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.

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

Author

Amiram Eldar, Apr 05 2024

Keywords

• nonn

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

• Amiram Eldar, Table of n, a(n) for n = 3..10000
• Eric Weisstein's World of Mathematics, Polygonal Number.
• Eric Weisstein's World of Mathematics, Poulet Number.
• Wikipedia, Polygonal number.
• Wikipedia, Pseudoprime.
• Index entries for sequences related to pseudoprimes.

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).