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.

A060394 Values of k associated with A060392.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 7, 1, 13, 8, 2, 0, 10, 16, 21, 28, 15, 18, 2, 3, 24, 5, 21, 34, 32, 39, 47, 20, 51, 41, 63, 31, 53, 74, 64, 69, 80, 5, 10, 59, 20, 74, 94, 5, 8, 96, 46, 36, 44, 9, 116, 0, 8, 105, 50
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2001

Keywords

Comments

Additional terms require finding the minimal values for the primepuzzles.net conjecture for primes >= 269.

Links

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 24 2003

A060380 Let f(m) = smallest prime that divides k^2 + k + m for k = 0,1,2,...; sequence gives smallest m >= 2 such that f(m) is the n-th prime, or -1 if no such m exists.

Original entry on oeis.org

2, 3, 5, 47, 11, 221, 17, 1217, 941, 2747, 8081, 9281, 41, 55661, 19421, 333491, 1262201, 601037, 5237651, 9063641, 12899891, 26149427, 24073871, 28537121, 352031501, 398878547, 160834691, 67374467, 146452961, 24169417397
Offset: 1

Views

Author

Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Apr 03 2001

Keywords

Comments

Chris Nash (see the Prime Puzzles link) has shown that such an m always exists.
For n>2, least odd number d such that the Legendre symbol (1-4d/prime(k)) = -1 for k = 2,...,n, but not for n+1. See A060392. - T. D. Noe, Apr 19 2004

Examples

			k^2 + k + 2 takes the values 2, 4, 8, 14, ... for k = 0,1,2,...; the smallest prime divisor of these numbers is 2, so f(2) = 2.

References

  • R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082

Links

Crossrefs

Cf. A060392-A060398. A060393 gives associated values of k.

Programs

  • Mathematica
    (* This program is not convenient beyond a(24) *) a[1] = 2; a[2] = 3; a[n_] := For[d = 1, True, d = d+2, If[And @@ (# == -1 & /@ Table[JacobiSymbol[1 - 4d, Prime[k]], {k, 2, n}]) && JacobiSymbol[1 - 4d, Prime[n+1]] != -1, Return[d]]]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Oct 14 2013, after T. D. Noe *)

Extensions

Corrected by T. D. Noe, Apr 19 2004

A060395 Smallest prime that divides k^2 + k + n for k = 0, 1, 2, ....

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 11, 2, 3, 2, 3, 2, 17, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 41, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 11
Offset: 0

Views

Author

N. J. A. Sloane, Apr 04 2001

Keywords

Examples

			To obtain a(7), note that x^2+x+7 takes the values 7,9,13,19,... for k=0,1,2,... and the smallest prime dividing these numbers is 3.

Links

Crossrefs

Cf. A060380, A060392-A060398. A060396 gives values of k.

Programs

  • Mathematica
    a[n_] := Switch[n, 0, 2, 1, 3, , Module[{f, kmax0 = 2}, f[kmax] := f[kmax] = MinimalBy[Table[{k, FactorInteger[k^2 + k + n][[1, 1]]}, {k, 0, kmax}], Last, 1]; f[kmax = kmax0]; f[kmax = 2 kmax]; While[f[kmax] != f[kmax/2], kmax = 2 kmax]; f[kmax][[1, 2]]]];
Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Aug 15 2022 *)

Formula

a(n)=2 if n is equal to 0, 2 or 4 modulo 6; a(n)=3 if n is equal to 1 or 3 modulo 6.

Extensions

More terms from Matthew Conroy, Apr 18 2001

A060397 Smallest prime that divides k^2 + k + 2n + 1 for k = 0,1,2,....

Original entry on oeis.org

3, 3, 5, 3, 3, 11, 3, 3, 17, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 41, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 11, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 11, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 11, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 7, 3, 3
Offset: 0

Views

Author

N. J. A. Sloane, Apr 04 2001

Keywords

Comments

Bisection of A060395.

Examples

			To obtain a(3), note that x^2+x+7 takes the values 7,9,13,19,... for k=0,1,2,... and the smallest prime dividing these numbers is 3.

Links

Crossrefs

Cf. A060380, A060392-A060398. A060398 gives values of k.

Programs

  • Mathematica
    a[n_] := Switch[n, 0, 3, , Module[{f, kmax0 = 2}, f[kmax] := f[kmax] = MinimalBy[Table[{k, FactorInteger[k^2 + k + 2 n + 1][[1, 1]]}, {k, 0, kmax}], Last, 1]; f[kmax = kmax0]; f[kmax = 2 kmax]; While[f[kmax] != f[kmax/2], kmax = 2 kmax]; f[kmax][[1, 2]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 15 2022 *)

Formula

a(n)=3 if n is equal to 0 or 1 mod 3.

Extensions

More terms from Matthew Conroy, Apr 18 2001

A181667 Least integer m > 0 such that none of the first n primes divides any value of the polynomial x^2 + x + m.

Original entry on oeis.org

1, 5, 11, 11, 17, 17, 41, 41, 41, 41, 41, 41, 19421, 19421, 333491, 601037, 601037, 5237651, 9063641, 12899891, 24073871, 24073871, 28537121, 67374467, 67374467, 67374467, 67374467, 146452961, 13236860171, 13236860171, 17959429571, 57391479317, 57391479317
Offset: 1

Views

Author

Esteban Crespi de Valldaura, Feb 04 2011

Keywords

Comments

All the elements of this sequence with n > 2 are congruent mod 30 to one of the polynomials x^2 + x + 11 or x^2 + x + 17.
The elements of the sequence have been taken from A060392, see below.

Examples

			x^2 + x + 11 takes the values 11, 13, 17, 23, 31, 41, 53, 67, 83, ... never divisible by any of the primes 2, 3, or 5.

Links

Crossrefs

a(n) equals min_{k > n} A060392(k).

Extensions

a(29) corrected and more terms added by William P. Orrick, Mar 17 2017
Showing 1-5 of 5 results.

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

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