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 1-3 of 3 results.

A188459 Numbers k such that 4*k^2 + 4*k + 653 is a prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 64, 66, 68, 69, 70, 71, 74, 76, 77, 78, 79, 80, 83, 87, 88, 89, 90, 91, 92, 94, 97, 98, 99, 102, 106, 107, 108, 113, 114
Offset: 1

Views

Author

Vincenzo Librandi, Apr 01 2011

Keywords

Links

Crossrefs

Cf. A145202 (the corresponding primes).

Programs

  • Magma
    [n: n in [0..250] | IsPrime(4*n^2+4*n+653)]; // Vincenzo Librandi, Apr 01 2011
  • Mathematica
    Select[Range[0,150],PrimeQ[4#^2+4#+653]&]  (* Harvey P. Dale, Apr 02 2011 *)

Extensions

Offset changed to 1 by Georg Fischer, Sep 27 2022

A145125 Primes of the form 4n^2-4n+653.

Original entry on oeis.org

653, 653, 661, 677, 701, 733, 773, 821, 877, 941, 1013, 1093, 1181, 1277, 1381, 1493, 1613, 1741, 1877, 2333, 2677, 2861, 3253, 3461, 3677, 4133, 4373, 4621, 4877, 5413, 5693, 5981, 6277, 6581, 7213, 7541, 7877, 8221, 8573, 8933, 9677, 10061, 10453
Offset: 1

Views

Author

Bobby Kramer (panthar1(AT)gmail.com), Oct 02 2008

Keywords

Comments

These primes are in A000414. [Bruno Berselli, Apr 20 2014]

Links

Crossrefs

A145202 is essentially the same sequence.

Programs

  • Magma
    [ a: n in [0..900] | IsPrime(a) where a is 4*n^2-4*n+653]; // Vincenzo Librandi, Nov 25 2010
  • Mathematica
    Select[Table[4 n^2 - 4 n + 653, {n, 0, 300}], PrimeQ] (* Vincenzo Librandi, Apr 21 2014 *)

Extensions

More terms from Vincenzo Librandi, Apr 28 2010

A356567 Numbers that generate increasing numbers of consecutive primes when doubled and added to the sequence of odd squares. (Positions of records in A354499.)

Original entry on oeis.org

1, 2, 11, 29, 326
Offset: 1

Views

Author

Steven M. Altschuld, Aug 12 2022

Keywords

Comments

With a(1)=1, a(n) such that (2*x-1)^2 + 2*a(n) gives prime numbers for x=1 to k where the k for a(n) exceeds the k for a(n-1), a(n-2), ..., a(1).
Conjecture: this list is complete, since primes get farther apart as numbers increase. (2x-1)^2 + 2*29 generates many primes, with 38 of the first 43 and 105 of the first 156 values of x generating primes.
For the (2x-1)^2 + 2*29 values that are not prime, there seems to be a restriction on the factors. No values with prime factors below 29 were seen, nor were 41, 43, 53, 71, 73, 89, 97, ... For each of the other a(n) (or indeed any other natural number K), it seems there is a list of acceptable prime factors for the (2x-1)^2 + 2*K value. This gives a curious connection between addition and prime factors.
a(6) > 10^8, if it exists. - Amiram Eldar, Aug 15 2022

Examples

			a(1)=1, because 1^2+2*1=3 and 3^2+2*1=11 are prime but 5^2+2*1=27 is not, and thus k=2.
a(2)=2, because 1^2+2*2=5 ... 7^2+2*2=53 are prime but 9^2+2*2=85 is not, thus k=4.
a(3)=11, because 1^2+2*11=23 ... 9^2+2*11=103 are prime, thus k=5.
a(4)=29, because 1^2+2*29=59 ... 27^2+2*29=787 are all prime, thus k=14.
a(5)=326 because 1^2+2*326=653 ... 35^2+2*352=1877 are all prime, thus k=18.

Crossrefs

Cf. A016754 (odd squares), A354499 (consecutive primes generated by adding 2n to odd squares).
Cf. A145202 and A188459 (related to last term).

Programs

  • Mathematica
    f[n_] := Module[{k = 1}, While[PrimeQ[k^2 + 2*n], k += 2]; (k - 1)/2]; s = {}; fm = 0; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^3}]; s (* Amiram Eldar, Aug 15 2022 *)
  • PARI
    f(n) = my(k=1); while (isprime(k^2+2*n), k+=2); (k-1)/2; \\ A354499
lista(nn) = my(m=0); for (n=1, nn, my(x = f(n)); if (x > m, m = x; print1(n, ", "));); \\ Michel Marcus, Aug 16 2022
Showing 1-3 of 3 results.

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