A125855 -id:A125855 - cp's OEIS frontend

A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

1, 10, 19, 82, 148, 187, 208, 325, 346, 565, 943, 1300, 1564, 1573, 1606, 1804, 1891, 1942, 2101, 2227, 2530, 3172, 3484, 4378, 5134, 5533, 6298, 6721, 6949, 7222, 7726, 7969, 8104, 8272, 8881, 9784, 9913, 10111, 10984, 11653, 11929, 12220, 13546, 14416, 15727

Offset: 1

N. J. A. Sloane and J. H. Conway, Mar 15 1996

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A024912, A102338, A102342, A102700, A007530, A014561, A008471, A032352, A216292, A216293, A125855.

Cf. A010051, A245304, A245305.

a007811 n = a007811_list !! (n-1)
a007811_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [10, 10, 10, 10]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n: n in [0..10000] | forall{10*n+r: r in [1,3,7,9] | IsPrime(10*n+r)}]; // Bruno Berselli, Sep 04 2012

for n from 1 to 10000 do m := 10*n: if isprime(m+1) and isprime(m+3) and isprime(m+7) and isprime(m+9) then print(n); fi: od: quit

Select[ Range[ 1, 10000, 3 ], PrimeQ[ 10*#+1 ] && PrimeQ[ 10*#+3 ] && PrimeQ[ 10*#+7 ] && PrimeQ[ 10*#+9 ]& ]
Select[Range[15000], And @@ PrimeQ /@ ({1, 3, 7, 9} + 10#) &] (* Ray Chandler, Jan 12 2007 *)

p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==8 && r-p==6 && q-p==2 && p%10==1, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Mar 21 2013

use ntheory ":all"; my @s = map { ($-1)/10 } sieve_prime_cluster(10,1e9, 2,6,8); say for @s; # _Dana Jacobsen, May 04 2017

a(n) = 3*A014561(n) + 1. - Zak Seidov, Sep 21 2009

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

Original entry on oeis.org

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

A245304 Numbers m such that m+1, m+3, m+7, m+9 and m+13 are all primes.

Original entry on oeis.org

4, 10, 100, 1480, 16060, 19420, 21010, 22270, 43780, 55330, 144160, 165700, 166840, 195730, 201820, 225340, 247600, 268810, 326140, 347980, 361210, 397750, 465160, 518800, 536440, 633460, 633790, 661090, 768190, 795790, 829720, 857950, 876010, 958540

Offset: 1

Reinhard Zumkeller, Jul 18 2014

nonn

W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #82, variant.

Reinhard Zumkeller and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 120 terms from Zumkeller)

Cf. A010051, A022006, A245305, A007811, subsequence of A125855.

a245304 n = a245304_list !! (n-1)
a245304_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [1, 1, 1, 1, 1]) [1, 3, 7, 9, 13]

[n: n in [0..10^6] | IsPrime(n+1) and IsPrime(n+3) and IsPrime(n+7) and IsPrime(n+9) and IsPrime(n+13)]; // Vincenzo Librandi, Jun 15 2015

Select[Range[10^6],AllTrue[#+{1,3,7,9,13},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 07 2015 *)

forprime(p=2, 10^7, m=p-1; if(isprime(m+3)&&isprime(m+7)&&isprime(m+9)&&isprime(m+13), print1(m", "))) \\ Jens Kruse Andersen, Jul 18 2014

a(n) = A022006(n)-1. - Jens Kruse Andersen, Jul 18 2014

A245305 Numbers k such that 4k+1, 4k+3, and 6k+5 are all primes.

Original entry on oeis.org

1, 4, 7, 37, 142, 154, 202, 214, 307, 424, 469, 487, 499, 559, 577, 664, 742, 814, 847, 979, 982, 1054, 1129, 1159, 1162, 1252, 1369, 1522, 1612, 1642, 1672, 1837, 1987, 2107, 2134, 2149, 2209, 2242, 2359, 2407, 2419, 2482, 2632, 2677, 2767, 2887, 2929, 2944

Offset: 1

Reinhard Zumkeller, Jul 18 2014

nonn

Sequence is infinite (Sierpiński).

Infinitude of the sequence would follow from Dickson's (unproved) conjecture. - Jens Kruse Andersen, Jul 18 2014

W. Sierpiński, A Selection of Problems in the Theory of Numbers. Pergamon, 1964, p. 52, #15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A010051, A007811, A125855, A245304.

a245305 n = a245305_list !! (n-1)
a245305_list = map ((`div` 4) . (subtract 1) . head) $
   filter (all (== 1) . map a010051') $
          iterate (zipWith (+) [4, 4, 6]) [1, 3, 5]

[n: n in [0..3*10^3] | IsPrime(4*n+1) and IsPrime(4*n+3) and IsPrime(6*n+5)]; // Vincenzo Librandi, Jun 15 2015

Select[Range[0, 3000], PrimeQ[4 # + 1] && PrimeQ[4 # + 3] && PrimeQ[6 # + 5] &] (* Vincenzo Librandi, Jun 15 2015 *)

isok(k) = isprime(4*k+1) && isprime(4*k+3) && isprime(6*k+5); \\ Michel Marcus, Jan 24 2022

A125856 a(n) = least number k such that k^(2^n)+1, k^(2^n)+3, k^(2^n)+7 and k^(2^n)+9 are all prime.

Original entry on oeis.org

4, 2, 83270, 5241160, 57171410, 359829200

Offset: 0

Artur Jasinski, Dec 12 2006

nonn

In 1958, Schinzel showed that for each n>0, there are infinitely many primes among the numbers k^(2^n)+{1,3,7, or 9}.

Sierpinski, W. Elementary theory of numbers. Warszawa 1964 Monografie Matematyczne Vol. 42.

Cf. A125855, A057015, A125779, A125780.

a(n) = {k = 1; while(!isprime(k^(2^n)+1) || !isprime(k^(2^n)+3) || !isprime(k^(2^n)+7) || !isprime(k^(2^n)+9), k++); k;} \\ Michel Marcus, Nov 03 2013

Edited by Don Reble, Dec 16 2006

One more term from Farideh Firoozbakht, Jan 01 2007

cp's OEIS Frontend

A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Perl

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

A245304 Numbers m such that m+1, m+3, m+7, m+9 and m+13 are all primes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

A245305 Numbers k such that 4k+1, 4k+3, and 6k+5 are all primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

A125856 a(n) = least number k such that k^(2^n)+1, k^(2^n)+3, k^(2^n)+7 and k^(2^n)+9 are all prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

PARI

Extensions