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.
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
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Feb 07 2010
EXTENSIONS
Edited and extended beyond a(9) by Klaus Brockhaus, Feb 09 2010
STATUS
approved