15.

For n >= 2, a(n) = 10*A007811(n-1) + 5.

a(n) = A007530(n) + 4.

a(n) = A125855(n) + 5. (End)

Example

9 is a term because 9-4 = 5 is prime, 9-2 = 7 is prime, 9+2 = 11 is prime and 9+4 = 13 is prime.

Mathematica

Select[Range[100000], AllTrue[#+{4, 2, -2, -4}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 30 2015 *)

Prog

(Magma) [ p+4: p in PrimesUpTo(100000) | IsPrime(p) and IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8) ]; // Klaus Brockhaus, Feb 09 2010
(PARI) is(n)=isprime(n-4) && isprime(n-2) && isprime(n+2) && isprime(n+4) \\ Charles R Greathouse IV, Sep 24 2015
(Python)
from sympy import primerange
def aupto(limit):
    p, q, r, alst = 2, 3, 5, []
    for s in primerange(7, limit+5):
        if p+2 == q and p+6 == r and p+8 == s: alst.append(p+4)
        p, q, r = q, r, s
    return alst
print(aupto(10**5)) # Michael S. Branicky, Feb 03 2022

Crossrefs

Cf. A001359, A006512, A007530, A007811, A014561, A023200, A038800, A046132, A087680, A112540, A125855, A164385.

Sequence in context: A219129 A226927 A152219 * A029712 A136353 A136354

Adjacent sequences: A173034 A173035 A173036 * A173038 A173039 A173040

Keyword

nonn

Author

Juri-Stepan Gerasimov, Feb 07 2010

Extensions

Edited and extended beyond a(9) by Klaus Brockhaus, Feb 09 2010

Status

approved