A039949 -id:A039949 - cp's OEIS frontend

A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

1, 12, 1, 6, 1, 3, 1, 6, 2, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 2, 1, 2, 6, 1, 1, 1, 1, 1, 6, 2, 1, 2, 4, 3, 2, 2, 4, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 3, 2, 2, 4, 2, 2, 1, 1, 4, 2, 1, 1, 4, 1, 3, 2, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

nonn

Primes can be classified according to their remainder modulo 30: remainder 1 (A136066), 7 (A132231), 11 (A132232), 13 (A132233), 17 (A039949), 19 (A132234), 23 (A132235), or 29 (A132236). In the sequence A007775 of all numbers (prime or nonprime) in any of these remainder classes, we look for runs of numbers that are successively prime or nonprime and place the lengths of these runs in this sequence.

			Groups of runs in A007775 are (1), (7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47), (49), (53, 59, 61, 67, 71, 73), (77), (79, 83,...), which is 1 nonprime followed by 12 primes followed by 1 nonprime followed by 6 primes etc.

Cf. A136066, A132231, A132232, A132233, A039919, A132234, A132235, A132236.

A007775 := proc(n) option remember ; local a; if n = 1 then 1; else for a from A007775(n-1)+1 do if a mod 2 <>0 and a mod 3 <>0 and a mod 5 <> 0 then RETURN(a) ; fi ; od: fi ; end: A := proc() local al,isp,n; al := 0: isp := false ; n := 1: while n< 300 do a := A007775(n) ; if isprime(a) <> isp then printf("%d,",al) ; al := 1; isp := not isp ; else al := al+1 ; fi ; n := n+1: od: end: A() ; # R. J. Mathar, Jun 16 2008

Edited by R. J. Mathar, Jun 16 2008

A140533 Primes congruent to 13 or 17 mod 30.

Original entry on oeis.org

13, 17, 43, 47, 73, 103, 107, 137, 163, 167, 193, 197, 223, 227, 257, 283, 313, 317, 347, 373, 433, 463, 467, 523, 557, 587, 613, 617, 643, 647, 673, 677, 733, 797, 823, 827, 853, 857, 883, 887, 947, 977, 1033, 1063, 1093, 1097, 1123, 1153, 1187, 1213, 1217

Offset: 1

Juri-Stepan Gerasimov, Jul 28 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A132233, A039949.

Cf. A000040.

[p: p in PrimesUpTo(3000)|p mod 30 in {13,17}]; // Vincenzo Librandi, Dec 18 2010

A140533 := proc(n) local a; if n = 1 then 13; else a := nextprime(procname(n-1)) ; while not a mod 30 in {13,17} do a := nextprime(a) ; end do: return a; end if; end: seq(A140533(n),n=1..80) ; # R. J. Mathar, Oct 22 2009

Select[Prime[Range[300]],MemberQ[{13,17},Mod[#,30]]&] (* Vincenzo Librandi, Aug 15 2012 *)

A229947 Primes congruent to {1, 11, 13, 17, 19, 29} mod 30.

Original entry on oeis.org

11, 13, 17, 19, 29, 31, 41, 43, 47, 59, 61, 71, 73, 79, 89, 101, 103, 107, 109, 131, 137, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 257, 269, 271, 281, 283, 311, 313, 317, 331, 347, 349, 359, 373, 379, 389

Offset: 1

Omar E. Pol, Oct 27 2013

For twin primes congruent to {1, 11, 13, 17, 19, 29} mod 30 see A132247.

Complement of A132237, primes congruent to 7 or 23 (mod 30), in the set of primes > 5. - M. F. Hasler, Nov 02 2013

Omar E. Pol, Prime numbers in a sieve with 30 columns

Cf. A000040, A001097, A039949, A132230, A132232-A132234, A132236, A132238-A132259, A230462.

