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 1-7 of 7 results.

A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

Original entry on oeis.org

13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249, 129793, 139969, 142573, 147853, 169933
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Primes p such that p = 1 + 3*m^2 for some integer m (A111051). - Michael Somos, Sep 15 2005

Examples

			193 is a term since 193 = (9^3 - 7^3)/(9 - 7) is a prime.

References

  • A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002407, A111051 (values of m).
A subsequence of A007645.

Programs

  • Magma
    [a: n in [0..400] | IsPrime(a) where a is 3*n^2+1]; // Vincenzo Librandi, Dec 02 2011
  • Mathematica
    Select[Table[3n^2+1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)
  • PARI
    {a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( cMichael Somos, Sep 15 2005 */

Formula

a(n) = 3*A111051(n)^2 + 1. - Paul F. Marrero Romero, Nov 03 2023

Extensions

Entry revised by N. J. A. Sloane, Jan 29 2013

A121259 Numbers k such that (3*k^2 + 1)/4 is prime.

Original entry on oeis.org

3, 5, 7, 9, 13, 19, 21, 23, 27, 29, 35, 47, 49, 51, 55, 57, 61, 65, 69, 75, 77, 83, 85, 91, 97, 99, 105, 111, 117, 125, 127, 133, 135, 149, 161, 163, 173, 177, 181, 183, 187, 191, 211, 217, 239, 247, 251, 257, 259, 273, 281, 285, 295, 307, 313, 315, 317, 329, 331, 341
Offset: 1

Views

Author

Zak Seidov, Aug 23 2006

Keywords

Examples

			(3*5^2 + 1)/4 = 19 is the 2nd prime of this form, so a(2) = 5.
(3*13^2 + 1)/4 = 127 is the 5th prime of this form, so a(5) = 13.
(3*19^2 + 1)/4 = 271 is the 6th prime of this form, so a(6) = 19.

Links

Crossrefs

Cf. comment by Michael Somos in A002407.
Cf. A002504, A111251, A111051 (simpler variant).

Programs

Formula

a(n) = sqrt((4*A002407(n) - 1)/3). [corrected by Rémi Guillaume, Dec 07 2023]
a(n) = 2*A002504(n) - 1. - Hugo Pfoertner, Oct 07 2023
a(n) = 2*A111251(n) + 1. - Rémi Guillaume, Dec 06 2023

A207838 Numbers k such that 5*k^4 + 1 is prime.

Original entry on oeis.org

6, 12, 114, 120, 324, 336, 390, 420, 498, 504, 540, 672, 756, 768, 840, 852, 876, 1014, 1062, 1092, 1122, 1170, 1188, 1248, 1266, 1314, 1344, 1398, 1440, 1470, 1524, 1758, 1770, 1818, 1860, 1968, 2028, 2046, 2088, 2184, 2190, 2232, 2262, 2268, 2304, 2382, 2430
Offset: 1

Views

Author

Bruno Berselli, Feb 21 2012

Keywords

Comments

All terms are multiples of 6.

Links

Crossrefs

Cf. A207837 (primes of the form 5*k^4+1).
Cf. A005097, A111051, A115104.

Programs

  • Magma
    [6*n: n in [1..406] | IsPrime(6480*n^4+1)];
  • Mathematica
    Select[Range[2440], PrimeQ[5 #^4 + 1] &] (* by definition *)
  • PARI
    for(n=1, 406, r=6480*n^4+1; if(isprime(r), print1(6*n", ")));

Formula

a(n) = ((A207837(n) - 1)/5)^1/4. - Paul F. Marrero Romero, Dec 07 2023

A375390 Numbers k such that k^2 + 1, k^2 + 3 and k^2 + 5 are semiprimes.

Original entry on oeis.org

44, 102, 104, 108, 152, 188, 226, 234, 296, 328, 426, 526, 586, 692, 720, 842, 846, 856, 926, 994, 1076, 1278, 1284, 1386, 1426, 1484, 1498, 1574, 1704, 1746, 1764, 1822, 1826, 1848, 1952, 2058, 2114, 2128, 2142, 2148, 2164, 2186, 2386, 2416, 2442, 2484, 2640, 2704, 2904, 2948, 3108, 3142, 3164
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Aug 15 2024

Keywords

Comments

All terms are even.
a(n)^2 + 3 or a(n)^2 + 5 is 3 times a prime. In the first case, a(n)/3 is in A111051.

