A038026 -id:A038026 - 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

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

A038025 Winner of n-th Littlewood Frog Race.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 1, 1, 9, 10, 1, 12, 1, 1, 9, 1, 1, 1, 1, 4, 21, 22, 1, 24, 25, 1, 27, 27, 1, 1, 1, 16, 1, 16, 35, 32, 1, 38, 9, 10, 25, 33, 25, 1, 45, 27, 1, 25, 49, 44, 25, 24, 1, 1, 9, 34, 27, 1, 49, 24, 1, 58, 57, 64, 49, 8, 49, 65, 51, 48, 49, 72, 69, 68

Offset: 1

Christian G. Bower from a problem by David W. Wilson

nonn

For 0 < k <= n, gcd(n,k) = 1, let P(n,k) be the smallest prime of the form a*n+k, with a >= 0. "Frog" k0 is said to win "race" n if P(n,k0) is largest of the phi(n) values P(n,k).

In case of draws of P(n,k) values take the largest k. - R. J. Mathar, Jul 26 2015

Cf. A038026, A038029 (records).

A038025P := proc(n,k)
    local a;
    for a from 0 do
        if isprime(a*n+k) then
            return a;
        end if;
    end do:
end proc:
A038025 := proc(n)
    local a,phimax,phi,k ;
    a :=0 ;
    phimax := 0 ;
    for k from 1 to n do
        if igcd(k,n) = 1 then
            phi := A038025P(n,k) ;
            if phi >= phimax then
                a := k;
                phimax := phi;
            end if;
        end if;
    od;
    a ;
end proc:
seq(A038025(n),n=1..100) ; # R. J. Mathar, Jul 26 2015

A038025P[n_, k_] := Module[{a}, For[a = 0, True, a++, If[PrimeQ[a n + k], Return[a]]]];
A038025[n_] := Module[{a = 0, phiMax = 0, phi, k}, For[k = 1, k <= n, k++, If [GCD[k, n] == 1, phi = A038025P[n, k]; If[phi >= phiMax, a = k; phiMax = phi]]]; a];
Array[A038025, 100] (* Jean-François Alcover, Apr 16 2020, after R. J. Mathar *)

A085420 For each n, let p(n,b) be the smallest prime in the arithmetic progression k*n+b, with k > 0. Then a(n) = max(p(n,b)) with 0 < b < n and gcd(b,n) = 1.

Original entry on oeis.org

2, 3, 7, 7, 19, 11, 29, 23, 43, 19, 71, 23, 103, 53, 43, 43, 103, 53, 191, 59, 97, 79, 233, 73, 269, 103, 173, 83, 317, 79, 577, 151, 227, 193, 239, 157, 439, 191, 233, 157, 587, 107, 467, 257, 389, 307, 967, 191, 613, 269, 421, 601, 659, 199, 353, 233, 433, 317, 709

Offset: 1

T. D. Noe, Jun 29 2003

nonn

Linnik proved that there are n0 and L such that a(n) < n^L for all n > n0. It has been conjectured that a(n) < n^2. The sequence A034694 has the primes p(n,1).

a(n) is also the maximum term in row n of A060940, which defines a(1). A007918(n+1) is the minimum term in row n of A060940. - Seiichi Manyama, Apr 02 2018

			a(5) = 19 because p(5,1) = 11, p(5,2) = 7, p(5,3) = 13 and p(5,4) = 19.

P. Ribenboim, The New Book of Prime Number Records, Springer, 1996, pp. 277-284.

T. D. Noe, Table of n, a(n) for n = 1..10000 (corrected by Michel Marcus, Jan 19 2019)
A. Granville, Least primes in arithmetic progressions, Théorie des nombres / Number Theory (ed. J. M. De Koninck & C. Levesque), (de Gruyter, New York, 1989), 306-321.
Eric Weisstein's World of Mathematics, Linnik's Theorem

A038026 is a lower bound.

