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

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

1,1

Comments

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.

Links

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.

Example

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

Mathematica

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

Prog

(PARI) isok(n) = my(t = n*(n+1)*(n+2)/6); (t != 1) && (Mod(2, t)^t == 2); \\ Michel Marcus, Nov 20 2018

Crossrefs

Cf. A000292, A001567, A293622, A293624.

Sequence in context: A234893 A251485 A082242 * A235835 A235830 A235586

Adjacent sequences: A321863 A321864 A321865 * A321867 A321868 A321869

Keyword

nonn

Author

Amiram Eldar, Nov 20 2018

Status

approved