A141849 -id:A141849 - 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

A296920 Rational primes that decompose in the quadratic field Q(sqrt(-11)).

Original entry on oeis.org

3, 5, 23, 31, 37, 47, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 179, 181, 191, 199, 223, 229, 251, 257, 269, 311, 313, 317, 331, 353, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521, 577, 587, 599, 617, 619, 631, 641, 643, 647, 653, 661, 683, 691, 709, 719

Offset: 1

N. J. A. Sloane, Dec 25 2017

Primes that are 1, 3, 5, 9, or 15 mod 22. - Charles R Greathouse IV, Mar 18 2018

(Which means: union of A141849, A141850, A141852, A141856 and A141851. - R. J. Mathar, Apr 15 2024)

Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.

Cf. A191060, A056874.

# In the quadratic field Q(sqrt(D)), for squarefree D<0, compute lists of:
# rational primes that decompose (SD),
# rational primes that are inert (SI),
# primes p such that D is a square mod p (QR), and
# primes p such that D is a nonsquare mod p (NR),
# omitting the latter if it is the same as the inert primes.
# Consider first M primes p.
# Reference: Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.
with(numtheory):
HH := proc(D,M)
    local SD,SI,QR,NR,p,q,i,t1;
    # if D >= 0 then error("D must be negative"); fi;
    if not issqrfree(D) then
        error("D must be squarefree");
    end if;
    q:=-D;
    SD:=[]; SI:=[]; QR:=[]; NR:=[];
    if (D mod 8) = 1 then
        SD:=[op(SD),2];
    end if;
    if (D mod 8) = 5 then
        SI:=[op(SI),2];
    end if;
    for i from 2 to M do
        p:=ithprime(i);
        if (D mod p) <> 0 and legendre(D,p)=1 then
            SD:=[op(SD),p];
        end if;
        if (D mod p) <> 0 and legendre(D,p)=-1 then
            SI:=[op(SI),p];
        end if;
    end do;
    for i from 1 to M do
        p:=ithprime(i);
        if legendre(D,p) >= 0 then
            QR:=[op(QR),p];
        else
        NR:=[op(NR),p];
        end if;
    end do:
    lprint("Primes that decompose:", SD);
    lprint("Inert primes:", SI);
    lprint("Primes p such that Legendre(D,p) = 0 or 1: ", QR);
    if SI <> NR then
        lprint("Note: SI <> NR here!");
        lprint("Primes p such that Legendre(D,p) = -1: ", NR);
    end if;
end proc:
HH(-11,200); # produces the present sequence (A296920), A191060, and A056874.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[KroneckerSymbol[-11, p] == 1, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Apr 29 2019 *)

list(lim)=my(v=List()); forprime(p=2,lim, if(kronecker(-11,p)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018

a(n) ~ 2n log n. - Charles R Greathouse IV, Mar 18 2018

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

A141850 Primes congruent to 3 mod 11.

Original entry on oeis.org

3, 47, 113, 157, 179, 223, 311, 421, 443, 487, 509, 619, 641, 751, 773, 839, 883, 971, 1103, 1213, 1279, 1301, 1367, 1433, 1499, 1543, 1609, 1697, 1741, 1873, 2027, 2137, 2203, 2269, 2357, 2423, 2467, 2621, 2687, 2731, 2753, 2797, 2819, 3061, 3083, 3259, 3347

Offset: 1

N. J. A. Sloane, Jul 11 2008

Primes congruent to 3 mod 22. - Chai Wah Wu, Apr 29 2025

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

Cf. A141849, A090187.

[ p: p in PrimesUpTo(5000) | p mod 11 eq 3 ]; // Vincenzo Librandi, Apr 19 2011

Select[Range[3,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)
Select[Prime[Range[500]],Mod[#,11]==3&] (* Harvey P. Dale, Jul 14 2021 *)

is(n)=isprime(n) && n%11==3 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A141851 Primes congruent to 4 mod 11.

Original entry on oeis.org

37, 59, 103, 191, 257, 367, 389, 433, 499, 521, 587, 631, 653, 719, 829, 983, 1049, 1093, 1181, 1291, 1423, 1489, 1511, 1621, 1709, 1753, 1907, 1951, 1973, 2017, 2039, 2083, 2237, 2281, 2347, 2633, 2677, 2699, 2897, 2963, 3271, 3359, 3469, 3491, 3557, 3623

Offset: 1

N. J. A. Sloane, Jul 11 2008

Primes congruent to 15 mod 22. - Chai Wah Wu, Apr 29 2025

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

Cf. A000040, A144562, A141849, A090187, A141850.

[ p: p in PrimesUpTo(5000) | p mod 11 eq 4 ]; // Vincenzo Librandi, Apr 19 2011

Select[Range[4,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==4 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A090187 Primes of the form 11*n+2.

Original entry on oeis.org

2, 13, 79, 101, 167, 211, 233, 277, 409, 431, 541, 563, 607, 673, 739, 761, 827, 937, 1069, 1091, 1201, 1223, 1289, 1399, 1487, 1531, 1553, 1597, 1619, 1663, 1861, 1949, 1993, 2081, 2213, 2389, 2411, 2477, 2521, 2543, 2609, 2719, 2741, 2851, 2917, 2939, 3049

Offset: 1

Giovanni Teofilatto, Feb 04 2004

Primes congruent to 2 mod 11. - Vincenzo Librandi, Aug 06 2012

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

Cf. A141849.

