A038700 -id:A038700 - 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

A034694 Smallest prime == 1 (mod n).

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 103, 19, 191, 41, 43, 23, 47, 73, 101, 53, 109, 29, 59, 31, 311, 97, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 89, 181, 47, 283, 97, 197, 101, 103, 53, 107, 109, 331, 113, 229, 59, 709, 61, 367, 311

Offset: 1

Labos Elemer, David W. Wilson, Spring 1998

Thangadurai and Vatwani prove that a(n) <= 2^(phi(n)+1)-1. - T. D. Noe, Oct 12 2011
Conjecture: a(n) < n^2 for n > 1. - Thomas Ordowski, Dec 19 2016
Eric Bach and Jonathan Sorenson show that, assuming GRH, a(n) <= (1 + o(1))*(phi(n)*log(n))^2 for n > 1. See the abstract of their paper in the Links section. - Jianing Song, Nov 10 2019
a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order n. - Joerg Arndt, Oct 18 2020

			If n = 7, the smallest prime in the sequence 8, 15, 22, 29, ... is 29, so a(7) = 29.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 2.12, pp. 127-130.
P. Ribenboim, The Book of Prime Number Records. Chapter 4,IV.B.: The Smallest Prime In Arithmetic Progressions, 1989, pp. 217-223.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Mathematics of Computation, 65(216) (1996), 1717-1735.
Steven R. Finch, Linnik's Constant
S. Graham, On Linnik's Constant, Acta Arithm. 39, 1981, pp. 163-179.
I. Niven and B. Powell, Primes in Certain Arithmetic Progressions, Amer. Math. Monthly 83(6) (1976), 467-469.
R. Thangadurai and A. Vatwani, The least prime congruent to one modulo n, Amer. Math. Monthly 118(8) (2011), 737-742.

Cf. A034693, A034780, A034782, A034783, A034784, A034785, A034846, A034847, A034848, A034849, A038700, A085420.

Records: A120856, A120857.

a034694 n = until ((== 1) . a010051) (+ n) (n + 1)
-- Reinhard Zumkeller, Dec 17 2013

a[n_] := Block[{k = 1}, If[n == 1, 2, While[Mod[Prime@k, n] != 1, k++ ]; Prime@k]]; Array[a, 64] (* Robert G. Wilson v, Jul 08 2006 *)
With[{prs=Prime[Range[200]]},Flatten[Table[Select[prs,Mod[#-1,n]==0&,1],{n,70}]]] (* Harvey P. Dale, Sep 22 2021 *)

a(n)=if(n<0,0,s=1; while((prime(s)-1)%n>0,s++); prime(s))

a(n) = min{m: m = k*n + 1 with k > 0 and A010051(m) = 1}. - Reinhard Zumkeller, Dec 17 2013

a(n) = n * A034693(n) + 1. - Joerg Arndt, Oct 18 2020

A053989 Smallest k such that nk-1 is prime.

Original entry on oeis.org

3, 2, 1, 1, 4, 1, 2, 1, 2, 2, 4, 1, 8, 1, 2, 2, 4, 1, 2, 1, 2, 2, 6, 1, 6, 4, 2, 3, 6, 1, 2, 1, 4, 2, 4, 2, 2, 1, 6, 2, 4, 1, 6, 1, 2, 3, 6, 1, 2, 3, 2, 2, 4, 1, 2, 3, 2, 3, 6, 1, 8, 1, 4, 2, 6, 2, 6, 1, 2, 2, 4, 1, 14, 1, 2, 2, 4, 3, 2, 1, 8, 2, 4, 1, 6, 3, 2, 3, 16, 1, 2, 4, 6, 3, 4, 2, 2, 1, 2, 2

Offset: 1

Henry Bottomley, Apr 04 2000

			a(5)=4 because the smallest prime in the sequence 5k-1 (4,9,14,19,24...) is 19 when k=4

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A034693, A038700, A071558, A010051, A103689, A200996.

a053989 n = head [k | k <- [1..], a010051' (k * n - 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

a(n)=my(j);while(!isprime(j++*n-1),);j \\ Charles R Greathouse IV, Apr 18 2013

a(n) = (A038700(n)+1)/n.

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

A093868 Smallest prime that differs from a multiple of n by unity.

