A007528 -id:A007528 - cp's OEIS frontend

A045309 Primes congruent to {0, 2} mod 3.

2, 3, 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

Offset: 1

N. J. A. Sloane

Also, primes p such that the equation x^3 == y (mod p) has a unique solution x for every choice of y. - Klaus Brockhaus, Mar 02 2001; Michel Drouzy (DrouzyM(AT)noos.fr), Oct 28 2001

2, 3 and primes congruent to 5 mod 6. - Chai Wah Wu, Apr 28 2025

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

Cf. A040028, A014752, A060121, A003627, A007528, A045410.

[ p: p in PrimesUpTo(1000) | #[ x: x in ResidueClassRing(p) | x^3 eq 2 ] eq 1 ]; // Klaus Brockhaus, Apr 11 2009

Select[Prime[Range[150]],MemberQ[{0,2},Mod[#,3]]&] (* Harvey P. Dale, Jun 14 2011 *)

is(n)=isprime(n) && n%3!=1 \\ Charles R Greathouse IV, Apr 20 2015

from itertools import count, islice
from sympy import isprime
def A045309_gen(): # generator of terms
    yield from (2,3)
    yield from filter(isprime, count(5,6))
A045309_list = list(islice(A045309_gen(),48)) # Chai Wah Wu, Apr 28 2025

a(n) ~ 2n log n. - Charles R Greathouse IV, Apr 20 2015

Edited by N. J. A. Sloane, Apr 11 2009

A132231 Primes congruent to 7 (mod 30).

Original entry on oeis.org

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

A132232 Primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 131, 191, 251, 281, 311, 401, 431, 461, 491, 521, 641, 701, 761, 821, 881, 911, 941, 971, 1031, 1061, 1091, 1151, 1181, 1301, 1361, 1451, 1481, 1511, 1571, 1601, 1721, 1811, 1871, 1901, 1931, 2081, 2111, 2141, 2351, 2381, 2411, 2441, 2531

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, A068231, A039949.

Cf. A132230, A132231, A132233, A132234, A132235, A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 11 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{11},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[11,2600,30],PrimeQ] (* Harvey P. Dale, Mar 07 2020 *)

is(n)=isprime(n) && n%30==11 \\ Charles R Greathouse IV, Jul 01 2016

From Ray Chandler, Apr 07 2009: (Start)
a(n) = A158614(n)*30 + 11.
Intersection of A030430 and A007528. (End)

Extended by Ray Chandler, Apr 07 2009

A090686 Primes of the form 6n^2 - 1.

Original entry on oeis.org

5, 23, 53, 149, 293, 383, 599, 863, 1013, 1733, 2399, 2903, 4373, 4703, 5399, 6143, 7349, 8663, 11093, 12149, 16223, 18149, 20183, 21599, 23063, 23813, 25349, 27743, 29399, 31973, 33749, 35573, 40343, 41333, 45413, 51893, 56453, 59999, 62423

Offset: 1

Cino Hilliard, Dec 18 2003

Subset of A007528. The values of n for which 6*n^2 - 1 is prime are 1, 2, 3, 5, 7, 8, 10, 12, 13, 17, 20, 22, 27, 28, 30, 32, 35, 38, 43, 45, 52, 55, 58, 60, 62, 63, 65, 68, 70, 73, 75, 77, 82, 83, 87, 93, 97, 100, ... - Jonathan Vos Post, Aug 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A007528, A090687.

[a: n in [0..500] | IsPrime(a) where a is 6*n^2-1]; // Vincenzo Librandi, Dec 05 2011

lst={};Do[p=6*n^2-1;If[PrimeQ[p],AppendTo[lst,p]],{n,0,3*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[6n^2-1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Dec 05 2011 *)

mx2pmp(n) = { for(x=1,n, y = 6*x^2-1; if(isprime(y),print1(y",")) ) }

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 14 2008

A132235 Primes congruent to 23 (mod 30).

Original entry on oeis.org

23, 53, 83, 113, 173, 233, 263, 293, 353, 383, 443, 503, 563, 593, 653, 683, 743, 773, 863, 953, 983, 1013, 1103, 1163, 1193, 1223, 1283, 1373, 1433, 1493, 1523, 1553, 1583, 1613, 1733, 1823, 1913, 1973, 2003, 2063, 2153, 2213, 2243, 2273, 2333, 2393

Offset: 1

Omar E. Pol, Aug 15 2007

Primes (excluding 3) ending in 3 with (SOD-1)/3 non-integer where SOD is sum of digits. - Ki Punches
The sequence is infinite by Dirichlet's theorem. - Arkadiusz Wesolowski, Apr 02 2014
Terms are non-twin primes A007510. - Omar E. Pol, Jul 25 2019

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

Cf. A000040, A039949.

Cf. A132230, A132231, A132232, A132233, A132234, A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 23 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{23},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[23,2400,30],PrimeQ] (* Harvey P. Dale, Jan 27 2020 *)

is(n)=isprime(n) && n%30==23 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A158791(n)*30 + 23. - Ray Chandler, Apr 07 2009

Intersection of A030431 and A007528. - Ray Chandler, Apr 07 2009

Extended by Ray Chandler, Apr 07 2009

A154577 Primes of the form 2n^2+14n+5.

