A068229 -id:A068229 - cp's OEIS frontend

A116615 Values of n such that prime(2n) mod 12 = 7.

2, 4, 7, 11, 17, 18, 19, 23, 24, 29, 45, 57, 69, 94, 101, 105, 111, 112, 116, 121, 129, 133, 136, 137, 138, 141, 150, 157, 162, 164, 170, 172, 174, 177, 184, 187, 197, 203, 207, 209, 220, 231, 235, 239, 245, 250, 251, 252, 254, 255, 260, 261, 270, 273, 276, 283

Offset: 1

Roger L. Bagula, Mar 29 2006

nonn

			23 is in the sequence because the 46th prime is 199 and 199 mod 12=7.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A116602, A116610, A116612-A116617, A160591-A160594. [M. F. Hasler, May 22 2009]

a:=proc(n) if ithprime(2*n) mod 12 = 7 then n else fi end: seq(a(n),n=1..300);

Select[Range[1, 500], Mod[Prime[2*#], 12] == 7 &] (* G. C. Greubel, Nov 19 2017 *)

for( n=1,999, prime(2*n)%12==7 & print1(n",")) \\ M. F. Hasler, May 22 2009

Equals (1/2) * { even terms in A160592 = A000720(A068229) }. - M. F. Hasler, May 22 2009

Edited by N. J. A. Sloane, Apr 05 2006

A124988 Primes of the form 12k+7 generated recursively. Initial prime is 7. General term is a(n)=Min {p is prime; p divides 3+4Q^2; Mod[p,12]=7}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

7, 199, 7761799, 487, 67, 103, 1482549740515442455520791, 31, 139, 787, 19, 39266047, 1955959, 50650885759, 367, 185767, 62168707

Offset: 1

Nick Hobson, Nov 18 2006

nonn

All prime divisors of 3+4Q^2 are congruent to 1 modulo 6.
At least one prime divisor of 3+4Q^2 is congruent to 3 modulo 4 and hence to 7 modulo 12.
The first six terms are the same as those of A057204.

			a(3) = 1482549740515442455520791 is the smallest prime divisor congruent to 7 mod 12 of 3+4Q^2 = 5281642303363312989311974746340327 = 3562539697 * 1482549740515442455520791, where Q = 7 * 199 * 7761799 * 487 * 67 * 103.

Tyler Busby, Table of n, a(n) for n = 1..21

Cf. A000945, A068229, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={7}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4*q^2+3][[All,1]],Mod[#,12]==7 &]]];
    ];
a (* Robert Price, Jul 15 2015 *)

A141682 Number of isomorphism classes of (2n+1)-reflexive polygons.

Original entry on oeis.org

16, 1, 12, 29, 1, 61, 81, 1, 113, 131, 2, 163, 50, 2, 215, 233, 2, 34, 285, 3, 317, 335, 2, 367, 182, 3, 419, 72, 4, 469, 489, 3, 93, 539, 4, 571, 591, 3, 185, 641, 5, 673, 131, 5, 725, 240, 6, 148, 795, 5, 827, 845, 3, 877, 897, 7, 929, 186, 6, 338, 656, 7, 240, 1049, 8, 1081, 393, 5, 1133, 1151, 8, 542, 245, 7, 1235, 1253

Offset: 0

Benjamin Nill, Jul 02 2012

nonn

There are no l-reflexive polygons for even index l.

			a(0)=16 equals the number of isomorphism classes of (1-)reflexive polygons, A090045(2).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..99 (from the Graded Ring Database)
Dimitrios I. Dais, On the Twelve-Point Theorem for l-Reflexive Polygons, arXiv:1806.08351 [math.CO], 2018.
A. M. Kasprzyk and B. Nill, Reflexive polytopes of higher index and the number 12, arXiv:1107.4945 [math.AG], 2011.
A. M. Kasprzyk and B. Nill, Toric l-reflexive surfaces at Graded Ring Database.

Cf. A090045.

It seems that for n > 2, a(n) = 17*n - k where k = 21, 22, 23, 24 iff 2*n+1 is a prime from A068228, A068229, A040117, A068231, respectively. - Andrey Zabolotskiy, Apr 21 2022

A160592 Indices of primes congruent to 7 modulo 12.

Original entry on oeis.org

4, 8, 11, 14, 19, 22, 27, 31, 34, 36, 38, 46, 47, 48, 58, 61, 63, 67, 73, 75, 85, 90, 93, 95, 99, 101, 105, 111, 114, 115, 117, 125, 129, 131, 133, 138, 141, 143, 149, 153, 155, 157, 163, 167, 175, 177, 179, 181, 188, 193, 202, 207, 210, 213, 217, 222, 224, 229, 232

Offset: 1

M. F. Hasler, May 22 2009

nonn

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 02 2021

			a(1) = 4 since the 4th prime, A000040(4) = 7, is the first one to be equal to 7 (mod 12).
a(2) = 8 since the 8th prime, A000040(8) = 19, is the second one to be equal to 7 (mod 12).

