A045392 -id:A045392 - cp's OEIS frontend

A179474 a(n) = position of A045392(n) in A042997.

1, 8, 10, 18, 24, 29, 32, 37, 43, 56, 57, 61, 63, 67, 73, 74, 79, 84, 87, 100, 105, 110, 118, 125, 126, 130, 134, 143, 150, 157, 160, 165, 175, 177, 184, 185, 187, 188, 193, 200, 205, 215, 230, 233, 235, 242, 248, 256, 258, 261, 262, 265, 272, 276, 278, 279, 290

Offset: 1

Zak Seidov, Jul 16 2010

nonn

Positions of primes congruent to {2} mod 7 among primes congruent to {2,3,4,5,6} mod 7.

Cf. A042997 Primes congruent to {2, 3, 4, 5, 6} mod 7, A045392 Primes congruent to {2} mod 7.

pr=Prime[Range[500]];A042997=Select[pr,1A045392=Select[A042997,2==Mod[ #,7]&];
s=Flatten[Position[A042997,# ]&/@A045392]

A007528 Primes of the form 6k-1.

Original entry on oeis.org

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

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

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A045368 Primes congruent to {2, 5} mod 7.

Original entry on oeis.org

2, 5, 19, 23, 37, 47, 61, 79, 89, 103, 107, 131, 149, 163, 173, 191, 229, 233, 257, 271, 313, 317, 331, 359, 373, 383, 397, 401, 439, 443, 457, 467, 499, 509, 523, 541, 569, 593, 607, 653, 677, 691, 709, 719, 733

Offset: 1

N. J. A. Sloane

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

Cf. A045392, A045458.

[ p: p in PrimesUpTo(1500) | p mod 7 in {2, 5} ]; // Vincenzo Librandi, Aug 06 2012

Select[Prime[Range[200]],MemberQ[{2,5},Mod[#,7]]&] (* Harvey P. Dale, Apr 28 2012 *)

A045383 Primes congruent to {0, 2} mod 7.

Original entry on oeis.org

2, 7, 23, 37, 79, 107, 149, 163, 191, 233, 317, 331, 359, 373, 401, 443, 457, 499, 541, 569, 653, 709, 751, 821, 863, 877, 919, 947, 1031, 1087, 1129, 1171, 1213, 1283, 1297, 1367, 1381, 1409, 1423, 1451, 1493

Offset: 1

N. J. A. Sloane

{7} UNION A045392.

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

[ p: p in PrimesUpTo(2000) | p mod 7 in {0, 2} ]; // Vincenzo Librandi, Aug 07 2012

Select[Prime[Range[700]],MemberQ[{0,2},Mod[#,7]]&] (* Vincenzo Librandi, Aug 07 2012 *)
Select[Flatten[#+{0,2}&/@(7*Range[0,300])],PrimeQ] (* Harvey P. Dale, Oct 17 2021 *)

A023223 Primes p such that 7*p + 2 is also prime.

Original entry on oeis.org

3, 5, 11, 23, 47, 53, 71, 101, 107, 131, 167, 173, 197, 251, 257, 293, 311, 317, 353, 383, 431, 461, 467, 563, 587, 593, 683, 701, 773, 797, 821, 827, 863, 887, 911, 953, 977, 983, 1031, 1091, 1097, 1103, 1151, 1181, 1187, 1193, 1217, 1223, 1277, 1301, 1307, 1373

Offset: 1

David W. Wilson

nonn

Subsequence of A105772. Except for the first term all others are congruent to 5 (mod 6) because 7*(6n+1)+2 is divisible by 3. - John Cerkan, Jul 08 2016

			3 is in the sequence because 7 * 3 + 2 = 23, which is prime.
5 is in the sequence because 7 * 5 + 2 = 37, which is prime.
7 is not in the sequence because 7 * 7 + 2 = 51 = 3 * 17.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A045392, A105772.

[n: n in [0..100000] | IsPrime(n) and IsPrime(7*n+2)]; // Vincenzo Librandi, Nov 19 2010

Select[Prime[Range[250]], PrimeQ[7# + 2] &] (* Alonso del Arte, Apr 08 2015 *)

lista(nn) = forprime(p=2, nn, if(isprime(7*p+2), print1(p, ", "))); \\ Altug Alkan, Jul 08 2016

A024904 Numbers k such that 7*k - 5 is prime.

