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Cipolla pseudoprimes to base 3: (9^p-1)/8 for any odd prime p.
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%I #36 Sep 08 2022 08:46:01

%S 91,7381,597871,3922632451,317733228541,2084647712458321,

%T 168856464709124011,1107867264956562636991,

%U 588766087155780604365200461,47690053059618228953581237351,25344449488056571213320166359119221,166284933091139163730593611482181209801

%N Cipolla pseudoprimes to base 3: (9^p-1)/8 for any odd prime p.

%C This is the case a=3 of Theorem 1 in the paper of Hamahata and Kokubun (see Links section).

%D Michele Cipolla, Sui numeri composti P che verificano la congruenza di Fermat a^(P-1) = 1 (mod P), Annali di Matematica 9 (1904), p. 139-160.

%H Bruno Berselli, <a href="/A210461/b210461.txt">Table of n, a(n) for n = 1..50</a>

%H Umberto Cerruti, <a href="/A210461/a210461.pdf">Pseudoprimi di Fermat e numeri di Carmichael</a> (in Italian), 2013.

%H 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).

%e 91 is in the sequence because 91=((3^3-1)/2)*((3^3+1)/4), even if p=3 divides 3*(3^2-1), and 3^90 = (91*8+1)^15 == 1 (mod 91).

%e 7381 is in the sequence because 7381=((3^5-1)/2)*((3^5+1)/4) and 3^7380 = (7381*472400+1)^369 == 1 (mod 7381).

%p P:=proc(q)local n;

%p for n from 2 to q do print((9^ithprime(n)-1)/8);

%p od; end: P(100); # _Paolo P. Lava_, Oct 11 2013

%t (9^# - 1)/8 & /@ Prime[Range[2, 12]]

%o (Magma) [(9^NthPrime(n)-1)/8: n in [2..12]];

%o (Maxima)

%o Prime(n) := block(if n = 1 then return(2), return(next_prime(Prime(n-1))))$

%o makelist((9^Prime(n)-1)/8, n, 2, 12);

%o (Haskell)

%o a210461 = (`div` 8) . (subtract 1) . (9 ^) . a065091

%o -- _Reinhard Zumkeller_, Jan 22 2013

%Y Cf. A005935, A002452, A065091, A210454, A217853.

%K nonn

%O 1,1

%A _Bruno Berselli_, Jan 22 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)