[p: p in PrimesUpTo(3000) | p mod 11 eq 2]; // Vincenzo Librandi, Aug 06 2012

Select[Range[2,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

A102656 Numbers k such that 11*k + 1 is prime.

Original entry on oeis.org

2, 6, 8, 18, 30, 32, 36, 38, 42, 56, 60, 62, 66, 78, 80, 86, 90, 92, 102, 116, 120, 128, 132, 146, 162, 170, 182, 188, 192, 198, 206, 210, 212, 216, 218, 230, 242, 246, 248, 260, 266, 270, 276, 288, 290, 296, 300, 302, 308, 312, 318, 330, 336, 338, 350, 356, 366

Offset: 1

Parthasarathy Nambi, Feb 02 2005

nonn

			At n=2, 11*2 + 1 = 23 (prime).
At n=60, 11*60 + 1 = 661 (prime).
At n=120, 11*120 + 1 = 1321 (prime).

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

Cf. A141849, A215667.

Filtered([1..380], n-> IsPrime(11*n+1)); # G. C. Greubel, Apr 03 2019

[n: n in [1..380] | IsPrime(11*n+1)]; // G. C. Greubel, Apr 03 2019

select(t -> isprime(11*t+1), [seq(i,i=2..1000,2)]); # Robert Israel, Apr 03 2019

Select[ Range[2, 367, 2], PrimeQ[11# + 1] &] (* Robert G. Wilson v, Feb 04 2005 *)

is(n)=isprime(11*n+1) \\ Charles R Greathouse IV, Jun 06 2017

[n for n in (1..380) if is_prime(11*n+1)] # G. C. Greubel, Apr 03 2019

a(n) = (A141849(n) -1)/11 = 2*A215667(n). - Robert Israel, Apr 03 2019

More terms from Robert G. Wilson v, Feb 04 2005

A142302 Primes congruent to 23 mod 44.

Original entry on oeis.org

23, 67, 199, 331, 419, 463, 683, 727, 859, 947, 991, 1123, 1607, 1783, 1871, 2003, 2179, 2267, 2311, 2399, 2531, 2663, 2707, 2927, 2971, 3191, 3323, 3499, 3631, 3719, 3851, 4027, 4159, 4423, 4643, 4951, 5039, 5171, 5303, 5347, 5479, 5743, 6007, 6271, 6359

Offset: 1

N. J. A. Sloane, Jul 11 2008

Subsequence of A141849. - Zak Seidov, Sep 22 2020

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

Cf. A000040, A141849.

[p: p in PrimesUpTo(8000) | p mod 44 eq 23]; // Vincenzo Librandi, Aug 25 2012

Select[Prime[Range[2500]],MemberQ[{23},Mod[#,44]]&] (* Vincenzo Librandi, Aug 25 2012 *)

is(n)=isprime(n) && n%44==23 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 20n log n. - Charles R Greathouse IV, Jul 02 2016

A215667 22n+1 is prime.

Original entry on oeis.org

1, 3, 4, 9, 15, 16, 18, 19, 21, 28, 30, 31, 33, 39, 40, 43, 45, 46, 51, 58, 60, 64, 66, 73, 81, 85, 91, 94, 96, 99, 103, 105, 106, 108, 109, 115, 121, 123, 124, 130, 133, 135, 138, 144, 145, 148, 150, 151, 154, 156, 159, 165, 168, 169, 175, 178, 183, 184, 186

Offset: 1

Zak Seidov, Aug 20 2012

nonn

First terms that are not in A125874: 19, 31, 39, 43, 46, 51, 58, 73, 85, 91 (terms with prime factor > 11).

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

Cf. A141849, A125874.

Select[Range[200],PrimeQ[22#+1]&] (* Harvey P. Dale, May 25 2025 *)

isok(n) = isprime(22*n+1); \\ Michel Marcus, Oct 19 2013

a(n) = (A141849(n) - 1)/22.

A352494 Primes congruent to 0, 1, or 10 mod 11.

Original entry on oeis.org

11, 23, 43, 67, 89, 109, 131, 197, 199, 241, 263, 307, 331, 353, 373, 397, 419, 439, 461, 463, 571, 593, 617, 659, 661, 683, 727, 769, 857, 859, 881, 947, 967, 991, 1013, 1033, 1123, 1187, 1231, 1277, 1297, 1319, 1321, 1409, 1429, 1451, 1453, 1583, 1607, 1627, 1693, 1759, 1783, 1847, 1871, 1913, 1979, 2003, 2069

Offset: 1

Charles R Greathouse IV, Mar 18 2022

Possible prime factors of x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1.

Primes that are 0 or +-1 mod p: A000040 (2, 3), A038872 (5), A045463 (7), this sequence (11).

Cf. A141849 (subsequence), A141857 (subsequence).

Select[Prime[Range[300]], MemberQ[{0, 1, 10}, Mod[#, 11]] &] (* Amiram Eldar, Mar 18 2022 *)

a(n) ~ 5n log n. - Charles R Greathouse IV, Mar 18 2022

cp's OEIS Frontend

A007528 Primes of the form 6k-1.

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A296920 Rational primes that decompose in the quadratic field Q(sqrt(-11)).

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A140444 Primes congruent to 1 (mod 14).

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A141850 Primes congruent to 3 mod 11.

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A141851 Primes congruent to 4 mod 11.

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A090187 Primes of the form 11*n+2.

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A102656 Numbers k such that 11*k + 1 is prime.