Original entry on oeis.org

5, 41, 293, 401, 461, 593, 821, 1181, 1493, 1721, 3593, 4493, 5081, 7793, 12941, 16001, 16361, 17093, 21821, 28541, 29021, 31481, 33521, 36161, 39461, 45281, 48341, 54101, 56093, 65141, 66593, 74093, 75641, 76421, 83621, 92861, 98993, 101681

Offset: 1

Vincenzo Librandi, Jan 12 2009

Primes in A154576. [Omar E. Pol, Aug 05 2009]
Subsequence of A007528, A040117. [Bruno Berselli, Jun 18 2014]
2*a(n) + 39 is a square. - Vincenzo Librandi, Apr 10 2015

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

Cf. A007528, A040117, A154576.

[a: n in [0..300] | IsPrime(a) where a is 2*n^2+14*n+5]; // Vincenzo Librandi, Jul 23 2012

Select[Table[2n^2+14n+5,{n,0,15001}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)

for (n=0, 300, if (isprime (k=2*n^2+14*n+5), print1 (k, ",  "))); \\ Vincenzo Librandi, Jul 23 2012

a(30)=66593 inserted, definition corrected and edited by Omar E. Pol, Aug 05 2009, Aug 06 2009

Added the first term 5 by Vincenzo Librandi, Jul 23 2012

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

A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 36, 42, 54, 48, 60, 84, 66, 78, 72, 126, 90, 102, 108, 114, 96, 120, 150, 138, 162, 132, 144, 168, 246, 156, 180, 186, 240, 204, 192, 216, 198, 210, 174, 258, 252, 222, 234, 228, 318, 282, 264, 276, 342, 306, 294, 312, 270, 354, 372

Offset: 1

Alexei Kourbatov, May 05 2020

nonn

Contains A268928 as a subsequence. First differs from A268928 at a(5)=24.
Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 has not occurred before, so a(2)=12.

Alexei Kourbatov, Table of n, a(n) for n = 1..160
Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.

Cf. A007528, A014320, A058320, A268928, A330853, A334544, A334545.

isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

a(n) = A334545(n) - A334544(n).

A343430 Part of n composed of prime factors of the form 3k-1.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 11, 4, 1, 2, 5, 16, 17, 2, 1, 20, 1, 22, 23, 8, 25, 2, 1, 4, 29, 10, 1, 32, 11, 34, 5, 4, 1, 2, 1, 40, 41, 2, 1, 44, 5, 46, 47, 16, 1, 50, 17, 4, 53, 2, 55, 8, 1, 58, 59, 20, 1, 2, 1, 64, 5, 22, 1, 68, 23, 10, 71, 8, 1, 2, 25, 4, 11, 2, 1, 80, 1, 82, 83, 4, 85

Offset: 1

Peter Munn, Jun 08 2021

Largest term of A004612 that divides n.
Modulo 6, the prime numbers are partitioned into 4 nonempty sets: {2}, {3}, primes of the form 6k-1 (A007528) and primes of the form 6k+1 (A002476). The modulo 3 partition is nearly the same, but unites the only even prime, 2, with primes of the form 6k-1 in the set of primes we use here.
A positive integer m is a Loeschian number (a term of A003136) if and only if a(A007913(m)) = 1, that is the squarefree part of m has no prime factors of the form 3k-1.

			n = 60 has prime factorization 60 = 2 * 2 * 3 * 5. Factors 2 = 3*1 - 1 and 5 = 3*2 - 1 have form 3k-1, whereas 3 does not (having form 3k). Multiplying the factors of form 3k-1, we get 2 * 2 * 5 = 20. So a(60) = 20.

Equivalent sequences for prime factors of other forms: A006519 (2 only), A000265 (2k+1), A038500 (3 only), A038502 (3k+/-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A343431 (6k-1).
Range of terms: A004612 (closure under multiplication of A003627).
Cf. A002476, A007528, squarefree part (A007913) of terms of A003136.
First 28 terms are the same as A247503.

f[p_, e_] := If[Mod[p, 3] == 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 11 2021 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 3, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343430(n): return prod(p**e for p, e in factorint(n).items() if p%3==2) # Chai Wah Wu, Dec 23 2022

Completely multiplicative with a(p) = p if p is of the form 3k-1, otherwise a(p) = 1.
For k >= 1, a(n) = a(k*n) / gcd(k, a(k*n)).
a(n) = A006519(n) * A343431(n).
a(n) = (n / A038500(n)) / A248909(n) = A038502(n) / A248909(n).
A006519(a(n)) = a(A006519(n)) = A006519(n).
A343431(a(n)) = a(A343431(n)) = A343431(n).
A038500(a(n)) = a(A038500(n)) = 1.
A248909(a(n)) = a(A248909(n)) = 1.

A045410 Primes congruent to {3, 5} mod 6.

Original entry on oeis.org

3, 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

Offset: 1

N. J. A. Sloane

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

Essentially the same as A007528.

Cf. A000040.

[ p: p in PrimesUpTo(600) | p mod 6 in {3,5} ]; // Vincenzo Librandi, Aug 12 2012

Select[Prime[Range[200]],MemberQ[{3,  5},Mod[#,6]]&] (* Vincenzo Librandi, Aug 12 2012 *)

is(n)=(n%6==5 && isprime(n)) || n==3 \\ Charles R Greathouse IV, Jul 20 2015

Extended by Charles R Greathouse IV, Mar 19 2010

cp's OEIS Frontend

A045309 Primes congruent to {0, 2} mod 3.

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A132231 Primes congruent to 7 (mod 30).

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A132232 Primes congruent to 11 (mod 30).

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A090686 Primes of the form 6n^2 - 1.

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A132235 Primes congruent to 23 (mod 30).

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A154577 Primes of the form 2n^2+14n+5.

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A141849 Primes congruent to 1 mod 11.

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