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A182291 Number of bases in which 2n+1 is a strong pseudoprime. 0
2, 4, 6, 2, 10, 12, 2, 16, 18, 2, 22, 4, 2, 28, 30, 2, 2, 36, 2, 40, 42, 2, 46, 6, 2, 52, 2, 2, 58, 60, 2, 6, 66, 2, 70, 72, 2, 2, 78, 2, 82, 6, 2, 88, 18, 2, 2, 96, 2, 100, 102, 2, 106, 108, 2, 112, 2, 2, 2, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The subsequence related to composite 2n+1 is characterized with records in A195328 and associated 2n+1 tabulated in A141768.

Let N = 2n+1 = product_{i=1..s} p_i^r_i be the prime factorization of the odd 2n+1. Related odd parts q and q_i are defined by N-1=2^k*q and p_i-1 = 2^(k_i)*q_i, with sorting such that k_1 <= k_2 <=k_3... Then a(n) = (1+sum_{j=0..k1-1} 2^(j*s)) *product_{i=1..s} gcd(q,qi).

Reduces to A006093 if 2n+1 is prime.

This might be correlated with 2*A195508(n).

LINKS

Table of n, a(n) for n=1..60.

F. Arnault, The Rabin-Monier theorem for Lucas pseudoprimes, Mathematics of Computation, vol. 66, no 218, April 1997, pp. 869-881.

Index entries for sequences related to pseudoprimes

MAPLE

A000265 := proc(n)

        n/2^padic[ordp](n, 2) ;

end proc:

a := proc(n)

        q := A000265(n-1) ;

        B := 1;

        s := 0 ;

        k1 := 10000000000000 ;

        for pf in ifactors(n)[2] do

                pi := op(1, pf) ;

                qi := A000265(pi-1) ;

                ki := ilog2((pi-1)/qi) ;

                k1 := min(k1, ki) ;

                B := B*igcd(q, qi) ;

                s := s+1 ;

        end do:

        1+add(2^(j*s), j=0..k1-1) ;

        return B*% ;

end proc:

seq(a(2*n+1), n=1..60) ;

CROSSREFS

Sequence in context: A094752 A214061 A260300 * A071294 A060684 A056134

Adjacent sequences:  A182288 A182289 A182290 * A182292 A182293 A182294

KEYWORD

nonn

AUTHOR

R. J. Mathar, Jul 03 2012

STATUS

approved

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Last modified October 20 15:14 EDT 2017. Contains 293612 sequences.