[p: p in PrimesUpTo(500) | p mod 30 in {1,11,13,17, 19,29} ]; // Vincenzo Librandi, Apr 05 2015

Select[Flatten[Table[30n + {1, 11, 13, 17, 19, 29}, {n, 0, 11}]], PrimeQ] (* Alonso del Arte, Nov 01 2013 *)
Select[Prime@Range[100], MemberQ[{1, 11, 13, 17, 19, 29}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)

is(n)=isprime(n) && setsearch([1,11,13,17,19,29], n%30) \\ Charles R Greathouse IV, Mar 08 2015

a(n) ~ 4/3 n log n. - Charles R Greathouse IV, Mar 08 2015

A132239 Primes congruent to {17, 19} mod 30.

Original entry on oeis.org

17, 19, 47, 79, 107, 109, 137, 139, 167, 197, 199, 227, 229, 257, 317, 347, 349, 379, 409, 439, 467, 499, 557, 587, 617, 619, 647, 677, 709, 739, 769, 797, 827, 829, 857, 859, 887, 919, 947, 977, 1009, 1039, 1069, 1097, 1129, 1187

Offset: 1

Omar E. Pol, Aug 15 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A001097, A001359, A006512, A039949.

[ p: p in PrimesUpTo(1300) | p mod 30 in {17, 19} ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{17,19},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

A132246 Twin primes congruent to {1, 17, 19, 29} mod 30.

Original entry on oeis.org

17, 19, 29, 31, 59, 61, 107, 109, 137, 139, 149, 151, 179, 181, 197, 199, 227, 229, 239, 241, 269, 271, 347, 349, 419, 421, 569, 571, 599, 601, 617, 619, 659, 661, 809, 811, 827, 829, 857, 859, 1019, 1021, 1049, 1051, 1229, 1231

Offset: 1

Omar E. Pol, Aug 17 2007

C. K. Caldwell, The Prime Pages.
C. K. Caldwell, Twin Primes.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A001097, A001359, A006512, A039949, A129805.

Select[Union[Flatten[Select[Partition[Prime[Range[300]],2,1],#[[2]]- #[[1]] == 2&]]],MemberQ[{1,17,19,29},Mod[#,30]]&] (* Harvey P. Dale, Feb 09 2015 *)

A220081 Primes of the form 15k^2 - 15k + 17.

Original entry on oeis.org

17, 47, 107, 197, 317, 467, 647, 857, 1097, 1367, 1667, 1997, 2357, 3167, 3617, 5147, 5717, 6317, 6947, 7607, 8297, 9767, 12197, 13967, 14897, 18917, 19997, 21107, 22247, 23417, 25847, 27107, 29717, 33857, 36767, 41357, 51347, 53117, 54917, 56747, 60497

Offset: 1

Vincenzo Librandi, Dec 17 2012

The formula gives consecutive primes for k from 0 to 13.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A000040, A005846, A156252.

Subsequence of A030432, A039949, A141860.

[a: n in [1..100] | IsPrime(a) where a is 15*n^2 - 15*n + 17 ];

Select[Table[15 n^2 - 15 n + 17, {n, 1, 100}], PrimeQ]

A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

Original entry on oeis.org

5, 17, 617, 857, 1277, 1427, 1697, 2087, 2687, 3557, 4217, 5417, 5477, 7307, 8837, 9437, 10067, 13877, 17657, 18287, 20747, 21587, 23537, 25577, 27917, 28547, 30467, 32117, 32297, 35507, 37337, 37547, 40127, 41177, 41387, 41957

Offset: 1

Karl V. Keller, Jr., Jan 09 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n, n+2, n+24, n+26).
Excluding 5, this is a subsequence of each of the following: A128468 (a(n) = 30*n + 17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).
A253624 is a subsequence of this sequence.

			For n = 17, the numbers 17, 19, 41, 43 are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A253624.

a245568[n_] := Select[Prime@ Range@ n, And[PrimeQ[# + 2], PrimeQ[# + 24], PrimeQ[# + 26]] &]; a245568[5000] (* Michael De Vlieger, Jan 11 2015 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+24) and isprime(n+26): print(n,end=', ')

A248474 Numbers congruent to 13 or 17 mod 30.