A116615 lists the even terms of this sequence, divided by 2.

Cf. A000720, A068229.

for( n=1,999, prime(n)%12==7 & print1(n","))

a(n) = A000720(A068229(n)).

A116616 Values of n such that prime(2n+1) mod 12 = 7.

Original entry on oeis.org

5, 9, 13, 15, 23, 30, 31, 33, 36, 37, 42, 46, 47, 49, 50, 52, 55, 57, 58, 62, 64, 65, 66, 70, 71, 74, 76, 77, 78, 81, 83, 87, 88, 89, 90, 96, 103, 106, 108, 114, 116, 117, 121, 123, 124, 130, 134, 142, 144, 148, 151, 152, 160, 163, 166, 167, 175, 182, 185, 191, 192

Offset: 1

Roger L. Bagula, Mar 29 2006

nonn

			33 is in the sequence because the 67th prime is 331 and 331 mod 12=7.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A116602, A116610, A116612-A116617, A160591-A160594. [M. F. Hasler, May 22 2009]

a:=proc(n) if ithprime(2*n+1) mod 12 = 7 then n else fi end: seq(a(n),n=0..215);

Select[Range[220], Mod[Prime[2# + 1], 12] == 7 &] (* Stefan Steinerberger, Apr 08 2006 *)

for(n=1,999, prime(2*n+1)%12==7 & print1(n",")) \\ M. F. Hasler, May 22 2009

Equals { odd terms in A160592 = A000720(A068229) } / 2, rounded towards zero. - M. F. Hasler, May 22 2009

Edited by N. J. A. Sloane, Apr 05 2006

A142786 Primes congruent to 7 mod 60.

Original entry on oeis.org

7, 67, 127, 307, 367, 487, 547, 607, 727, 787, 907, 967, 1087, 1327, 1447, 1567, 1627, 1747, 1867, 1987, 2287, 2347, 2467, 2647, 2707, 2767, 2887, 3067, 3187, 3307, 3547, 3607, 3727, 3847, 3907, 3967, 4027, 4327, 4447, 4507, 4567, 4987, 5107, 5167, 5227

Offset: 1

N. J. A. Sloane, Jul 11 2008

Comment from Joshua S.M. Weiner, Oct 12 2012 (Start)
Intersection of A068229 and A141882. Subsequence of A132231.
Congruence classes of primes mod 60: A088955 (1), (this sequence 7),  A117047 (11), A142787 (13), A142788 (17), A142789 (19), A142790 (23), A142791 (29), A142792 (31), A142793 (37), A142794 (41), A142795 (43), A142796 (47), A142797 (49), A142798 (53), A142799 (59). (End)

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

Cf. A000040, A142322.

[p: p in PrimesUpTo(6000) | p mod 60 eq 7 ]; // Vincenzo Librandi, Sep 04 2012

Select[Prime[Range[1000]], Mod[#, 60] == 7 &] (* T. D. Noe, Oct 12 2012 *)
Select[Range[7,5300,60],PrimeQ] (* Harvey P. Dale, Nov 21 2018 *)

A164622 Primes p such that pfloor(p/2) - 4 and pfloor(p/2) + 4 are prime numbers.

Original entry on oeis.org

151, 463, 571, 631, 643, 991, 1063, 1171, 1831, 2083, 2311, 4951, 5023, 6211, 6703, 6763, 7723, 7951, 9043, 11383, 12163, 12391, 13183, 14851, 15031, 17431, 19231, 19543, 20143, 22051, 23143, 25951, 26371, 27283, 28351, 29131, 30643, 32803

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

nonn

151*75-4=11321 (prime), 151*75+4=11329 (prime), ..

Subsequence of A068229. Cf. A008846, A020882, A068229, A086519, A158708, A164620, A164621

lst={};Do[p=Prime[n];If[PrimeQ[p*Floor[p/2]-4]&&PrimeQ[p*Floor[p/2]+4],AppendTo[lst,p]],{n,8!}];lst

Edited by Charles R Greathouse IV, Nov 02 2009

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.

Original entry on oeis.org

13, 157, 673, 1069, 1117, 1153, 1213, 1597, 2029, 2089, 2437, 2713, 2833, 3613, 4057, 4909, 5653, 6337, 6529, 7549, 7993, 8053, 9613, 10789, 11497, 11689, 12073, 12373, 13309, 13669, 13789, 14173, 15289, 15937, 16249, 18097, 18637, 19249, 19993

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Primes A000040(k) such that A008837(k)+-5 are also prime numbers.

			13 is in the sequence because 13*6-5=73 and 13*6+5=83 are both prime.

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

Cf. A008846, A020882, A068228, A068229, A082539, A086519, A107159, A158708, A139494, A164620, A164621, A164622, A023203.