Cf. A034694.

minP[n_, a_] := Module[{k, p}, If[GCD[n, a]>1, p=0, k=1; While[ !PrimeQ[k*n+a], k++ ]; p=k*n+a]; p]; Table[Max[Table[minP[n, i], {i, n-1}]], {n, 2, 100}]

p(n,b)=while(!isprime(b+=n),); ba(n)=my(t=p(n,1));for(b=2,n-1,if(gcd(n,b)==1,t=max(t,p(n,b))));t \\ Charles R Greathouse IV, Sep 08 2012

a(1) defined via A060940 by Seiichi Manyama, Apr 02 2018

A358240 Consider all invertible residues mod n. For each residue, find the smallest product of three primes (A014612) which is in that residue class mod n. a(n) is the greatest of these.

Original entry on oeis.org

8, 27, 28, 45, 66, 175, 45, 105, 76, 171, 102, 325, 165, 261, 124, 273, 230, 385, 188, 369, 268, 255, 175, 475, 284, 549, 436, 477, 285, 1309, 332, 385, 430, 927, 318, 1127, 442, 639, 610, 657, 595, 1075, 742, 805, 724, 637, 646, 1705, 642, 741, 670, 1005, 885, 1435, 801, 1705, 1105, 873, 1004, 2821, 938, 873, 844

Offset: 1

Charles R Greathouse IV, Jan 18 2023

nonn

			The least product of 3 primes = 1 mod 3 is 28, while the least = 2 mod 3 is 8, so a(2) = 28.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ramachandran Balasubramanian, Olivier Ramaré, and Priyamvad Srivastav, Product of three primes in large arithmetic progressions, arXiv:2208.04031 [math.NT], 2022.

All terms are in A014612.

Cf. A038026, A085420.

firstTri(m)=my(mod=m.mod); forprime(p=2,, if(mod%p==0, next); forprime(q=2,p, if(mod%q==0, next); forprimestep(r=2,q,m/p/q, return(p*q*r))))
a(n)=my(r=8); for(k=1,n-1, if(gcd(k,n)>1, next); r=max(firstTri(Mod(k,n)),r)); r

A result of Balasubramanian, Ramaré, & Srivastav proves that a(n) < n^e for each e > 9/2 and large enough n depending on e.

Corrected by Charles R Greathouse IV, May 10 2023

A220533 a(n) is minimal number such that the set of all composite numbers <= a(n) contains complete residue system modulo n.

Original entry on oeis.org

4, 9, 8, 15, 12, 35, 14, 27, 20, 33, 24, 65, 26, 45, 32, 45, 36, 77, 38, 63, 44, 63, 46, 95, 48, 69, 56, 87, 60, 187, 62, 93, 64, 105, 72, 175, 74, 117, 80, 123, 84, 215, 86, 117, 92, 135, 94, 245, 96, 153, 104, 141, 106, 245, 108, 165, 116

Offset: 1

Vladimir Shevelev, Feb 20 2013

nonn

Cf. A038026, A210537, A211204.

A220533 := proc(n)
    local sco,c,ai,scor ;
    sco := {} ;
    for ai from 1 do
        sco := sco union {A002808(ai)} ;
        scor := convert( [seq(c mod n,c=sco)], set) ;
        if nops(scor) = n then
            return A002808(ai) ;
        end if;
    end do:
end proc:
seq(A220533(n),n=1..60) ; # R. J. Mathar, Feb 27 2013

A002808[n_] := A002808[n] = Module[{a}, If[ n == 1 , 4, For[a = A002808[n-1] + 1 , True, a++, If[! PrimeQ[a], Return [a]]]]]; A220533[n_] := Module[{ sco, c, ai, scor}, sco = {}; For[ai = 1, True, ai++, AppendTo[sco, A002808[ai]] ; scor = Mod[#, n]& /@ sco // Union; If[Length[scor] == n , Return[A002808[ai]]]]]; Table[A220533[n], {n, 1, 57}] (* Jean-François Alcover, Feb 28 2013, translated from R. J. Mathar's Maple program *)

For odd n, a(n) <= 2n + 2; the equality holds if and only if n + 2 is prime.

a(n)<2*n for 25,33,49,... - R. J. Mathar and Vladimir Shevelev, Feb 28 2013

Corrected and extended by R. J. Mathar, Feb 27 2013

A358242 Consider all invertible residues k mod n. For each such k, find the product of three primes pqr = k (mod n) with the smallest max {p, q, r}. Then a(n) is the largest such p over the considered k.

