cp's OEIS Frontend

A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.

%I A210454 #47 Sep 08 2022 08:46:01
%S A210454 341,5461,1398101,22369621,5726623061,91625968981,23456248059221,
%T A210454 96076792050570581,1537228672809129301,6296488643826193618261,
%U A210454 1611901092819505566274901,25790417485112089060398421,6602346876188694799461995861
%N A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.
%C A210454 This is the case a=2 of Theorem 1 in the paper of Hamahata and Kokubun (see Links section).
%C A210454 Named after the Italian mathematician Michele Cipolla (1880-1947). - _Amiram Eldar_, Jun 15 2021
%H A210454 Bruno Berselli, <a href="/A210454/b210454.txt">Table of n, a(n) for n = 1..50</a>
%H A210454 Umberto Cerruti, <a href="/A210454/a210454.pdf">Pseudoprimi di Fermat e numeri di Carmichael</a> (in Italian), 2013. The sequence is on page 3.
%H A210454 Michele Cipolla, <a href="https://doi.org/10.1007/BF02419871">Sui numeri composti P, che verificano la congruenza di Fermat a^(P-1) = 1 (mod P)</a>, Annali di Matematica, Vol. 9 (1904), pp. 139-160.
%H A210454 Y. Hamahata and Y. Kokubun, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata2/hamahata44.html"> Cipolla Pseudoprimes</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.6.
%p A210454 P:=proc(q)local n;
%p A210454 for n from 3 to q do print((4^ithprime(n)-1)/3);
%p A210454 od; end: P(100); # _Paolo P. Lava_, Oct 11 2013
%t A210454 (4^# - 1)/3 & /@ Prime[Range[3, 15]]
%o A210454 (Magma) [(4^NthPrime(n)-1)/3: n in [3..15]];
%o A210454 (Maxima)
%o A210454 Prime(n) := block(if n = 1 then return(2), return(next_prime(Prime(n-1))))$
%o A210454 makelist((4^Prime(n)-1)/3, n, 3, 15);
%o A210454 (Haskell)
%o A210454 a210454 = (`div` 3) . (subtract 1) . (4 ^) . a000040 . (+ 2)
%o A210454 -- _Reinhard Zumkeller_, Jan 22 2013
%o A210454 (PARI) a(n)=4^prime(n+2)\3 \\ _Charles R Greathouse IV_, Jul 09 2015
%Y A210454 Cf. A001567, A002450, A210461, A215848, A216170.
%K A210454 nonn
%O A210454 1,1
%A A210454 _Bruno Berselli_, Jan 21 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)