Select[Prime[Range[2300]], PrimeQ[# (# - 1)/2 - 5] && PrimeQ[# (# - 1)/2 + 5] &]

forprime(p=2,10^6,my(b=binomial(p,2));if(isprime(b-5)&isprime(b+5),print1(p,", "))); /* Joerg Arndt, Apr 10 2013 */

Edited by R. J. Mathar, Aug 20 2009

Mathematica code adapted to the definition by Bruno Berselli, Apr 10 2013

A180217 a(n) = (n-th prime modulo 3) + (n-th prime modulo 4).

Original entry on oeis.org

4, 3, 3, 4, 5, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 5, 2, 4, 5, 2, 4, 5, 3, 2, 3, 4, 5, 2, 3, 4, 5, 3, 4, 3, 4, 2, 4, 5, 3, 5, 2, 5, 2, 3, 4, 4, 4, 5, 2, 3, 5, 2, 5, 3, 5, 3, 4, 2, 3, 4, 3, 4, 5, 2, 3, 4, 2, 5, 2, 3, 5, 4, 2, 4, 5, 3, 2, 3, 2, 5, 2, 5, 2, 4, 5, 3, 2, 3, 4, 5, 5, 4, 5, 4, 5, 3, 3, 4, 2, 4, 3

Offset: 1

Zak Seidov, Jan 16 2011

nonn

a(n) = 2 iff prime(n) == 1 (mod 12); a(n) = 2 for prime(n) = 13, 37, 61, 73, 97, 109, ... (A068228).
a(n) = 5 iff prime(n) == 11 (mod 12); a(n) = 5 for prime(n) = 11, 23, 47, 59, 71, 83, ... (A068231).
For n > 2, a(n) = 3 iff prime(n) == 5 (mod 12); a(n) = 3 for prime(n) = 5, 17, 29, 41, 53, 89, ... (A040117).
For n > 2, a(n) = 4 iff prime(n) == 7 (mod 12); a(n) = 4 for prime(n) = 7, 19, 31, 43, 67, 79, ... (A068229).

Equals A039701 + A039702.

Cf. A068228, A068231, A040117, A068229, A039710.

A180217:=func< n | p mod 3 + p mod 4 where p is NthPrime(n) >; [ A180217(n): n in [1..105] ]; // Klaus Brockhaus, Jan 18 2011

Mod[#,3]+Mod[#,4]&/@Prime[Range[110]] (* Harvey P. Dale, Nov 09 2011 *)

A274507 Primes one more than the sum over a pair of prime numbers that differ by 8.

Original entry on oeis.org

19, 31, 67, 127, 151, 211, 271, 307, 547, 727, 787, 811, 907, 967, 991, 1447, 1531, 1831, 1867, 2131, 2467, 2647, 2887, 2971, 3967, 5107, 5227, 5407, 5431, 5827, 6091, 6427, 6451, 6607, 6907, 6991, 7411, 8191, 8431, 8707, 9511, 10111

Offset: 1

Debapriyay Mukhopadhyay, Jun 25 2016

nonn

Any prime p in this sequence is such that p = (p-9)/2 + (p+7)/2 + 1, where (p-9)/2 and (p+7)/2 are also primes and they differ by 8.

This sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Jul 08 2016

			19 = 5 + 13 + 1. Note that, (19-9)/2 = 5 and (19+7)/2 = 13 and the prime pairs 5 and 13 differ by 8.
31 = 11 + 19 + 1. Note that, (31-9)/2 = 11 and (31+7)/2 = 19 and the prime pairs 11 and 19 differ by 8.

Cf. A023202, A274506.

A subsequence of A068229 and also of A145472.

Select[2 # + 9 &@ Select[Prime@ Range[10^3], PrimeQ[# + 8] &], PrimeQ] (* Michael De Vlieger, Jun 26 2016 *)

lista(nn)=forprime(p=3, nn, if (isprime(p+8) && isprime(q=2*p+9), print1(q, ", "))); \\ Michel Marcus, Jun 25 2016

cp's OEIS Frontend

A116615 Values of n such that prime(2n) mod 12 = 7.

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A124988 Primes of the form 12k+7 generated recursively. Initial prime is 7. General term is a(n)=Min {p is prime; p divides 3+4Q^2; Mod[p,12]=7}, where Q is the product of previous terms in the sequence.

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Mathematica

A141682 Number of isomorphism classes of (2n+1)-reflexive polygons.

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A160592 Indices of primes congruent to 7 modulo 12.

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PARI

Formula

A116616 Values of n such that prime(2n+1) mod 12 = 7.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A142786 Primes congruent to 7 mod 60.

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Magma

Mathematica

A164622 Primes p such that p*floor(p/2) - 4 and p*floor(p/2) + 4 are prime numbers.

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Mathematica

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A164623 Primes p such that p*(p-1)/2-5 and p*(p-1)/2+5 are also prime numbers.

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A164622 Primes p such that pfloor(p/2) - 4 and pfloor(p/2) + 4 are prime numbers.

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.