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.

A242702 Semiprimes n such that n^2+n+41 is also semiprime.

Original entry on oeis.org

49, 65, 82, 87, 91, 121, 122, 123, 143, 155, 159, 161, 178, 185, 187, 201, 205, 209, 213, 215, 217, 218, 237, 249, 259, 265, 278, 287, 289, 291, 295, 298, 299, 301, 302, 309, 314, 321, 326, 327, 329, 334, 361, 381, 395, 407, 422, 427, 445, 451, 454, 466, 471
Offset: 1

Views

Author

K. D. Bajpai, May 20 2014

Keywords

Comments

n^2+n+41 is sometimes referred to as Euler's polynomial.
Subsequence of A228184.

Examples

			65 = 5 * 13 is semiprime and 65^2 + 65 + 41 = 4331 = 61 * 71 is also semiprime so 65 is in the sequence.
87 = 3 * 29 is semiprime and 87^2 + 87 + 41 = 7697 = 43 * 179 is also semiprime so 87 is in the sequence.
6 = 2 * 3 is semiprime but 6^2+6+41 = 83 is prime (not semiprime) so 6 is not in the sequence.

Links

Crossrefs

Cf. A001358, A228183, A238242, A228184.

Programs

  • Maple
    with(numtheory): A242702:= proc();  if bigomega(n)=2 and bigomega(n^2+n+41)=2 then RETURN (n); fi; end: seq(A242702 (), n=1..1000);
  • Mathematica
    c = 0; Do[If [PrimeOmega[n] == 2 && PrimeOmega[n^2 + n + 41] == 2, c++; Print[c, "  ", n]], {n, 1, 10^5}];
Select[Range[500],PrimeOmega[#]==PrimeOmega[#^2+#+41]==2&] (* Harvey P. Dale, Nov 07 2016 *)

A238203 Squares s such that s^2+s+41 is prime.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 64, 100, 144, 169, 196, 225, 324, 400, 441, 484, 529, 576, 625, 841, 900, 961, 1089, 1444, 1521, 1849, 2209, 2601, 2704, 2809, 3025, 3136, 3249, 3364, 3721, 3844, 4096, 4225, 4356, 4489, 5476, 5625, 5776, 6241, 7056, 7921, 8464, 8836, 9025
Offset: 1

Views

Author

K. D. Bajpai, Feb 20 2014

Keywords

Comments

n^2+n+41: Euler’s prime generating polynomial.
First 6 terms in the sequence are first 6 consecutive squares.

Examples

			9 is in the sequence because 9 = 3^2 and 9^2+9+41 = 131 is prime.
36 is in the sequence because 36 = 6^2 and 36^2+36+41 = 1373 is prime.

Links

Crossrefs

Cf. A000040, A097823, A238242.

Programs

  • Maple
    with(numtheory):KD := proc() local a,b; a:=(n^2);b:=a^2+a+41; if isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..500);
  • Mathematica
    Select[Table[k = n^2, {n, 100}], PrimeQ[#^2 + # + 41] &] (* or *) c = 0; Do[k = n^2; If[PrimeQ[k^2 + k + 41], c = c + 1; Print[c, " ", k]], {n, 1, 10000}];
Select[Range[100]^2,PrimeQ[#^2+#+41]&] (* Harvey P. Dale, Dec 13 2021 *)

A242708 Primes p such that p^2 + p + 41 is semiprime.

Original entry on oeis.org

41, 89, 109, 127, 163, 173, 239, 251, 271, 283, 331, 347, 349, 367, 373, 383, 389, 401, 409, 421, 443, 449, 463, 467, 487, 547, 557, 563, 569, 571, 577, 587, 593, 613, 643, 661, 701, 727, 733, 739, 761, 769, 773, 797, 823, 827, 853, 857, 881, 907, 937, 947, 971
Offset: 1

Views

Author

K. D. Bajpai, May 21 2014

Keywords

Comments

n^2+n+41 is sometimes referred to as Euler's polynomial.
Subsequence of A228184.
A242702 is for semiprimes such that n^2+n+41 is also semiprime.

Examples

			41 is prime and 41^2 + 41 + 41 = 1763 = 41 * 43 is semiprime. Hence, 41 is in the sequence.
127 is prime and 127^2 + 127 + 41 = 16297 = 43 * 379 is semiprime. Hence, 127 is in the sequence.
43 is prime and 43^2 + 43 + 41 = 1933 which is prime (not semiprime). Hence, 43 is not in the sequence.

Links

Crossrefs

Cf. A000040, A001358, A228184, A238242, A242702.

Programs

  • Maple
    with(numtheory): A242708:= proc();  if isprime(n) and bigomega(n^2+n+41)=2 then RETURN (n);  fi;  end: seq(A242708 (), n=1..1000);
  • Mathematica
    c = 0; Do[If[PrimeQ[n] && PrimeOmega[n^2 + n + 41] == 2, c++; Print[c, "  ", n]], {n, 1, 3*10^5}];
Showing 1-3 of 3 results.

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