A112540 -id:A112540 - cp's OEIS frontend

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

Juri-Stepan Gerasimov, Feb 07 2010

nonn

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

			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.

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

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

[ 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

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 *)

is(n)=isprime(n-4) && isprime(n-2) && isprime(n+2) && isprime(n+4) \\ Charles R Greathouse IV, Sep 24 2015

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

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)

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

A176002 Numbers n such that 15*prime(n)+{-4,-2,2,4} are all primes.

Original entry on oeis.org

4, 6, 34, 176, 608, 1023, 1338, 1377, 1555, 1980, 2054, 2850, 2893, 3061, 3263, 3572, 3977, 4029, 4244, 4405, 6099, 6548, 7203, 7348, 7350, 7572, 7574, 9028, 10657, 11976, 12215, 12874, 13247, 13388, 13432, 14537, 14813, 15115, 15412, 15509

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2010

nonn

Numbers n such that 15*prime(n)-4, 15*prime(n)-2, 15*prime(n)+2 and 15*prime(n)+4 are primes.

			a(1)=4 because 15*prime(4)-4=101, 15*prime(4)-2=103, 15*prime(4)+2=107 and 15*prime(4)+4=109.

Cf. A112540, A173037, A173092.

p15Q[n_]:=And@@PrimeQ/@(15 Prime[n]+{-4,-2,2,4}); Select[Range[16000], p15Q]  (* Harvey P. Dale, Mar 20 2011 *)

A000040(a(n))=A112540(k).

More terms from R. J. Mathar, Apr 16 2010

A215473 Number of prime quadruples with smallest member < 2^n.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 3, 4, 4, 5, 7, 10, 11, 16, 23, 28, 43, 62, 106, 177, 309, 483, 795, 1305, 2105, 3525, 5923, 10096, 17259, 30004

Offset: 1

Alex Ratushnyak, Aug 12 2012

nonn

Prime quadruples (A007530) are numbers n such that n, n+2, n+6, n+8 are all prime.

			a(3) = 1 because there is only one prime quadruple below 2^3, namely {5, 7, 11, 13}.
a(4) = 2 because there are two prime quadruples below 2^4: the aforementioned and {11, 13, 17, 19}.

Cf. A050258, similar definition but with powers of 10 instead of 2.
Cf. A007530, A014561, A112540.
Cf. A033843, A007053.

(* First run program for A007530 *) Table[Length[Select[A007530, # < 2^n &]], {n, 14}] (* Alonso del Arte, Aug 12 2012 *)

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

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A176002 Numbers n such that 15*prime(n)+{-4,-2,2,4} are all primes.

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A215473 Number of prime quadruples with smallest member < 2^n.

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Mathematica