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.

A078402 Numbers k such that k^2 + 5 is prime.

Original entry on oeis.org

0, 6, 12, 36, 48, 72, 78, 96, 114, 126, 162, 168, 198, 204, 246, 258, 294, 336, 342, 372, 414, 432, 456, 462, 492, 504, 516, 534, 552, 576, 588, 594, 624, 666, 714, 726, 756, 768, 786, 792, 798, 804, 834, 852, 876, 888, 918, 954, 996
Offset: 1

Views

Author

Cino Hilliard, Dec 26 2002

Keywords

Comments

The sum of the reciprocals of the primes generated from these indices converges to 0.2332142.. The sum of the reciprocals of these indices cannot be computed.
All terms are divisible by 6. - Zak Seidov, Dec 27 2014

Links

Crossrefs

For the primes see A056905(n).
Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), this sequence (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

Programs

  • Magma
    [n: n in [0..1000]| IsPrime(n^2+5)]; // Vincenzo Librandi, Jul 15 2012
  • Mathematica
    Select[Range[0,1000],PrimeQ[#^2+5]&] (* Vincenzo Librandi, Jul 13 2012 *)
  • PARI
    for(n=0,10^4,q=n^2+5;if(isprime(q),print1(n,", ")));

Formula

a(n) = 6 * A056906(n).

Extensions

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A056905 Primes of the form k^2 + 5.

Original entry on oeis.org

5, 41, 149, 1301, 2309, 5189, 6089, 9221, 13001, 15881, 26249, 28229, 39209, 41621, 60521, 66569, 86441, 112901, 116969, 138389, 171401, 186629, 207941, 213449, 242069, 254021, 266261, 285161, 304709, 331781, 345749, 352841, 389381, 443561
Offset: 1

Views

Author

Henry Bottomley, Jul 07 2000

Keywords

Comments

Except for a(0), a(n) mod 180 = 41 or 149 since k must be a multiple of 6 without being a multiple of 30 for k^2+5 to be prime.

Examples

			a(2)=149 since 12^2 + 5 = 149, which is prime.

Links

Crossrefs

Cf. A002496, A056899, A049423, A005473.

Programs

  • Magma
    [a: n in [0..700] | IsPrime(a) where a is n^2+5]; // Vincenzo Librandi, Nov 30 2011
  • Mathematica
    Select[Table[n^2+5,{n,0,7000}],PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *)
  • PARI
    is(n) = ispseudoprime(n) && issquare(n-5) \\ Felix Fröhlich, May 25 2018

Formula

a(n) = 36 * A056906(n) + 5.

A056908 Numbers k such that 36*k^2 + 36*k + 13 is prime.

Original entry on oeis.org

0, 2, 4, 5, 7, 9, 14, 19, 22, 24, 29, 30, 34, 40, 42, 44, 50, 59, 62, 70, 72, 74, 75, 79, 80, 82, 84, 95, 102, 110, 119, 125, 132, 135, 139, 149, 150, 157, 160, 165, 172, 180, 197, 199, 200, 209, 210, 212, 224, 225, 227, 229, 230, 232, 235, 240, 244, 249
Offset: 1

Views

Author

Henry Bottomley, Jul 07 2000

Keywords

Comments

36*k^2 + 36*k + 13 = (6*k+3)^2 + 4, which is 4 more than a square.

Examples

			a(2)=4 since 36*4^2 + 36*4 + 13 = 733, which is prime (as well as being four more than a square).

Links

Crossrefs

This sequence and formula, together with A056907 and its formula, generate all primes of the form k^2+4, i.e., A005473.
Cf. A056900, A056902, A056904, A056906.

Programs

A056907 Numbers k such that 36*k^2 + 12*k + 5 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, 1, 2, -3, -6, 6, -8, -11, 11, 12, 14, -16, 16, 17, 19, -21, -23, -26, 27, -28, 32, -34, -36, 36, -39, 39, -41, 42, 44, -46, 46, -48, -49, 51, 52, -53, -58, 62, 64, 67, -68, -71, 71, -76, 77, 79, 81, -84, -89, 91, 96, -99, -101, 101, 102, -104, -111, 111, -113
Offset: 0

Views

Author

Henry Bottomley, Jul 07 2000

Keywords

Comments

36*k^2 + 12*k + 5 = (6*k+1)^2 + 4, which is four more than a square. Except for a(0), a(n) is never a multiple of 5.

Examples

			a(3)=2 since 36*2^2 + 12*2 + 5 = 173 which is prime (as well as being four more than a square).

Crossrefs

This sequence and formula, together with A056908 and its formula, generate all primes of the form k^2+4, i.e., A005473. Except for the first term, this sequence is a subsequence of A047201. Cf. A056900, A056902, A056904, A056906.

A056910 Numbers k such that 36*k^2 + 12*k + 7 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, -2, 3, 4, 5, -6, 10, -11, 13, -15, 15, 18, -22, 24, 25, 29, -31, 33, -37, -45, -55, 55, 59, -67, -72, 74, 80, -81, 85, -86, 88, -90, -95, 99, -101, -102, 108, -116, 118, -122, 129, -130, 143, 148, -151, -155, -157, 158, 159, -162, 164, 165
Offset: 0

Views

Author

Henry Bottomley, Jul 07 2000

Keywords

Comments

36*k^2 + 12*k + 7 = (6*k+1)^2 + 6, which is six more than a square.

Examples

			a(2)=-2 since 36*(-2)^2 + 12*(-2) + 7 = 127, which is prime (as well as being six more than a square).

Crossrefs

This sequence and formula generate all primes of the form k^2+6, i.e., A056909. Except for the first term, none of the a(n) are a multiple of 7 and so the rest of this sequence is a subsequence of A047304. Cf. A056900, A056902, A056904, A056906, A056907, A056908.

Formula

a(n) = (-1 +- sqrt(A056909(n) - 6))/6, choosing +- to give an integer result for each n.
Showing 1-5 of 5 results.

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