A061242 -id:A061242 - 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

A038194 Iterated sum-of-digits of n-th prime; or digital root of n-th prime; or n-th prime modulo 9.

Original entry on oeis.org

2, 3, 5, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 8, 5, 7, 4, 8, 1, 7, 2, 8, 7, 2, 4, 8, 1, 5, 1, 5, 2, 4, 5, 7, 4, 1, 5, 2, 8, 1, 2, 4, 8, 1, 4, 7, 2, 4, 8, 5, 7, 8, 5, 2, 8, 1, 7, 2, 4, 5, 1, 5, 7, 2, 7, 4, 5, 7, 2, 8, 7, 4, 1, 5, 2, 1, 5, 4, 5, 7, 8, 1, 7, 2, 8

Offset: 1

Den Roussel (DenRoussel(AT)webtv.net) and Clark Kimberling

Integers with iterated sum-of-digits 3, 6 or 9 are divisible by 3, so 3 is the only prime with iterated sum-of-digits 3 and there are no primes with iterated sum-of-digits 6 or 9.
The remaining values are very evenly distributed: these are the number of appearances in the first 1007933 primes: 1:167878; 2:168079; 4:167984; 5:168027; 7:167906; 8:168058. - Carmine Suriano, Jun 22 2015
Asymptotically, the ratios (number of primes <= n and == i mod 9)/(number of primes <= n and == j mod 9) go to 1 as n -> infinity for all i,j in {1,2,4,5,7,8} by the Prime Number Theorem for Arithmetic Progressions. For more detailed analysis, see the Granville-Martin link. - Robert Israel, Jul 08 2015

			Prime(5) = 11, 1 + 1 = 2 hence a(5) = 2.
a(297)=7 because the 297th prime is 1951 and 1+9+5+1 = 16 -> 1+6 = 7.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004

Cf. A007605, A010888, A061237 - A061242, A139413, A153110.

a038194 = flip mod 9 . a000040  -- Reinhard Zumkeller, Dec 10 2014

[p mod 9: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

A038194 := proc(n) return ithprime(n) mod 9: end: seq(A038194(n), n=1..100); # Nathaniel Johnston, May 04 2011

Table[Mod[Prime[n], 9], {n, 200}]
Mod[Prime[Range[100]], 9] (* Vincenzo Librandi, May 06 2014 *)

PARI
```
forprime(p=2,600,print1(p%9,","))
```

a(n) = A010888(A000040(n)).

Sum_k={1..n} a(k) ~ (9/2)*n. - Amiram Eldar, Dec 11 2024

Edited by Klaus Brockhaus, Feb 16 2002

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

A061237 Prime numbers == 1 (mod 9).

Original entry on oeis.org

19, 37, 73, 109, 127, 163, 181, 199, 271, 307, 379, 397, 433, 487, 523, 541, 577, 613, 631, 739, 757, 811, 829, 883, 919, 937, 991, 1009, 1063, 1117, 1153, 1171, 1279, 1297, 1423, 1459, 1531, 1549, 1567, 1621, 1657, 1693, 1747, 1783, 1801, 1873, 1999

Offset: 1

Amarnath Murthy, Apr 23 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A010888, A061238, A061239, A061240, A061241, A061242.

[ p: p in PrimesUpTo(2000) | p mod 9 in {1} ]; // Vincenzo Librandi, Dec 25 2010

Select[ Range[ 2000 ], PrimeQ[ # ] && Mod[ #, 9 ] == 1 & ]
Select[Prime[Range[350]],Mod[#,9]==1&] (* Harvey P. Dale, Jan 06 2013 *)

isok(p) = { isprime(p) && p%9 == 1 } \\ Harry J. Smith, Jul 19 2009

A010888(a(n)) = 1. - Reinhard Zumkeller, Feb 25 2005

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001

A061238 Prime numbers == 2 (mod 9).

