A129807 -id:A129807 - cp's OEIS frontend

A132231 Primes congruent to 7 (mod 30).

7, 37, 67, 97, 127, 157, 277, 307, 337, 367, 397, 457, 487, 547, 577, 607, 727, 757, 787, 877, 907, 937, 967, 997, 1087, 1117, 1237, 1297, 1327, 1447, 1567, 1597, 1627, 1657, 1747, 1777, 1867, 1987, 2017, 2137, 2287, 2347, 2377, 2437, 2467, 2557, 2617, 2647

Offset: 1

Omar E. Pol, Aug 15 2007

Primes ending in 7 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 07 2009
Intersection of A030432 and A002476. - Ray Chandler, Apr 07 2009
Only from 4927 on, there are more composite numbers than primes in {7+30k}, see A227869. - M. F. Hasler, Nov 02 2013
Terms are non-twin primes A007510, except for 7. - Jonathan Sondow, Oct 27 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages
Omar E. Pol, Prime numbers in a sieve with 30 columns
Omar E. Pol, Sobre el patrón de los números primos

Cf. A007510, A007528, A010051, A068229, A117047, A129807, A039949, A132230-A132236, A214360.

a132231 n = a132231_list !! (n-1)
a132231_list = [x | k <- [0..], let x = 30 * k + 7, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

[p: p in PrimesUpTo(3000) | p mod 30 eq 7 ]; // Vincenzo Librandi, Aug 14 2012

Select[30*Range[0,100]+7,PrimeQ] (* Harvey P. Dale, Feb 01 2012 *)
Select[Prime[Range[1000]],MemberQ[{7},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

forstep(p=7,1999,30,isprime(p)&&print1(p",")) \\ M. F. Hasler, Nov 02 2013

a(n) = A158573(n)*30 + 7. - Ray Chandler, Apr 07 2009

a(n) = A211890(4,n-1) for n <= 5. - Reinhard Zumkeller, Jul 13 2012

Extended by Ray Chandler, Apr 07 2009

A129805 Primes congruent to +-1 mod 18.

Original entry on oeis.org

17, 19, 37, 53, 71, 73, 89, 107, 109, 127, 163, 179, 181, 197, 199, 233, 251, 269, 271, 307, 359, 379, 397, 431, 433, 449, 467, 487, 503, 521, 523, 541, 557, 577, 593, 613, 631, 647, 683, 701, 719, 739, 757, 773, 809, 811, 827, 829, 863, 881, 883, 919, 937, 953, 971, 991

Offset: 1

N. J. A. Sloane, May 22 2007

From Katherine E. Stange, Feb 03 2010: (Start)
Equivalently, primes p such that the smallest extension of F_p containing the cube roots of unity also contains the 9th roots of unity.
Equivalently, the primes p for which, if p^t = 1 mod 3, then p^t = 1 mod 9.
Equivalently, primes congruent to +/-1 modulo 9.
Membership or non-membership of the prime p in this sequence and sequence A002144 (primes congruent to 1 mod 4; equivalently, primes p such that the smallest extension of F_p containing the square roots of unity contains the 4th roots of unity) appear to determine the behavior of amicable pairs on the elliptic curve y^2 = x^3 + p (Silverman, Stange 2009). (End)
Primes in A056020. - Reinhard Zumkeller, Jan 07 2012
Primes congruent to (1,17) mod 18. - Vincenzo Librandi, Aug 14 2012
Equivalently, primes such that p^2 == 1 (mod 9). - M. F. Hasler, Apr 16 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emma Lehmer, On special primes, Pac. J. Math., 118 (1985), 471-478.
J. H. Silverman and K. E. Stange. Amicable pairs and aliquot cycles for elliptic curves, arxiv:0912.1831 [math.NT], 2009.

Cf. A000040, A010051.

