A141850 -id:A141850 - 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

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

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

A306279 Numbers congruent to 3 or 18 mod 22.

Original entry on oeis.org

3, 18, 25, 40, 47, 62, 69, 84, 91, 106, 113, 128, 135, 150, 157, 172, 179, 194, 201, 216, 223, 238, 245, 260, 267, 282, 289, 304, 311, 326, 333, 348, 355, 370, 377, 392, 399, 414, 421, 436, 443, 458, 465, 480, 487, 502, 509, 524, 531, 546, 553, 568

Offset: 1

Davis Smith, Feb 02 2019

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A020639, A042948, A091999, A105398, A131555, A141850, A158459, A273669, A306277, A306289.

Primes in this sequence: A141850.

Maple
```
seq(seq(22*i+j, j=[3, 18]), i=0..200);
```

Select[Range[200], MemberQ[{3, 18}, Mod[#, 22]] &]
Flatten[Table[{22n + 3, 22n + 18}, {n, 0, 43}]] (* Alonso del Arte, Feb 18 2019 *)

for(n=3, 678, if((n%22==3) || (n%22==18), print1(n, ", ")))

PARI
```
vector(62,n,11*n-6+2*(-1)^n)
```

Vec(x*(3 + 15*x + 4*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 07 2019

(3 to 949 by 22).union(18 to 942 by 22).sorted // Alonso del Arte, Feb 18 2019

a(n) = 11*n - 6 + 2*(-1)^n.
a(n) = 11*n - A105398(n + 4).
A007310(a(n) + 1) = 11*A007310(n).
From Colin Barker, Feb 07 2019: (Start)
G.f.: x*(3 + 15*x + 4*x^2) / ((1 - x)^2*(1 + x)).
a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3. (End)
E.g.f.: 4 + (11*x - 6)*exp(x) + 2*exp(-x). - David Lovler, Sep 08 2022

A262608 Primes p such that floor(10*p/Pi) mod 10 = 0.

Original entry on oeis.org

19, 41, 107, 151, 173, 239, 283, 349, 421, 443, 487, 509, 619, 641, 751, 773, 839, 883, 971, 1087, 1103, 1109, 1153, 1307, 1373, 1439, 1483, 1549, 1571, 1637, 1747, 1907, 1951, 1973, 2017, 2039, 2083, 2237, 2281, 2347, 2551, 2617, 2683, 2749, 2837, 2903, 2969

Offset: 1

Giovanni Teofilatto, Sep 26 2015

nonn

a(n) = A141855(n)   for  1 <= n <=  8;
a(n) = A141850(n-1) for  9 <= n <= 19;
a(n) = A141856(n-4) for 22 <= n <= 31;
a(n) = A141851(n-5) for 32 <= n <= 40;
a(n) = A141857(n-3) for 41 <= n <= 49.

			19 is a term because floor(19*10/Pi) = 60 and 60 mod 10 = 0.

Cf. A000796, A141850, A141851, A141855, A141856, A141857.

Select[Prime@ Range@ 432, Mod[Floor[10 #/Pi], 10] == 0 &] (* Michael De Vlieger, Dec 09 2015 *)

forprime(p=2, 1e4, if (10*(p\Pi) == 10*p\Pi , print1(p", "))) \\ Altug Alkan, Sep 26 2015

More terms and better definition from Altug Alkan, Sep 26 2015

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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A141849 Primes congruent to 1 mod 11.

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A141851 Primes congruent to 4 mod 11.

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A306279 Numbers congruent to 3 or 18 mod 22.

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A262608 Primes p such that floor(10*p/Pi) mod 10 = 0.

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