Original entry on oeis.org

2, 11, 29, 47, 83, 101, 137, 173, 191, 227, 263, 281, 317, 353, 389, 443, 461, 479, 569, 587, 641, 659, 677, 821, 839, 857, 911, 929, 947, 983, 1019, 1091, 1109, 1163, 1181, 1217, 1289, 1307, 1361, 1433, 1451, 1487, 1523, 1559, 1613, 1667, 1721, 1811, 1847

Offset: 1

Amarnath Murthy, Apr 23 2001

A010888(a(n)) = 2. - Reinhard Zumkeller, Feb 25 2005

Except for the first term "2", all current prime numbers are of the form: 18*n-7. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011

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

Cf. A061237..A061242.

[ p: p in PrimesUpTo(2000) | p mod 9 in {2} ]; // Vincenzo Librandi, Dec 25 2010

Select[ Range[ 2000 ], PrimeQ[ # ] && Mod[ #, 9 ] == 2 & ]
Join[{2}, Select[Range[11, 5000, 18], PrimeQ]] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *) (* ~30 times faster *)

select(n->n%9==2, primes(400)) \\ Charles R Greathouse IV, May 27 2014

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001

A061240 Prime numbers == 5 (mod 9).

Original entry on oeis.org

5, 23, 41, 59, 113, 131, 149, 167, 239, 257, 293, 311, 347, 383, 401, 419, 491, 509, 563, 599, 617, 653, 743, 761, 797, 887, 941, 977, 1013, 1031, 1049, 1103, 1193, 1229, 1283, 1301, 1319, 1373, 1409, 1427, 1481, 1499, 1553, 1571, 1607, 1697, 1733, 1787

Offset: 1

Amarnath Murthy, Apr 23 2001

A010888(a(n)) = 5. - Reinhard Zumkeller, Feb 25 2005

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

Cf. A061237..A061242.

[ p: p in PrimesUpTo(2000) | p mod 9 in {5} ]; // Vincenzo Librandi, Dec 25 2010

Select[ Range[ 2500 ], PrimeQ[ # ] && Mod[ #, 9 ] == 5 & ]
Select[Prime[Range[300]],Mod[#,9]==5&] (* Harvey P. Dale, Oct 13 2017 *)

select(p->p%9==5,primes(1000)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001

A061241 Prime numbers == 7 (mod 9).

Original entry on oeis.org

7, 43, 61, 79, 97, 151, 223, 241, 277, 313, 331, 349, 367, 421, 439, 457, 547, 601, 619, 673, 691, 709, 727, 853, 907, 997, 1033, 1051, 1069, 1087, 1123, 1213, 1231, 1249, 1303, 1321, 1429, 1447, 1483, 1609, 1627, 1663, 1699, 1753, 1789, 1861, 1879, 1933

Offset: 1

Amarnath Murthy, Apr 23 2001

A010888(a(n)) = 7. - Reinhard Zumkeller, Feb 25 2005

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A061237..A061242.

[p: p in PrimesUpTo(2000) | p mod 9 eq 7]; // Vincenzo Librandi, Dec 25 2010

select(isprime,[seq(9*i+7,i=0..10^5)]); # Robert Israel, Apr 20 2014

Select[Range[2500], PrimeQ[#] && Mod[#, 9] == 7 &]
Select[Prime[Range[300]], Mod[#, 9] == 7 & ] (* Harvey P. Dale, Apr 30 2011 *)

select(n->n%9==7, primes(400)) \\ Charles R Greathouse IV, May 27 2014

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001

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

A061239 Prime numbers == 4 (mod 9).

Original entry on oeis.org

13, 31, 67, 103, 139, 157, 193, 211, 229, 283, 337, 373, 409, 463, 499, 571, 607, 643, 661, 733, 751, 769, 787, 823, 859, 877, 967, 1021, 1039, 1093, 1129, 1201, 1237, 1291, 1327, 1381, 1399, 1453, 1471, 1489, 1543, 1579, 1597, 1669, 1723, 1741, 1759

Offset: 1

Amarnath Murthy, Apr 23 2001

A010888(a(n)) = 4. - Reinhard Zumkeller, Feb 25 2005

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A061237..A061242.

