A047713 Euler-Jacobi pseudoprimes: 2^((n-1)/2) == (2 / n) mod n, where (2 / n) is a Jacobi symbol.

561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 6601, 8321, 8481, 10585, 12801, 15841, 16705, 18705, 25761, 29341, 30121, 33153, 34945, 41041, 42799, 46657, 49141, 52633, 62745, 65281, 74665, 75361, 80581, 85489, 87249, 88357, 90751, 104653

Offset

1,1

Comments

Odd composite numbers n such that 2^((n-1)/2) == (-1)^((n^2-1)/8) mod n. - Thomas Ordowski, Dec 21 2013

Most terms are congruent to 1 mod 8 (cf. A006971). Among the first 1000 terms, the number of terms congruent to 1, 3, 5 and 7 mod 8 are 764, 47, 125 and 64, respectively. - Jianing Song, Sep 05 2018

References

R. K. Guy, Unsolved Problems in Number Theory, A12.

H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the subsequence A006971).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Eric Weisstein's World of Mathematics, Euler-Jacobi Pseudoprime.

Eric Weisstein's World of Mathematics, Pseudoprime.

Index entries for sequences related to pseudoprimes

Mathematica

Select[ Range[ 3, 105000, 2 ], Mod[ 2^((# - 1)/2) - JacobiSymbol[ 2, # ], # ] == 0 && ! PrimeQ[ # ] & ]

Prog

(PARI) is(n)=n%2 && Mod(2, n)^(n\2)==kronecker(2, n) && !isprime(n) \\ Charles R Greathouse IV, Dec 20 2013

Crossrefs

Cf. A002997, A001567, A048950.

Terms in this sequence satisfying certain congruence: A270698 (congruent to 1 mod 4), A270697 (congruent to 3 mod 4), A006971 (congruent to +-1 mod 8), A244628 (congruent to 3 mod 8), A244626 (congruent to 5 mod 8).

Sequence in context: A135721 A290486 A253595 * A006971 A270698 A218483

Adjacent sequences: A047710 A047711 A047712 * A047714 A047715 A047716

Keyword

nonn,nice

Author

N. J. A. Sloane, Richard Pinch and Robert G. Wilson v

Extensions

Corrected by Eric W. Weisstein; more terms from David W. Wilson

Status

approved