Cf. A129806, A129807.

a129805 n = a129805_list !! (n-1)
a129805_list = [x | x <- a056020_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[ p: p in PrimesUpTo(1300) | p mod 18 in {1, 17} ]; // Vincenzo Librandi, Aug 14 2012

Union[Join[Select[Range[-1, 3000, 18], PrimeQ], Select[Range[1, 3000, 18], PrimeQ]]] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2012 *)
Select[Prime[Range[4000]],MemberQ[{1,17},Mod[#,18]]&] (* Vincenzo Librandi, Aug 14 2012 *)

select( {is_A129805(n)=n^2%9==1&&isprime(n)}, primes(199)) \\ M. F. Hasler, Apr 16 2022

A129806 Primes congruent to +-5 mod 18.

Original entry on oeis.org

5, 13, 23, 31, 41, 59, 67, 103, 113, 131, 139, 149, 157, 167, 193, 211, 229, 239, 257, 283, 293, 311, 337, 347, 373, 383, 401, 409, 419, 463, 491, 499, 509, 563, 571, 599, 607, 617, 643, 653, 661, 733, 743, 751, 761, 769, 787, 797, 823, 859, 877, 887, 941, 967, 977, 1013

Offset: 1

N. J. A. Sloane, May 22 2007

Primes congruent to (5,13) mod 18. - Vincenzo Librandi, Aug 14 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Emma Lehmer, On special primes, Pac. J. Math., 118 (1985), 471-478.

Cf. A129805, A129807.

[ p: p in PrimesUpTo(1300) | p mod 18 in {5, 13} ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[4000]],MemberQ[{5,13},Mod[#,18]]&] (* Vincenzo Librandi, Aug 14 2012 *)

A132237 Primes congruent to {7, 23} mod 30.

Original entry on oeis.org

7, 23, 37, 53, 67, 83, 97, 113, 127, 157, 173, 233, 263, 277, 293, 307, 337, 353, 367, 383, 397, 443, 457, 487, 503, 547, 563, 577, 593, 607, 653, 683, 727, 743, 757, 773, 787, 863, 877, 907, 937, 953, 967, 983, 997, 1013, 1087, 1103

Offset: 1

Omar E. Pol, Aug 15 2007

Up to 4913, there are more primes of this form than composites. See also A132231 and A227869 (congruent to 7 only). - M. F. Hasler, Nov 02 2013

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, A007510, A039949, A068229, A129807.

[ p: p in PrimesUpTo(1300) | p mod 30 in [7, 23] ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{7,23},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

is_A132237(n)=setsearch([7,23],n%30)&&isprime(n) \\ - M. F. Hasler, Nov 02 2013

A129865 List of pairs {n-th prime == 7 mod 18, n-th pair == -7 mod 18}.

Original entry on oeis.org

7, 11, 43, 29, 61, 47, 79, 83, 97, 101, 151, 137, 223, 173, 241, 191, 277, 227, 313, 263, 331, 281, 349, 317, 367, 353, 421, 389, 439, 443, 457, 461, 547, 479, 601, 569, 619, 587, 673, 641, 691, 659, 709, 677, 727, 821, 853, 839, 907, 857, 997, 911, 1033, 929

Offset: 1

Zak Seidov, May 24 2007

nonn

Sum of pairs is multiple of 18k, with k=1,4,6,9,11,16,22,24,28,32,34,37,40,45,49,51,57,65,67,73,75,77,86,94,98,106.

Cf. A129807.

pp=Prime[Range[1000]];se7=Select[pp,Mod[ #,18]==7&];se11=Select[pp,Mod[ #,18]==11&];nn=Min[Length/@{se7,se11}];tr=Table[{se7[[i]],se11[[i]]},{i,nn}];A1=tr//Flatten;k=(Total/@tr)/18

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A132231 Primes congruent to 7 (mod 30).

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A129805 Primes congruent to +-1 mod 18.

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A129806 Primes congruent to +-5 mod 18.

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A132237 Primes congruent to {7, 23} mod 30.

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A129865 List of pairs {n-th prime == 7 mod 18, n-th pair == -7 mod 18}.

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