[ p: p in PrimesUpTo(2000) | p mod 9 in {4} ]; // Vincenzo Librandi, Dec 25 2010

Select[ Range[ 2000 ], PrimeQ[ # ] && Mod[ #, 9 ] == 4 & ]
Select[Prime[Range[300]],Mod[#,9]==4&] (* Harvey P. Dale, Aug 20 2015 *)

select(n->n%9==4, primes(400)) \\ Charles R Greathouse IV, May 27 2014

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001

A138918 Numbers n such that 18n-1 is prime.

Original entry on oeis.org

1, 3, 4, 5, 6, 10, 11, 13, 14, 15, 20, 24, 25, 26, 28, 29, 31, 33, 36, 38, 39, 40, 43, 45, 46, 48, 49, 53, 54, 59, 61, 64, 66, 68, 70, 71, 76, 80, 83, 84, 88, 89, 90, 91, 95, 104, 105, 106, 110, 111, 115, 116, 119, 123, 126, 130, 131, 133, 134, 136, 144, 145, 148, 150

Offset: 1

Zak Seidov, Apr 03 2008

Corresponding primes are in A061242.

No terms in this sequence end with 2 or 7 (18n-1 ends with 5 when the last digit of n is 2 or 7). - David Garber, Jun 25 2015

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A061242, A258663.

[n: n in [0..150] | IsPrime(18*n-1)]; // Vincenzo Librandi, Jun 27 2015

Select[Range[200],PrimeQ[18#-1]&]  (* Harvey P. Dale, Mar 09 2011 *)

for(n=1,10^3,if(isprime(18*n-1),print1(n,", "))) \\ Derek Orr, Sep 03 2014

from gmpy2 import divexact, t_mod
from sympy import prime
A138918 = [divexact(p+1,18) for p in (prime(n) for n in range(1,10**6)) if not t_mod(p+1,18)] # Chai Wah Wu, Sep 02 2014

A138905 a(n) is n-th prime == -1 (mod 6n).

Original entry on oeis.org

5, 23, 71, 167, 179, 431, 461, 863, 863, 839, 1583, 1511, 1949, 2099, 2339, 4127, 4283, 4751, 4673, 4919, 5669, 6599, 8693, 10079, 7349, 10607, 12149, 11087, 12527, 11159, 15809, 19583, 16829, 19583, 13859, 25703, 24197, 25307, 23633, 21839, 34439

Offset: 1

Zak Seidov, Apr 03 2008

nonn

			a(1) = 1st term in A007528 (Primes of form 6n-1)
a(2) = 2nd term in A068231 (Primes congruent to 11 (mod 12))
a(3) = 3rd term in A061242 (Primes of form 18n-1)
a(4) = 4th term in A134517 (Primes of form 24n-1)
a(5) = 5th term in A132236 (Primes congruent to 29 (mod 30))

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A007528, A061242, A068231, A093871, A132236, A134517, A138906, A138907.

a[n_]:=Module[{p=1,cnt=0},Until[cnt==n,If[Mod[Prime[p],6n]==6n-1,cnt++];p++];Prime[p-1]];Array[a,41] (* James C. McMahon, Jun 22 2025 *)

cp's OEIS Frontend

A007528 Primes of the form 6k-1.

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A038194 Iterated sum-of-digits of n-th prime; or digital root of n-th prime; or n-th prime modulo 9.

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A061237 Prime numbers == 1 (mod 9).

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A061238 Prime numbers == 2 (mod 9).

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A061240 Prime numbers == 5 (mod 9).

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A061241 Prime numbers == 7 (mod 9).

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