A045458 -id:A045458 - cp's OEIS frontend

A007528 Primes of the form 6k-1.

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

A045392 Primes congruent to 2 mod 7.

Original entry on oeis.org

2, 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

N. J. A. Sloane

2 and primes congruent to 9 mod 14. - Chai Wah Wu, Apr 28 2025

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

Cf. A045437, A045458, A045471.

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

Select[Range[2, 50000, 7], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2011 *)

is(n)=isprime(n) && n%7==2 \\ Charles R Greathouse IV, Jul 01 2016

A045471 Primes congruent to 4 mod 7.

Original entry on oeis.org

11, 53, 67, 109, 137, 151, 179, 193, 263, 277, 347, 389, 431, 487, 557, 571, 599, 613, 641, 683, 739, 809, 823, 907, 977, 991, 1019, 1033, 1061, 1103, 1117, 1187, 1201, 1229, 1327, 1439, 1453, 1481, 1523, 1579, 1607, 1621, 1663, 1733, 1747, 1789, 1831, 1873

Offset: 1

N. J. A. Sloane

Primes congruent to 11 mod 14. - Chai Wah Wu, Apr 28 2025

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

Cf. A042990 (complement), A045458, A045473.

[ p: p in PrimesUpTo(11000) | p mod 7 eq 4 ]; // Vincenzo Librandi, Aug 13 2012

Select[Range[4, 50000, 7], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2011 *)

is(n)=isprime(n) && n%7==4 \\ Charles R Greathouse IV, Jul 01 2016

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

A140444 Primes congruent to 1 (mod 14).

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 743, 757, 827, 883, 911, 953, 967, 1009, 1051, 1093, 1163, 1289, 1303, 1373, 1429, 1471, 1499, 1583, 1597, 1667, 1709, 1723, 1877, 1933, 2003, 2017, 2087

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

From Federico Provvedi, May 24 2018: (Start)
Also primes congruent to 1 (mod 7).
For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).
Conjecture: The monic polynomial P(x) = (x+1)^7/x - 1/x = ((x+1)^7-1)/x is irreducible but factorizable over Galois field (mod a(n)) with exactly 6 distinct irreducible factors of degree 1. Examples:
P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x)    (mod 29)
P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x)   (mod 43)
P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)
P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x)   (mod 113)
... (End).
Primes in A131877. - Eric Chen, Jun 14 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.

Cf. A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.
Cf. A131877.
Primes congruent to 1 (mod k): A000040 (k=1), A065091 (k=2), A002476 (k=3 and 6), A002144 (k=4), A030430 (k=5 and 10), this sequence (k=7 and 14), A007519 (k=8), A061237 (k=9 and 18), A141849 (k=11 and 22), A068228 (k=12), A268753 (k=13 and 26), A132230 (k=15 and 30), A094407 (k=16), A129484 (k=17 and 34), A141868 (k=19 and 38), A141881 (k=20), A124826 (k=21 and 42), A212374 (k=23 and 46), A107008 (k=24), A141927 (k=25 and 50), A141948 (k=27 and 54), A093359 (k=28), A141977 (k=29 and 58), A142005 (k=31 and 62), A133870 (k=32).

Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # Muniru A Asiru, Jun 27 2018

[p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // Vincenzo Librandi, Dec 18 2010

select(isprime,select(n->modp(n,14)=1,[$1..2300])); # Muniru A Asiru, Jun 27 2018

Select[Prime[Range[500]], Mod[#, 14] == 1 &]  (* Harvey P. Dale, Mar 21 2011 *)

is(n)=isprime(n) && n%14==1 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 02 2016

Simpler definition from N. J. A. Sloane, Jul 11 2008

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 *)

A111367 Numbers k such that 7*k + 5 is prime.

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 18, 24, 32, 36, 38, 44, 54, 56, 62, 66, 72, 74, 84, 86, 96, 98, 102, 104, 108, 122, 126, 132, 138, 144, 152, 156, 164, 168, 174, 176, 182, 186, 188, 204, 206, 212, 218, 222, 228, 236, 242, 248, 254, 258, 266, 278, 282, 284, 294, 308, 314, 324

Offset: 1

Parthasarathy Nambi, Nov 07 2005

			k=108 is a term because 7*k + 5 = 761 is prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A017041, A045458, A111224.

[n: n in [0..100000] |IsPrime(7*n+5)] // Vincenzo Librandi, Nov 13 2010

is(n)=isprime(7*n+5) \\ Charles R Greathouse IV, Jun 13 2017

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

A163603 Numbers k such that prime(k) == 5 (mod 7).

Original entry on oeis.org

3, 8, 15, 18, 24, 27, 32, 40, 50, 55, 58, 65, 76, 78, 85, 91, 97, 99, 108, 111, 123, 125, 128, 130, 135, 149, 154, 158, 164, 170, 180, 184, 191, 194, 200, 203, 207, 214, 216, 227, 229, 237, 242, 246, 252, 260, 266, 271

Offset: 1

Juri-Stepan Gerasimov, Aug 01 2009

nonn

The asymptotic density of this sequence is 1/6 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A000720, A045458.

Select[Range[300], Mod[Prime[#], 7] == 5 &] (* G. C. Greubel, Jul 29 2017 *)

isok(n) = (prime(n) % 7) == 5; \\ Michel Marcus, Jul 29 2017

a(n) = A000720(A045458(n)).

A000040(a(n)) = A045458(n).

Remainder in definition corrected by R. J. Mathar, Aug 01 2009

A355025 a(1)=2; for n > 1, a(n) is the least new prime such that a(n-1) + a(n) is a multiple of 7.

Original entry on oeis.org

2, 5, 23, 19, 37, 47, 79, 61, 107, 89, 149, 103, 163, 131, 191, 173, 233, 229, 317, 257, 331, 271, 359, 313, 373, 383, 401, 397, 443, 439, 457, 467, 499, 509, 541, 523, 569, 593, 653, 607, 709, 677, 751, 691, 821, 719, 863, 733, 877, 761

Offset: 1

Zak Seidov, Jun 15 2022

nonn

			2 + 3 = 5 is not a multiple of 7, but 2 + 5 = 7 is, so a(2) = 5.
5 + 2 = 7 is a multiple of 7, but 2 is already a term; 5 + 3 = 8, 5 + 7 = 12, ..., 5 + 19 = 24 are not multiples of 7, but 5 + 23 = 28 is, so a(3) = 23.
23 + 5 = 28 is a multiple of 7, but 5 is already a term; 19 is the next prime p such that 7 divides 23 + p, so a(4) = 19.

Cf. A045392, A045458.

s = {2}; Do[p = 3; a = s[[-1]]; While[MemberQ[s, p] || Mod[a + p, 7] != 0, p = NextPrime[p]]; AppendTo[s, p], {100}]; s

a(n) = A045392((n+1)/2) if n is odd, A045458(n/2) if n is even. - Jon E. Schoenfield, Jun 15 2022

cp's OEIS Frontend

A007528 Primes of the form 6k-1.

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A045392 Primes congruent to 2 mod 7.

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A045471 Primes congruent to 4 mod 7.

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A141849 Primes congruent to 1 mod 11.

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A140444 Primes congruent to 1 (mod 14).

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

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A111367 Numbers k such that 7*k + 5 is prime.

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