cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A007528 Primes of the form 6k-1.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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.

Links

Crossrefs

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.

Programs

  • GAP
    Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018
  • Haskell
    a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012
  • Maple
    select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018
  • Mathematica
    Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)
  • PARI
    forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011
  • PARI
    forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

Formula

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

Views

Author

N. J. A. Sloane, Jul 11 2008

Keywords

Comments

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

Links

Crossrefs

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)).

Programs

Formula

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

A263769 Smallest prime q such that q == -1 (mod prime(n)-1).

Original entry on oeis.org

2, 3, 3, 5, 19, 11, 31, 17, 43, 83, 29, 71, 79, 41, 137, 103, 173, 59, 131, 139, 71, 233, 163, 263, 191, 199, 101, 211, 107, 223, 251, 389, 271, 137, 443, 149, 311, 647, 331, 859, 1423, 179, 379, 191, 587, 197, 419, 443
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 25 2015

Keywords

Comments

a(n): A000040(1), A065091(1), A002145(1), A007528(1), A030433(1), A068231(1), A127576(1), A061242(1), A141857(1), A141976(1), A132236(1), A142111(1), A142198(1), A141898(1), A141926(1), A142531(1), A142004(1), A142799(1), A142068(1), A142099(1), ...
Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

Examples

			a(4) = 5 because 5 == -1 (mod prime(4)-1) and is prime.

Links

Crossrefs

Cf. A263730, A263770.
Cf. A000040, A065091, A002145, A007528, A030433, A068231, A127576, A061242, A141857, A141976, A132236, A142111, A142198, A141898, A141926, A142531, A142004, A142799, A142068, A142099.

Programs

  • Maple
    for n from 1 to 100 do
  k:= ithprime(n)-1;
  q:= 2;
  while (1 + q) mod k <> 0 do
    q:= nextprime(q)
  od;
  A[n]:= q;
od:
seq(A[i],i=1..1000); # Robert Israel, Oct 26 2015
  • Mathematica
    Table[q = 2; z = Prime@ n - 1; While[Mod[q, z] != z - 1, q = NextPrime@ q]; q, {n, 59}] (* Michael De Vlieger, Oct 26 2015 *)

Extensions

Corrected and edited by Robert Israel, Oct 26 2015,

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

Views

Author

Giovanni Teofilatto, Sep 26 2015

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Select[Prime@ Range@ 432, Mod[Floor[10 #/Pi], 10] == 0 &] (* Michael De Vlieger, Dec 09 2015 *)
  • PARI
    forprime(p=2, 1e4, if (10*(p\Pi) == 10*p\Pi , print1(p", "))) \\ Altug Alkan, Sep 26 2015

Extensions

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

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

Views

Author

Charles R Greathouse IV, Mar 18 2022

Keywords

Comments

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

Crossrefs

Primes that are 0 or +-1 mod p: A000040 (2, 3), A038872 (5), A045463 (7), this sequence (11).
Cf. A141849 (subsequence), A141857 (subsequence).

Programs

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

Formula

a(n) ~ 5n log n. - Charles R Greathouse IV, Mar 18 2022
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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