A293622 -id:A293622 - cp's OEIS frontend

A322130 Fermat pseudoprimes to base 2 that are hexagonal.

561, 2701, 4371, 8911, 10585, 18721, 33153, 49141, 93961, 104653, 115921, 157641, 226801, 289941, 314821, 334153, 534061, 665281, 721801, 831405, 873181, 915981, 1004653, 1373653, 1537381, 1755001, 1815465, 1987021, 2035153, 2233441, 2284453, 3059101, 3363121

Offset: 1

Amiram Eldar, Nov 27 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite. His proof is the same as that of triangular pseudoprimes, since all the triangular numbers that he generates are also hexagonal (see comment in A320599).
Intersection of A001567 and A000384.
Subsequence of A293622.
The corresponding indices of the hexagonal numbers are 17, 37, 47, 67, 73, 97, 129, 157, 217, 229, 241, 281, 337, 381, 397, 409, 517, 577, 601, 645, 661, 677, 709, 829, 877, 937, 953, 997, ...

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. A000384, A001567, A293622, A293623, A293624, A320599.

hex[n_] := n(2n-1); Select[hex[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]

isok(n) = ispolygonal(n, 6) && (Mod(2, n)^n==2) && !isprime(n) && (n>1); \\ Michel Marcus, Nov 28 2018

A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes.

Original entry on oeis.org

9, 24, 39, 57, 84, 144, 150, 165, 207, 219, 234, 249, 252, 267, 309, 324, 357, 402, 414, 507, 522, 534, 555, 570, 639, 654, 759, 765, 777, 792, 795, 882, 924, 927, 942, 969, 1044, 1065, 1089, 1155, 1200, 1215, 1227, 1389, 1395, 1437, 1509, 1530, 1554, 1557

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Rotkiewicz proved that if k is in this sequence then (4k + 1)*(8k + 1) is a triangular Fermat pseudoprime to base 2 (A293622), and thus under Schinzel's Hypothesis H there are infinitely many triangular Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 2701, 18721, 49141, 104653, 226801, 665281, 721801, ...

			9 is in the sequence since 4*9 + 1 = 37 and 8*9 + 1 = 73 are both primes.

Seiichi Manyama, 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. A000217, A001567, A293622.

Intersection of A005098 and A005123.

Select[Range[1000], PrimeQ[4#+1] && PrimeQ[8#+1] &]

isok(n) = isprime(4*n+1) && isprime(8*n+1); \\ Michel Marcus, Nov 20 2018

from sympy import isprime
def ok(n): return isprime(4*n + 1) and isprime(8*n + 1)
print(list(filter(ok, range(1558)))) # Michael S. Branicky, Sep 24 2021

A321866 Indices of tetrahedral numbers that are Fermat pseudoprimes to base 2.

Original entry on oeis.org

3457, 16705, 21169, 28297, 30577, 45481, 114601, 123121, 127297, 140977, 156601, 159337, 312841, 393121, 418177, 437977, 443017, 453601, 509737, 518017, 521137, 539401, 545161, 545617, 657841, 679297, 704161, 717817, 762121, 775057, 832801, 904801, 996601

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Numbers n such that n(n+1)(n+2)/6 is a Fermat pseudoprimes to base 2.
The corresponding tetrahedral Fermat pseudoprimes are 6891657409, 777080801185, 1581289265305, 3776730328549, 4765143438329, 15680770945781, 250856489370101, 311068284648121, 343806031110049, ...
Sierpinski asked for the existence of these numbers in 1965.

			3457 is in the sequence since A000292(3457) = 6891657409 is a Fermat pseudoprime to base 2.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

Cf. A000292, A001567, A293622, A293624.

fermatQ[n_,k_] := CompositeQ[n] && PowerMod[k,n-1,n]==1; p[n_] := n(n+1)(n+2)/6; seq={}; Do[p1=p[n]; If[fermatQ[p1,2], AppendTo[seq,n]], {n,1,1000000,2}]; seq

isok(n) = my(t = n*(n+1)*(n+2)/6); (t != 1) && (Mod(2, t)^t == 2); \\ 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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A322130 Fermat pseudoprimes to base 2 that are hexagonal.

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A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes.

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A321866 Indices of tetrahedral numbers that are Fermat pseudoprimes to base 2.

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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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