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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.
2
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
3,1
COMMENTS
The corresponding indices of the n-gonal numbers are 33, 1093, 73, 17, 97, ... (A371760).
LINKS
Eric Weisstein's World of Mathematics, Polygonal Number.
Eric Weisstein's World of Mathematics, Poulet Number.
Wikipedia, Polygonal number.
Wikipedia, Pseudoprime.
FORMULA
a(n) = ((n-2)*k^2 - (n-4)*k)/2, where k = A371760(n).
EXAMPLE
a(4) = A001220(1)^2 = 1093^2 = 1194649. The only known square base-2 pseudoprimes are the squares of the Wieferich primes (A001220).
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]
PROG
(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); }
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 05 2024
STATUS
approved