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.

A322161 Numbers k such that m = 8k^2 + 2k + 33 and 8m - 7 are both primes.

Original entry on oeis.org

1, 46, 58, 133, 145, 175, 208, 223, 241, 403, 430, 463, 526, 568, 808, 868, 985, 1015, 1021, 1105, 1120, 1360, 1465, 1501, 1600, 1918, 1978, 2236, 2350, 2413, 2908, 2965, 3043, 3211, 3265, 3523, 3556, 3568, 3601, 3721, 3811, 3868, 4066, 4291, 4300, 4336, 4831
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2018

Keywords

Comments

Rotkiewicz proved that if k is in this sequence, and m = 8k^2 + 2k + 33, then m*(8m - 7) is an octadecagonal Fermat pseudoprime to base 2 (A322160), and thus under Schinzel's Hypothesis H there are infinitely many decagonal Fermat pseudoprimes to base 2.
The corresponding pseudoprimes are 14491, 2326319101, 5858192341, 160881885091, 227198832571, 481700815831, 960833787841, ...

Examples

			1 is in the sequence since 8*1^2 + 2*1 + 33 = 43 and 8*43 - 7 = 337 are both primes.

Links

Crossrefs

Cf. A001567, A051870, A322160.

Programs

  • Mathematica
    Select[Range[1000], PrimeQ[8#^2 + 2# + 33] && PrimeQ[64#^2 + 16# + 257]  &]
  • PARI
    isok(n) = isprime(m = 8*n^2+2*n+33) && isprime(8*m-7); \\ Michel Marcus, Nov 29 2018

A371759 a(n) is the smallest n-gonal number that is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

Original entry on oeis.org

561, 1194649, 7957, 561, 23377, 341, 129889, 1105, 35333, 561, 204001, 31609, 2940337, 1105, 493697, 8481, 13981, 1905, 88561, 41665, 10680265, 1729, 107185, 264773, 449065, 6601, 2165801, 23001, 1141141, 13981, 272251, 4369, 17590957, 15841, 137149, 2821, 561
Offset: 3

Views

Author

Amiram Eldar, Apr 05 2024

Keywords

Comments

The corresponding indices of the n-gonal numbers are 33, 1093, 73, 17, 97, ... (A371760).

Examples

			a(4) = A001220(1)^2 = 1093^2 = 1194649. The only known square base-2 pseudoprimes are the squares of the Wieferich primes (A001220).

Links

Crossrefs

Cf. A001220, A001567, A293622, A293623, A322130, A321868, A321870, A322123, A322160, A371760.

Programs

  • Mathematica
    p[k_, n_] := ((n-2)*k^2 - (n-4)*k)/2; pspQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; a[n_] := Module[{k = 2}, While[! pspQ[p[k, n]], k++]; p[k, n]]; Array[a, 50, 3]
  • PARI
    p(k, n) = ((n-2)*k^2 - (n-4)*k)/2;
ispsp(n) = !isprime(n) && Mod(2, n)^(n-1) == 1;
a(n) = {my(k = 2); while(!ispsp(p(k, n)), k++); p(k, n);}

Formula

a(n) = ((n-2)*k^2 - (n-4)*k)/2, where k = A371760(n).

A371760 a(n) is the smallest number k such that the k-th n-gonal number is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

Original entry on oeis.org

33, 1093, 73, 17, 97, 11, 193, 17, 89, 11, 193, 73, 673, 13, 257, 33, 41, 15, 97, 65, 1009, 13, 97, 149, 190, 23, 401, 41, 281, 31, 133, 17, 1033, 31, 89, 13, 6, 59, 241, 157, 1217, 91, 145, 37, 937, 29, 1289, 73, 97, 41, 617, 19, 137, 151, 34, 103, 8641, 47, 82
Offset: 3

Views

Author

Amiram Eldar, Apr 05 2024

Keywords

Comments

The corresponding pseudoprimes are in A371759.

Links

Crossrefs

Cf. A001567, A293622, A293623, A322130, A321868, A321870, A322123, A322160, A371759.

Programs

  • Mathematica
    p[k_, n_] := ((n - 2)*k^2 - (n - 4)*k)/2; pspQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; a[n_] := Module[{k = 2}, While[! pspQ[p[k, n]], k++]; k]; Array[a, 100, 3]
  • PARI
    p(k, n) = ((n-2)*k^2 - (n-4)*k)/2;
ispsp(n) = !isprime(n) && Mod(2, n)^(n-1) == 1;
a(n) = {my(k = 2); while(!ispsp(p(k, n)), k++); k;}
Showing 1-3 of 3 results.

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