

A007530


Prime quadruples: numbers k such that k, k+2, k+6, k+8 are all prime.
(Formerly M3816)


109



5, 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, 9431, 13001, 15641, 15731, 16061, 18041, 18911, 19421, 21011, 22271, 25301, 31721, 34841, 43781, 51341, 55331, 62981, 67211, 69491, 72221, 77261, 79691, 81041, 82721, 88811, 97841, 99131
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OFFSET

1,1


COMMENTS

Except for the first term, 5, all terms == 11 (mod 30).  Zak Seidov, Dec 04 2008
Some further values: For k = 1, ..., 10, a(k*10^3) = 11721791, 31210841, 54112601, 78984791, 106583831, 136466501, 165939791, 196512551, 230794301, 265201421.  M. F. Hasler, May 04 2009
k is the first prime of 2 consecutive twin prime pairs.  Daniel Forgues, Aug 01 2009
The prime quadruples of form p + (0, 2, 6, 8) have the quadruple congruence class (1, +1, 1, +1) (mod 6).  Daniel Forgues, Aug 12 2009
s = (p+8)(p) = 8 is the smallest s giving an admissible prime quadruple form, for which the only admissible form is p + (0, 2, 6, 8), since (0, 2, 6, 8) is the only form not covering all the congruence classes for any prime <= 4. Since s is smallest, these prime quadruples are prime constellations (or prime quadruplets), i.e., they contain consecutive primes.  Daniel Forgues, Aug 12 2009
Except for the first term, 5, all prime quadruples are of the form (15k4, 15k2, 15k+2, 15k+4), with k >= 1, and so are centered on 15k.  Daniel Forgues, Aug 12 2009
Solutions of the equation n'+(n+2)'+(n+6)'+(n+8)'=4, where n' is the arithmetic derivative of n'.  Paolo P. Lava, Nov 09 2012
Subsequence of A022004.  R. J. Mathar, Feb 10 2013
The quadruplets are listed in A136162.  M. F. Hasler, Apr 20 2013
Starting at a(2) and examining the first 50 terms, (a(n)+4)/15 is a prime in 8 cases and a semiprime in 21; the last 18 terms have 2 primes and 11 semiprimes. Do the number of semiprimes continue to occur greater than mere chance?  J. M. Bergot, Apr 27 2015


REFERENCES

H. Rademacher, Lectures on Elementary Number Theory. Blaisdell, NY, 1964, p. 4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe).
C. K. Caldwell, The Prime Glossary, prime quadruple
Tony Forbes and Norman Luhn, prime ktuplets
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Norman Luhn, Table of n, a(n) for n = 1..1000000
Thomas R. Nicely, Enumeration to 1.6e15 of the prime quadruplets
H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, ISBN: 9780817682972, Chap. 4, see p. 65.
Eric Weisstein's World of Mathematics, Prime Quadruplet


FORMULA

a(n) = 11 + 30*A014561(n1) for n > 1.  M. F. Hasler, May 04 2009


EXAMPLE

From M. F. Hasler, May 04 2009: (Start)
a(1)=5 is the start of the first prime quadruplet, {5,7,11,13}.
a(2)=11 is the start of the second prime quadruplet, {11,13,17,19}, and all other prime quadruplets differ from this one by a multiple of 30.
a(100)=470081 is the start of the 100th prime quadruplet;
a(500)=4370081 is the start of the 500th prime quadruplet.
a(167)=1002341 is the least quadruplet prime beyond 10^6. (End)


MAPLE

A007530:=proc(q)
local n;
for n from 1 to q do
if isprime(n) and isprime (n+2) and isprime(n+6) and isprime (n+8) then print(n); fi;
od; end:
A007530(10000000000); # Paolo P. Lava, Jan 30 2013


MATHEMATICA

A007530 = Select[Range[1, 10^5  1, 2], Union[PrimeQ[# + {0, 2, 6, 8}]] == {True} &] (* Alonso del Arte, Sep 24 2011 *)
Select[Prime[Range[10000]], AllTrue[#+{2, 6, 8}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 11 2019 *)


PROG

(PARI) A007530( n, print_all=0, s=2 )={ my(p, q, r); until(!n, until( p+8==s=nextprime(s+2), p=q; q=r; r=s); print_all && print1(p", ")); p} \\ The optional 3rd argument can be used to obtain large values by starting from some precomputed point instead of zero, using a(n+k) = A007530(k+1, , a(n)) (or A007530(k, , a(n)1) for k>0); e.g., you get a(10^4+k) using A007530(k+1, , 265201421) (value of a(10^4) from the comments section).  M. F. Hasler, May 04 2009
(PARI) forprime(p=2, 10^5, if(isprime(p+2) && isprime(p+6) && isprime(p+8), print1(p, ", "))) \\ Felix Fröhlich, Jun 22 2014
(Magma) [ p: p in PrimesUpTo(11000) IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8)] // Vincenzo Librandi, Nov 18 2010
(Python)
from sympy import primerange
def aupto(limit):
p, q, r, alst = 2, 3, 5, []
for s in primerange(7, limit+9):
if p+2 == q and p+6 == r and p+8 == s: alst.append(p)
p, q, r = q, r, s
return alst
print(aupto(10**5)) # Michael S. Branicky, May 11 2021


CROSSREFS

Cf. A159910 (first differences divided by 30), A120120, A007811, A014561.
Sequence in context: A199325 A199305 A096473 * A157967 A088268 A030085
Adjacent sequences: A007527 A007528 A007529 * A007531 A007532 A007533


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Robert G. Wilson v


EXTENSIONS

More terms from Warut Roonguthai
Incorrect formula and Mathematica program removed by N. J. A. Sloane, Dec 04 2008, at the suggestion of Zak Seidov
Values up to a(1000) checked with the given PARI code by M. F. Hasler, May 04 2009


STATUS

approved