Original entry on oeis.org

2, 3, 2, 3, 11, 5, 13, 7, 17, 11, 23, 11, 53, 13, 29, 17, 67, 17, 37, 19, 41, 23, 47, 23, 101, 53, 53, 29, 59, 29, 61, 31, 67, 67, 71, 37, 73, 37, 79, 41, 83, 41, 173, 43, 89, 47, 281, 47, 97, 101, 101, 53, 107, 53, 109, 113, 113, 59, 353, 59, 367, 61, 127, 127, 131, 67, 269

Offset: 1

Amarnath Murthy, Apr 20 2004

Numbers n such that a(n-1)=a(n+1)=n are A025584 (primes p such that p-2 is not a prime). - Rick L. Shepherd, Aug 23 2004

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

Cf. A093869.

Cf. A034694 (Smallest prime == 1 (mod n)), A038700 (Smallest prime == -1 (mod n)).

f:= proc(n) local j,k;
  for k from 1 do
    for j in [-1,1] do
      if isprime(k*n+j) then return k*n+j fi
  od od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 07 2019

a[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[Divisible[p-1, n] || Divisible[p+1, n], Return[p]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 04 2023 *)

a(n) = forprime(p=2, , if (!((p+1) % n) || !((p-1) % n), return (p))); \\ Michel Marcus, Aug 08 2014

a(n) = min(A034694(n), A038700(n)) for all n >= 1. - Rick L. Shepherd, Aug 23 2004

More terms from Rick L. Shepherd, Aug 23 2004

A093871 a(n) is the n-th prime = -1 (mod n).

Original entry on oeis.org

2, 5, 11, 19, 89, 41, 181, 127, 251, 199, 571, 227, 1013, 433, 599, 751, 2039, 593, 2089, 859, 1637, 1429, 4001, 1103, 4049, 2053, 3779, 2267, 6263, 1499, 6571, 3583, 5279, 3943, 6089, 2879, 11321, 4597, 7331, 4919, 15497, 3779, 15307, 6599, 8009, 7681

Offset: 1

Amarnath Murthy, Apr 20 2004

nonn

Main diagonal of A093870.

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

Cf. A077317, A038700, A093870.

f:= proc(n) local p,count;
  count:= 0;
  for p from n-1 by n do
    if isprime(p) then
       count:= count+1;
       if count = n then return p fi
    fi
  od;
end proc:
map(f, [$1..100]); # Robert Israel, Nov 08 2019

Edited and extended by Franklin T. Adams-Watters, Aug 29 2006

A265647 Smallest k such that n divides k(k+1)(k+2)/6.

Original entry on oeis.org

1, 2, 7, 2, 3, 7, 5, 6, 25, 3, 9, 7, 11, 6, 8, 14, 15, 26, 17, 4, 7, 10, 21, 8, 23, 11, 79, 6, 27, 8, 29, 30, 9, 15, 5, 26, 35, 18, 25, 8, 39, 7, 41, 10, 25, 22, 45, 16, 47, 23, 16, 12, 51, 79, 9, 6, 17, 27, 57, 8, 59, 30, 26, 62, 13, 43, 65, 15, 44, 14, 69, 54, 71, 35, 25, 18, 20, 26, 77, 14, 241

Offset: 1

Ctibor O. Zizka, Dec 11 2015

nonn

More generally we can ask for the smallest k such that gcd(n,f(k)) = n. This sequence has f(k) = k*(k+1)*(k+2)/6. For other examples in the OEIS, see the crossrefencess.

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

Cf. A000292, A011772, A002034, A047930, A005179, A070982, A038700, A344005, A345989.

Table[k = 1; While[! Divisible[k (k + 1) (k + 2)/6, n], k++]; k, {n, 81}] (* Michael De Vlieger, Dec 11 2015 *)

a(n)=my(k=1);while((k*(k+1)*(k+2)/6)%n>0,k++);k \\ Anders Hellström, Dec 11 2015

first(n) = { my(todo = n, i = 1, res = vector(n)); while(todo > 0, d = select(x -> x <= n, divisors(binomial(i + 2, 3))); for(j = 1, #d, if(res[d[j]] == 0, res[d[j]] = i; todo-- ) ); i++ ); res } \\ David A. Corneth, Mar 22 2021

More terms from Michael De Vlieger, Dec 11 2015

A219109 The smallest k such that prime(k) == -1 (mod n).

