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A321866 Indices of tetrahedral numbers that are Fermat pseudoprimes to base 2. 1

%I

%S 3457,16705,21169,28297,30577,45481,114601,123121,127297,140977,

%T 156601,159337,312841,393121,418177,437977,443017,453601,509737,

%U 518017,521137,539401,545161,545617,657841,679297,704161,717817,762121,775057,832801,904801,996601

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

%C Numbers n such that n(n+1)(n+2)/6 is a Fermat pseudoprimes to base 2.

%C The corresponding tetrahedral Fermat pseudoprimes are 6891657409, 777080801185, 1581289265305, 3776730328549, 4765143438329, 15680770945781, 250856489370101, 311068284648121, 343806031110049, ...

%C Sierpinski asked for the existence of these numbers in 1965.

%H Andrzej Rotkiewicz, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21137.pdf">On some problems of W. Sierpinski</a>, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

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

%t 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

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

%Y Cf. A000292, A001567, A293622, A293624.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 20 2018

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Last modified April 19 03:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)