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 5, 7, 7, 11, 7, 11, 11, 11, 13, 11, 13, 11, 11, 13, 17, 13, 13, 13, 19, 17, 17, 17, 13, 17, 17, 17, 19, 19, 29, 17, 17, 13, 23, 19, 23, 19, 23, 17, 29, 17, 23, 23, 23, 19, 23, 19, 23, 17, 31, 23, 29, 19, 29, 29, 29, 19, 23

Offset: 1

Charles R Greathouse IV, Jan 18 2023

nonn

This is an analog of Linnik's theorem for 3-almost primes. - Charles R Greathouse IV, Jun 30 2024

			From _M. F. Hasler_, Jul 02 2024: (Start)
For n = 1, there is only one residue class in Z/nZ = {Z}, and it is invertible (since 0 = 1 in this case), and p = q = r = 2 satisfies p*q*r = k (mod n) and gives clearly the smallest possible prime to satisfy the conditions, so a(1) = 2.
For n = 2, there is only one invertible residue class in Z/nZ, namely that of k = 1, and none of p, q, r may equal 2 (= 0 in Z/2Z) to yield p*q*r = 1 (mod n), so p = q = r = 3 is the smallest solution to p*q*r = 1 (mod n), whence a(2) = 3.
For n = 3, the two invertible residues (mod n) are k = 1 and k = 2. For p = q = r = 2, we have p*q*r = 8 = 2 (mod 3), but for the residue k = 1, we need a different solution, which therefore can't have as largest prime p = 2 (=> k = 2) nor p = 3 = 0 (mod 3) nor p = 5 = 2 (mod 3), but p = 7 >= q = r = 2 does work, with 4*7 = 28 = 1 (mod 3), so a(3) = 7. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..100000
Ramachandran Balasubramanian, Olivier Ramaré, and Priyamvad Srivastav, Product of three primes in large arithmetic progressions, International Journal of Number Theory, Vol. 19, No. 4 (2023), pp. 843-857; arXiv preprint, arXiv:2208.04031 [math.NT], 2022.
Barnabás Szabó, On the existence of products of primes in arithmetic progressions, Bulletin of the London Mathematical Society 56 (3), pp. 1227-1243, 2024.

Cf. A358240, A038026, A085420, A014612.

do(m)=my(mod=m.mod); forprime(p=2,, if(mod%p==0, next); forprime(q=2,p, if(mod%q==0, next); forprimestep(r=2,q,m/p/q, return(p))))
a(n)=my(r=2); for(k=1,n-1, if(gcd(k,n)>1, next); r=max(do(Mod(k,n)),r)); r

a(n)=my(N,s=eulerphi(n)); forprime(p=2,, if(n%p==0, next); forprime(q=2,p, if(n%q==0, next); my(pq=p*q%n); forprime(r=2,q, if(n%r==0, next); my(pqr=pq*r%n); if(bittest(N,pqr)==0, N+=1<Charles R Greathouse IV, Jun 30 2024

Balasubramanian, Ramaré, & Srivastav prove that a(n) < n^e for each e > 3/2 and large enough n depending on e.

Szabó improves this to e > 6/5. - Charles R Greathouse IV, Jun 30 2024

Definition clarified by Charles R Greathouse IV and M. F. Hasler, Jul 02 2024

cp's OEIS Frontend

A038031 Position reached by frog in A038029. A038026(A038029(n)).

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A038028 Position reached by frog in A038027 or 0 if none. A038026(A038027(n)).

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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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A038025 Winner of n-th Littlewood Frog Race.

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A085420 For each n, let p(n,b) be the smallest prime in the arithmetic progression k*n+b, with k > 0. Then a(n) = max(p(n,b)) with 0 < b < n and gcd(b,n) = 1.

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A358240 Consider all invertible residues mod n. For each residue, find the smallest product of three primes (A014612) which is in that residue class mod n. a(n) is the greatest of these.

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A220533 a(n) is minimal number such that the set of all composite numbers <= a(n) contains complete residue system modulo n.

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A358242 Consider all invertible residues k mod n. For each such k, find the product of three primes pqr = k (mod n) with the smallest max {p, q, r}. Then a(n) is the largest such p over the considered k.