Original entry on oeis.org

1, 4, 6, 12, 16, 22, 24, 28, 34, 46, 48, 52, 54, 58, 64, 66, 72, 78, 82, 94, 102, 108, 118, 124, 126, 132, 136, 148, 156, 162, 168, 174, 184, 186, 196, 198, 202, 204, 208, 214, 222, 232, 252, 256, 258, 268, 274, 286, 288, 292, 294, 298, 306, 312, 316, 318, 334, 336, 342, 346

Offset: 1

Clark Kimberling

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

Cf. A045392 (associated primes).

[n: n in [1..1000] |IsPrime(7*n-5)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[400], PrimeQ[7 # - 5] &] (* Vincenzo Librandi, Sep 26 2012 *)

is(n)=isprime(7*n-5) \\ Charles R Greathouse IV, Jun 06 2017

A045393 Primes congruent to {0, 1, 3, 4, 5, 6} mod 7.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 151, 157, 167, 173, 179, 181, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 257, 263

Offset: 1

N. J. A. Sloane

Or, primes not congruent to 2 mod 7, i.e., primes not in A045392. - M. F. Hasler, Feb 21 2015

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

Cf. A000040.

[p: p in PrimesUpTo(400) | p mod 7 in [0, 1, 3, 4, 5, 6]]; // Vincenzo Librandi, Aug 11 2012

Select[Prime[Range[300]],MemberQ[{0,1,3,4,5,6},Mod[#,7]]&] (* Vincenzo Librandi, Aug 11 2012 *)

A140442 Primes congruent to 9 mod 14.

Original entry on oeis.org

23, 37, 79, 107, 149, 163, 191, 233, 317, 331, 359, 373, 401, 443, 457, 499, 541, 569, 653, 709, 751, 821, 863, 877, 919, 947, 1031, 1087, 1129, 1171, 1213, 1283, 1297, 1367, 1381, 1409, 1423, 1451, 1493, 1549, 1619, 1759, 1787, 1801, 1871, 1913, 1997

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

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

Primes arising in sequences A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.

[p: p in PrimesUpTo(2000)|p mod 14 in {9}] // Vincenzo Librandi, Dec 18 2010

Select[Prime[Range[500]],MemberQ[{9},Mod[#,14]]&] (* Vincenzo Librandi, Aug 07 2012 *)
Select[Range[9,2000,14],PrimeQ] (* Harvey P. Dale, Apr 17 2015 *)

is(n)=isprime(n) && n%14==9 \\ Charles R Greathouse IV, Jul 03 2016

a(n) = A045392(n+1) = A045383(n+2). - Zak Seidov, Mar 12 2014

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 03 2016

1451 inserted by R. J. Mathar, Sep 13 2008

A274202 Primes congruent to 31 mod 65.

Original entry on oeis.org

31, 421, 811, 941, 1201, 1721, 2111, 2371, 3541, 3671, 3931, 4451, 5101, 5231, 5881, 6011, 6271, 6661, 6791, 8221, 8741, 9001, 9391, 9521, 9781, 10301, 10691, 11471, 11731, 12251, 12511, 12641, 13291, 13421, 13681, 14071, 14461, 14591, 14851, 15241, 15761

Offset: 1

Vincenzo Librandi, Jun 13 2016

Subsequence of A030430 and A102732.

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

Cf. A000040, A030430, A102732.

Cf. similar sequences of the type primes congruent to k mod 2*k+3: A045392 (k=2), A102732 (k=5), A138629 (k=7), A141873 (k=8), A141914 (k=10), A141935 (k=11), A141989 (k=13), A142018 (k=14), A142086 (k=16), A142126 (k=17), A142216 (k=19), A142269 (k=20), A142373 (k=22), A142433 (k=23), A142555 (k=25), A142619 (k=26), A142755 (k=28), A142827 (k=29), this sequence (k=31), A154621 (k=32), A154624 (k=34), A154628 (k=35).

[p: p in PrimesUpTo(20000) | p mod 65 eq 31];

Select[Prime[Range[2000]], MemberQ[{31}, Mod[#, 65]] &]
Select[Range[31,16000,65],PrimeQ] (* Harvey P. Dale, May 06 2018 *)

cp's OEIS Frontend

A179474 a(n) = position of A045392(n) in A042997.

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A007528 Primes of the form 6k-1.

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A141849 Primes congruent to 1 mod 11.

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A045368 Primes congruent to {2, 5} mod 7.

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A045383 Primes congruent to {0, 2} mod 7.

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A023223 Primes p such that 7*p + 2 is also prime.

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A024904 Numbers k such that 7*k - 5 is prime.

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