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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A241267 Numbers n such that 4*n^2 + 4*n + 653 is not prime.

Original entry on oeis.org

18, 19, 21, 24, 28, 33, 39, 46, 54, 60, 61, 62, 63, 65, 67, 72, 73, 75, 81, 82, 84, 85, 86, 93, 95, 96, 100, 101, 103, 104, 105, 109, 110, 111, 112, 117, 121, 122, 123, 125, 130, 137, 138, 139, 141, 142, 143, 144, 147, 152, 153, 154, 159, 160, 161, 165, 166
Offset: 1

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Author

Vincenzo Librandi, Apr 20 2014

Keywords

Examples

			19 is in this sequence because 4*19^2 + 4*19 + 653 = 2173 = 41*53.

Links

Crossrefs

Complement of A188459.

Programs

  • Magma
    [n: n in [0..250] | not IsPrime(4*n^2+4*n+653)];
  • Mathematica
    Select[Range[0, 250], ! PrimeQ[4 #^2 + 4 # + 653] &]

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 1-2 of 2 results.

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

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