Examples

			a(3) = 104 is a term because 104^2 + 1 = 10817 = 29 * 373, 104^2 + 3 = 10819 = 31 * 349 and 104^2 + 5 = 10821 = 3 * 3607 are all semiprimes.

Links

Crossrefs

Cf. A001358, A111051. Intersection of A085722, A242331 and A242333.

Programs

  • Maple
    select(t -> andmap(s -> numtheory:-bigomega(t^2+s)=2, [1,3,5]), 2*[$1..2000]);
  • Mathematica
    Select[Range[3000], 2 == PrimeOmega[1 + #^2] == PrimeOmega[3 +
#^2] ==   PrimeOmega [5 + #^2] &]

A132190 Numbers n such that 7*n^2 + 1 is prime.

Original entry on oeis.org

2, 4, 8, 10, 12, 14, 18, 20, 22, 26, 30, 34, 44, 46, 58, 66, 68, 70, 74, 76, 78, 84, 90, 96, 100, 106, 108, 110, 120, 128, 134, 140, 146, 152, 154, 156, 158, 162, 164, 168, 174, 184, 186, 188, 196, 200, 202, 210, 216, 228, 232, 238, 250, 252, 260, 262, 264, 268
Offset: 1

Views

Author

Parthasarathy Nambi, Nov 05 2007

Keywords

Examples

			If n=2 then 7*n^2 + 1 = 29 (prime).
If n=100 then 7*n^2 + 1 = 70001 (prime).

Crossrefs

Cf. A111051.

Programs

Extensions

More terms from Stefan Steinerberger, Jan 02 2008

A132398 Numbers n such that 11*n^2 + 1 is prime.

Original entry on oeis.org

6, 24, 30, 54, 66, 84, 90, 96, 126, 144, 150, 186, 210, 234, 246, 276, 300, 324, 330, 360, 420, 426, 444, 450, 474, 480, 486, 516, 606, 624, 636, 684, 720, 750, 786, 804, 816, 864, 876, 900, 906, 924, 966, 996, 1014, 1020, 1056, 1074, 1104, 1110, 1116, 1194
Offset: 1

Views

Author

Parthasarathy Nambi, Nov 12 2007

Keywords

Comments

All terms are multiples of 6.

Examples

			If n=6 then 11*n^2 + 1 = 397 (prime).
If n=144 then 11*n^2 + 1 = 228097 (prime).

Crossrefs

Cf. A111051, A132190.

Programs

Extensions

More terms from Stefan Steinerberger, Nov 13 2007

A155962 Numbers n with property that 3*(2n)^2+1 and 1*(2n)^2+3 are primes.

Original entry on oeis.org

1, 4, 11, 32, 56, 73, 80, 109, 122, 143, 158, 175, 182, 217, 256, 262, 280, 284, 290, 308, 343, 347, 403, 431, 434, 437, 535, 581, 598, 619, 655, 665, 928, 973, 980, 1018, 1036, 1046, 1096, 1120, 1159, 1207, 1222, 1235, 1267, 1382, 1393, 1439, 1460, 1463, 1501
Offset: 1

Views

Author

Zak Seidov, Jan 31 2009

Keywords

Comments

2*A155962 is intersection of A049422 and A111051.

Examples

			n=1, {3*(2n)^2+1, 1*(2n)^2+3}={13,7};
n=4, {3*(2n)^2+1, 1*(2n)^2+3}={193,67};
n=11, {3*(2n)^2+1, 1*(2n)^2+3}={1453,487};
n=32, {3*(2n)^2+1,1*(2n)^2+3}={12289,4099}.
Resulting primes are congruent to 1 mod 3.

Links

Crossrefs

Cf. A049422, A111051.

Programs

  • Mathematica
    Select[Range[1600],AllTrue[{3(2#)^2+1,(2#)^2+3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 30 2016 *)

Extensions

All the terms in the b-file had to be divided by 2. Corrected by N. J. A. Sloane, Aug 31 2009.
Showing 1-7 of 7 results.

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

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