A321870 -id:A321870 - cp's OEIS frontend

A321871 Numbers k such that m = 4k^2 + 2k + 17 and 4m - 3 are both primes.

1, 3, 4, 9, 11, 14, 23, 36, 38, 49, 66, 101, 133, 134, 141, 154, 158, 191, 193, 196, 198, 213, 228, 241, 269, 283, 294, 313, 334, 339, 384, 394, 411, 413, 431, 453, 499, 511, 554, 558, 601, 619, 639, 649, 661, 686, 701, 704, 718, 758, 791, 804, 818, 821, 881

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Rotkiewicz proved that if k is in this sequence, and m = 4k^2 + 2k + 17, then m*(4m - 3) is a decagonal Fermat pseudoprime to base 2 (A321870), and thus under Schinzel's Hypothesis H there are infinitely many decagonal Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 2047, 13747, 31417, 514447, 1092547, 2746477, 18985627, 111202297, 137763037, ...

			1 is in the sequence since 4*1^2 + 2*1 + 17 = 23 and 4*23 - 3 = 89 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A001107, A001567, A321870.

Select[Range[1000], PrimeQ[4#^2 + 2# + 17] && PrimeQ[16#^2 + 8# + 65] &]

isok(n) = isprime(m=4*n^2 + 2*n + 17) && isprime(4*m-3); \\ Michel Marcus, Nov 20 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

Amiram Eldar, Apr 05 2024

nonn

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

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

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.

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

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]

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);}

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

Amiram Eldar, Apr 05 2024

nonn

The corresponding pseudoprimes are in A371759.

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.

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

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]

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;}

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A321871 Numbers k such that m = 4k^2 + 2k + 17 and 4m - 3 are both primes.

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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.

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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.

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