Original entry on oeis.org

1, 2, 1, 2, 8, 3, 6, 4, 7, 8, 14, 5, 27, 6, 10, 11, 19, 7, 12, 8, 13, 14, 33, 9, 35, 27, 16, 23, 40, 10, 18, 11, 32, 19, 34, 20, 21, 12, 51, 22, 38, 13, 55, 14, 24, 33, 60, 15, 25, 35, 26, 27, 47, 16, 29, 39, 30, 40, 71, 17, 93, 18, 54, 31, 77, 32, 79, 19, 33, 34, 61, 20, 172, 21, 35, 36

Offset: 1

Irina Gerasimova, Apr 11 2013

nonn

Numbers n such that a(n) + 1 = a(n + 1) where the a(n)-th prime is not the smaller prime in a twin prime pair: 1, 3, 122, 267, 356, 362, 392, 403, 416, 446, 514, ....

Primes p(n) such that p is not -1 mod n for all prime p < p(n): 2, 3, 11, 31, 41, 59, 83, 97, 101, 109, 167, 191, 211, 277, 283, 313, 331, 367, 419,... Also primes p(n) such that p(n) <= A038700(n).

			For n = 11, we see that the 14th prime (43), modulo 11 is 10, or -1, so a(11) = 14.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A038700, A221861.

a(n)=forstep(t=n-1,n^99,n,if(isprime(t),return(primepi(t)))) \\ Charles R Greathouse IV, Mar 17 2014

a(n) = A000720(A038700(n)). - Joerg Arndt, Apr 16 2013

A084730 Smallest prime of the form k*n! - 1.

Original entry on oeis.org

2, 3, 5, 23, 239, 719, 5039, 201599, 1088639, 14515199, 159667199, 479001599, 31135103999, 87178291199, 2615348735999, 188305108991999, 711374856191999, 192071211171839999, 3649353012264959999, 12164510040883199999, 2248001455555215359999, 2248001455555215359999

Offset: 2

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003

nonn

			a(8) = 161599 = 4*8! - 1 is a prime.

Robert Israel, Table of n, a(n) for n = 2..447

Cf. A083663, A038700.

f:= proc(n) local w,k;
w:= n!;
for k from 1 do
  if isprime(k*w-1) then return k*w-1 fi
od
end proc:
map(f, [$1..30]); # Robert Israel, Nov 23 2023

Do[k = 1; While[ !PrimeQ[k*n! - 1], k++ ]; Print[k*n! - 1], {n, 2, 20}]

a(n)=if(n<1,0,k=1; while(isprime(k*n!-1)==0,k++); k*n!-1)

a(n) = A083663(n)*n! - 1 = A038700(n!). - Robert Israel, Nov 23 2023

Corrected and extended by Benoit Cloitre, Jun 14 2003

A086507 If n is even, a(n) = smallest prime == 1 (mod n), If n is odd, a(n) = smallest prime == -1 (mod n).

Original entry on oeis.org

2, 3, 2, 5, 19, 7, 13, 17, 17, 11, 43, 13, 103, 29, 29, 17, 67, 19, 37, 41, 41, 23, 137, 73, 149, 53, 53, 29, 173, 31, 61, 97, 131, 103, 139, 37, 73, 191, 233, 41, 163, 43, 257, 89, 89, 47, 281, 97, 97, 101, 101, 53, 211, 109, 109, 113, 113, 59, 353, 61, 487, 311, 251

Offset: 1

Amarnath Murthy, Jul 29 2003

nonn

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

Cf. A034694, A038700, A086508.

f:= proc(n) local x;
if n::even then x:= 1 else x:= -1; fi;
do
  x:= x+n;
  if isprime(x) then return x fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Dec 09 2020

More terms from David Wasserman, Mar 09 2005

cp's OEIS Frontend

A007528 Primes of the form 6k-1.

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A034694 Smallest prime == 1 (mod n).

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A053989 Smallest k such that nk-1 is prime.

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A141849 Primes congruent to 1 mod 11.

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A093868 Smallest prime that differs from a multiple of n by unity.

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A093871 a(n) is the n-th prime = -1 (mod n).

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A265647 Smallest k such that n divides k(k+1)(k+2)/6.