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.

A097823 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") is not squarefree.

Original entry on oeis.org

40, 603, 798, 890, 917, 1245, 1253, 1318, 1640, 1651, 1721, 2010, 2069, 2251, 2452, 2606, 2649, 3094, 3099, 3321, 3402, 3527, 3607, 4123, 4239, 4301, 4819, 4943, 5002, 5083, 5308, 5372, 5425, 5736, 5790, 5930, 5958, 5998, 6150, 6416, 6511, 6683, 6764
Offset: 1

Views

Author

Hugo Pfoertner, Aug 26 2004

Keywords

Examples

			a(1)=40: p(40)=40^2+40+41=1681=41^2, a(2)=603: p(603)=364253=197*43^2, a(11)=1721: p(1721)=2963603=43*41^3, a(68)=10428: p(10428)=108753653=743^2*197, a(91)=14144: p(14144)=200066921=47^4*41.

Links

Crossrefs

Cf. A013929 n is not squarefree, A002837 n such that n^2-n+41 is prime, A007634 n such that n^2+n+41 is composite, A005846 primes of form n^2+n+41, A097822 n^2+n+41 has more than 2 prime factors.

Programs

  • Mathematica
    Select[Range[10000],!SquareFreeQ[#^2+#+41]&] (* Harvey P. Dale, Nov 06 2011 *)

A145293 a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.

Original entry on oeis.org

0, 41, 420, 2911, 38913, 707864, 6618260, 78776990, 725005500
Offset: 1

Views

Author

Artur Jasinski, Oct 07 2008

Keywords

Comments

The Euler polynomial gives primes for consecutive x from 0 to 39.
For numbers x for which x^2 + x + 41 is not prime, see A007634.
For composite numbers of the form x^2 + x + 41, see A145292.

Examples

			a(1)=0 because when x=0 then x^2+x+41=41 (1 distinct prime divisor);
a(2)=41 because when x=41 then x^2+x+41=1763=41*43 (2 distinct prime divisors);
a(3)=420 because when x=420 then x^2+x+41=176861=47*53*71 (3 distinct prime divisors);
a(4)=2911 because when x=2911 then x^2+x+41=8476873=41*47*53*83 (4 distinct prime divisors);
a(5)=38913 because when x=38913 then x^2+x+41=1514260523=43*47*61*71*173 (5 distinct prime divisors);
a(6)=707864 because when x=707864 then x^2+x+41=501072150401=41*43*47*53*71*1607 (6 distinct prime divisors);
a(7)=6618260 because when x=6618260 then x^2+x+41=43801372045901=41*43*47*61*83*131*797 (7 distinct prime divisors);
a(8)=78776990 because when x=78776990 then x^2+x+41=6205814232237131=41*43*61*71*97*131*167*383 (8 distinct prime divisors).
a(9)=725005500: a(9)^2 + a(9) + 41 = 525632975755255541 = 41*43*47*53*61*71*151*397*461. - _Hugo Pfoertner_, Mar 05 2018

Crossrefs

Cf. A005846, A007634, A097822, A145292, A145294, A145295.
Cf. A228122. - Zak Seidov, Feb 03 2016

Programs

  • Mathematica
    a = {}; Do[x = 1; While[Length[FactorInteger[x^2 + x + 41]] < k - 1, x++ ]; AppendTo[a, x]; Print[x], {k, 2, 10}]; a

Extensions

Corrected and edited, a(8) added by Zak Seidov, Jan 31 2016
Example for a(8) corrected by Hugo Pfoertner, Mar 02 2018
a(9) from Hugo Pfoertner, Mar 05 2018

A228184 Numbers k such that k^2 + k + 41 is semiprime.

Original entry on oeis.org

40, 41, 44, 49, 56, 65, 76, 81, 82, 84, 87, 89, 91, 96, 102, 104, 109, 117, 121, 122, 123, 126, 127, 130, 136, 138, 140, 143, 147, 155, 159, 161, 162, 163, 164, 170, 172, 173, 178, 184, 185, 186, 187, 190, 201, 204, 205, 207, 208, 209, 213, 215, 216, 217, 218
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 15 2013

Keywords

Comments

Subsequence of A007634. Numbers in A007634 but not in here are 420, 431, 491, 492, 514, 533, 573, etc. (A097822). - R. J. Mathar, Aug 17 2013

Examples

			a(3) = 44 is in the sequence because 44^2 + 44 + 41 = 43*47 is semiprime.

Links

Crossrefs

Cf. A002837, A001358, A007634.

Programs

  • Mathematica
    a = {}; Do[
If[PrimeOmega[x^2 + x + 41] == 2, AppendTo[a, x]], {x, 1, 500}]; a
Showing 1-3 of 3 results.

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