Original entry on oeis.org

13, 17, 43, 47, 73, 77, 103, 107, 133, 137, 163, 167, 193, 197, 223, 227, 253, 257, 283, 287, 313, 317, 343, 347, 373, 377, 403, 407, 433, 437, 463, 467, 493, 497, 523, 527, 553, 557, 583, 587, 613, 617, 643, 647, 673, 677, 703, 707, 733, 737, 763, 767, 793, 797

Offset: 1

Karl V. Keller, Jr., Oct 06 2014

The combination of A082369(30*n+13) and A128468(30*n+17) is the base sequence for A140533(Primes congruent to 13 or 17 mod 30).

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A082369 (30*n+13), A128468 (30*n+17).

Cf. A039949 (Primes of the form 30n-13), A132233 (Primes congruent to 13 mod 30), A140533 (Primes congruent to 13 or 17 mod 30).

Flatten[Table[{15n - 2, 15n + 2}, {n, 1, 41, 2}]] (* Alonso del Arte, Oct 06 2014 *)

Vec(x*(13*x^2+4*x+13)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Oct 07 2014

for n in range(1,101):
  print (n*30-17),
  print (n*30-13),

From Colin Barker, Oct 07 2014: (Start)
a(n) = (-15-11*(-1)^n+30*n)/2.
a(n) = a(n-1)+a(n-2)-a(n-3).
G.f.: x*(13*x^2+4*x+13) / ((x-1)^2*(x+1)). (End)
E.g.f.: 13 + ((30*x - 15)*exp(x) - 11*exp(-x))/2. - David Lovler, Sep 10 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2*(5+sqrt(5)))+sqrt(3)-sqrt(15))*Pi / (30*(sqrt(6*(5+sqrt(5)))+sqrt(5)-1)). - Amiram Eldar, Jul 30 2024

A248523 Initial members of prime quadruples (n, n+2, n+144, n+146).

Original entry on oeis.org

5, 137, 1787, 1997, 2237, 2657, 3527, 4127, 4337, 4787, 8087, 12107, 13757, 14447, 17987, 19697, 21377, 23057, 23687, 31247, 32297, 34157, 34367, 35447, 37547, 38567, 39227, 43397, 48677, 51197, 51827, 53087, 58907, 65027, 65837

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+144,n+146).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).

			For n=137, the numbers 137, 139, 281, 283, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605.

from sympy import isprime
for n in range(1,10000001,2):
    if isprime(n) and isprime(n+2) and isprime(n+144) and isprime(n+146): print(n,end=', ')

A248661 Initial members of prime quadruples (n, n+2, n+54, n+56).

Original entry on oeis.org

5, 17, 137, 227, 827, 1427, 1667, 1877, 2027, 2087, 2657, 3527, 3767, 4217, 4967, 10037, 11117, 11777, 12107, 13877, 17987, 19697, 20717, 21557, 22037, 23687, 24977, 27527, 27737, 34157, 37307, 41177, 42017, 42407, 47657, 48677

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+54,n+56).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (primes, 30n-13), A181605 (twin primes, end 7), and A092340 (prime n, where n^2+2*n divides (fibonacci(n^2)+fibonacci(2*n))).

			For n=17, the numbers 17, 19, 71, 73, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A092340.

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+54) and isprime(n+56): print(n,end=', ')

cp's OEIS Frontend

A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

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A140533 Primes congruent to 13 or 17 mod 30.

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Magma

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Mathematica

A229947 Primes congruent to {1, 11, 13, 17, 19, 29} mod 30.

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Magma

Mathematica

PARI

Formula

A132239 Primes congruent to {17, 19} mod 30.

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Magma

Mathematica

A132246 Twin primes congruent to {1, 17, 19, 29} mod 30.

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Mathematica

A220081 Primes of the form 15*k^2 - 15*k + 17.

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Magma

Mathematica

A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

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Python

A248474 Numbers congruent to 13 or 17 mod 30.

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Mathematica

A220081 Primes of the form 15k^2 - 15k + 17.