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.

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A068232 a(n) is the smallest prime p such that p and the next n-1 primes are all == 1 (mod 12).

Original entry on oeis.org

13, 661, 8317, 12829, 586153, 1081417, 7790917, 7790917, 370861009, 370861009, 370861009, 5637496849, 289391626057, 469257742237, 628337233501, 84424712545429, 155494152002017, 341821313785729
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Keywords

Comments

Dickson's conjecture implies that a(n) exists for all n.

Links

Crossrefs

Cf. A068227, A068228, A068229, A040117, A068231, A068233, A068234, A068235.

Programs

  • Mathematica
    For[i=n=1, True, Null, For[j=0, jHarvey P. Dale, Dec 24 2020 *)
  • PARI
    {i=n=1; while(1,j=0; while(j

Extensions

Edited by Dean Hickerson, Mar 06 2002
a(12)-a(15) from Giovanni Resta, Feb 18 2006
a(16)-a(18) from Giovanni Resta, Aug 04 2013

A068233 a(n) is the smallest prime p such that p and the next n-1 primes are all == 7 (mod 12).

Original entry on oeis.org

7, 199, 199, 32443, 180799, 180799, 4338787, 84885631, 472798219, 1786054267, 6024282871, 64791932287, 592175010019, 6265824724519, 7816088451907, 24660781037467
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Keywords

Comments

Dickson's conjecture implies that a(n) exists for all n.

Links

Crossrefs

Cf. A068227, A068228, A068229, A040117, A068231, A068232, A068234, A068235.

Programs

  • Mathematica
    For[i=n=1, True, Null, For[j=0, j
  • PARI
    {i=n=1; while(1,j=0; while(j

Extensions

Edited by Dean Hickerson, Mar 06 2002
More terms from Giovanni Resta, Feb 18 2006
a(16) from Giovanni Resta, Aug 04 2013

A068234 a(n) is the smallest prime p such that p and the next n-1 primes are all == 5 (mod 12).

Original entry on oeis.org

5, 509, 4397, 42509, 647417, 647417, 1248869, 13175609, 234946997, 1039154933, 7114719473, 32021552837, 32021552837, 1237381737257, 2904797643617, 2904797643617, 2904797643617
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Keywords

Comments

Dickson's conjecture implies that a(n) exists for all n.
a(18) > 4*10^14. - Giovanni Resta, Aug 04 2013

Links

Crossrefs

Cf. A068227, A068228, A068229, A040117, A068231, A068232, A068233, A068235.

Programs

  • Mathematica
    For[i=n=1, True, Null, For[j=0, jHarvey P. Dale, Feb 02 2022 *)
  • PARI
    {i=n=1; while(1,j=0; while(j

Extensions

Edited by Dean Hickerson, Mar 06 2002
More terms from Giovanni Resta, Feb 18 2006

A068235 a(n) is the smallest prime p such that p and the next n-1 primes are all == 11 (mod 12).

Original entry on oeis.org

11, 467, 1499, 16763, 260339, 2003387, 7722419, 20221283, 927161471, 4284484931, 7355362139, 84805717127, 478527373859, 2046207697631, 7302359785151, 21104656617827, 21104656617827
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Keywords

Comments

Dickson's conjecture implies that a(n) exists for all n.
a(18) > 4*10^14. - Giovanni Resta, Aug 04 2013

Links

Crossrefs

Cf. A068227, A068228, A068229, A040117, A068231, A068232, A068233, A068234.

Programs

  • Mathematica
    For[i=n=1, True, Null, For[j=0, j
  • PARI
    {i=n=1; while(1,j=0; while(j

Extensions

Edited by Dean Hickerson, Mar 06 2002
More terms from Giovanni Resta, Feb 18 2006
a(16)-a(17) from Giovanni Resta, Aug 04 2013

A167056 Numbers k such that 12*k + 7 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 16, 17, 18, 22, 23, 25, 27, 30, 31, 36, 38, 40, 41, 43, 45, 47, 50, 51, 52, 53, 57, 60, 61, 62, 65, 67, 68, 71, 73, 75, 76, 80, 82, 86, 87, 88, 90, 93, 97, 102, 106, 107, 108, 110, 116, 118, 120, 121, 122, 123, 127, 128, 130, 131, 135, 138
Offset: 1

Views

Author

Michael B. Porter, Oct 27 2009

Keywords

Comments

Corresponds to odd numbers in A024899.

Examples

			2 is in the sequence since 12*2+7 = 31 is prime.

Links

Crossrefs

A110801, A167055, A167057, A024899, primes are in A068229.

Programs

  • Magma
    [n: n in [1..150] | IsPrime(12*n+7)]; // Vincenzo Librandi, May 20 2014
  • Mathematica
    Select[Range[0, 200], PrimeQ[12 # + 7] &] (* Vincenzo Librandi, May 20 2014 *)
  • PARI
    isA167056(n) = isprime(12*n+7)

A132237 Primes congruent to {7, 23} mod 30.

Original entry on oeis.org

7, 23, 37, 53, 67, 83, 97, 113, 127, 157, 173, 233, 263, 277, 293, 307, 337, 353, 367, 383, 397, 443, 457, 487, 503, 547, 563, 577, 593, 607, 653, 683, 727, 743, 757, 773, 787, 863, 877, 907, 937, 953, 967, 983, 997, 1013, 1087, 1103
Offset: 1

Views

Author

Omar E. Pol, Aug 15 2007

Keywords

Comments

Up to 4913, there are more primes of this form than composites. See also A132231 and A227869 (congruent to 7 only). - M. F. Hasler, Nov 02 2013

Links

Crossrefs

Cf. A000040, A007510, A039949, A068229, A129807.

