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

Author

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

Keywords

• nonn
• nice

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

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).

Programs

Mathematica

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

PARI

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

Extensions

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