Programs

  • Magma
    [ p: p in PrimesUpTo(1300) | p mod 30 in [7, 23] ]; // Vincenzo Librandi, Aug 14 2012
  • Mathematica
    Select[Prime[Range[1000]],MemberQ[{7,23},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
  • PARI
    is_A132237(n)=setsearch([7,23],n%30)&&isprime(n) \\ - M. F. Hasler, Nov 02 2013

A374912 Primes p such that (p - 1)^p == p (mod 2*p - 1).

Original entry on oeis.org

3, 7, 19, 31, 79, 139, 199, 211, 271, 307, 331, 367, 379, 439, 499, 547, 607, 619, 691, 727, 811, 967, 1171, 1279, 1399, 1459, 1531, 1627, 1759, 1867, 2011, 2131, 2179, 2311, 2467, 2539, 2551, 2707, 2719, 2791, 2851, 3019, 3067, 3187, 3319, 3331, 3391, 3499, 3607, 3739, 3967
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 23 2024

Keywords

Links

Crossrefs

Aside from the first term, a subsequence of A068229.
Cf. A374913, A374914.

Programs

  • Magma
    [p: p in PrimesUpTo(10^4) | (p-1)^p mod (2*p-1) eq p];
  • Mathematica
    Select[Prime[Range[1000]], PowerMod[# - 1, #, 2*# - 1] == # &] (* Paolo Xausa, Jul 24 2024 *)
  • PARI
    list(lim)=my(v=List([3])); forprimestep(p=7,lim\1,12, if(Mod(p-1,2*p-1)^p==p, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 23 2024

Formula

a(n) == 7 (mod 12) for n>1. - Hugo Pfoertner, Jul 24 2024

A164621 Primes p such that p*floor(p/2)-2 and p*floor(p/2)+2 are also prime numbers.

Original entry on oeis.org

7, 31, 79, 211, 271, 751, 787, 1231, 1447, 1459, 2347, 2551, 3727, 5119, 6427, 6691, 8467, 8707, 9007, 10099, 10531, 10567, 10831, 11959, 18691, 21487, 22039, 22567, 23059, 23167, 23371, 24379, 24499, 25171, 26371, 27967, 28579, 28591, 29287
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Keywords

Examples

			7*3-2=13, 7*3+2=17,..

Links

Crossrefs

Cf. A068229, A158708, A164620

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p*Floor[p/2]-2]&&PrimeQ[p*Floor[p/2]+2],AppendTo[lst,p]],{n,2*7!}];lst
Select[Prime[Range[3200]],AllTrue[# Floor[#/2]+{2,-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 04 2020 *)

A193143 Primes which are the sum of 5 distinct positive squares in more than one way.

Original entry on oeis.org

103, 127, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 229, 239, 241, 251, 263, 271, 277, 281, 283, 307, 311, 313, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 16 2011

Keywords

Comments

All terms from 103 onwards in A068229 are primes which are the sum of 5 distinct positive squares in more than one way.

Examples

			103 = 1^2 + 2^2 + 3^2 + 5^2 + 8^2 = 2^2 + 3^2 + 4^2 + 5^2 + 7^2.
127 = 1^2 + 2^2 + 3^2 + 7^2 + 8^2 = 1^2 + 4^2 + 5^2 + 6^2 + 7^2 = 1^2 + 2^2 + 4^2 + 5^2 + 9^2.

Crossrefs

Cf. A068229, A085317, A193141, A193142.

Programs

  • Mathematica
    sum5sqP = {}; Do[Do[Do[Do[Do[p = a^2 + b^2 + c^2 + d^2 + e^2; If[PrimeQ[p], AppendTo[sum5sqP, p]], {e, d - 1, 1, -1}], {d, c - 1, 1, -1}], {c, b - 1, 1, -1}], {b, a - 1, 1, -1}], {a, 6, 30}]; a = Take[Sort[sum5sqP], 1000]; a = Select[Table[If[a[[n]] == a[[n - 1]] && a[[n]] != a[[n - 2]], a[[n]], ""], {n, 3, Length[a]}], IntegerQ]

A020677 Numbers of form 3*x^2 + 4*y^2.

Original entry on oeis.org

0, 3, 4, 7, 12, 16, 19, 27, 28, 31, 36, 39, 43, 48, 52, 63, 64, 67, 75, 76, 79, 84, 91, 100, 103, 108, 111, 112, 124, 127, 139, 144, 147, 148, 151, 156, 163, 171, 172, 175, 183, 192, 196, 199, 208, 211, 219, 223, 228, 243, 244, 247, 252, 256, 259, 268, 271, 279, 283, 291
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Each of these numbers is congruent to 0, 3, 4 or 7 mod 12. Therefore, except for 3, all the primes in this sequence are of the form 12k + 7. - Alonso del Arte, Jan 23 2014

Links

Crossrefs

Cf. A068229.

Programs

  • Mathematica
    max = 300; Select[Union[Flatten[Table[3x^2 + 4y^2, {x, 0, Ceiling[Sqrt[max/3]]}, {y, 0, Ceiling[Sqrt[max/4]]}]]], # < max &] (* Alonso del Arte, Jan 23 2014 *)
Previous Showing 11-20 of 33 results. Next

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