A073503 Numbers n such that the number of solutions to x^4 == 1 (mod n) is twice the number of solutions of x^2 == 1 (mod n).

5, 10, 13, 15, 16, 17, 20, 25, 26, 29, 30, 32, 34, 35, 37, 39, 40, 41, 45, 48, 50, 51, 52, 53, 55, 58, 60, 61, 64, 68, 70, 73, 74, 75, 78, 82, 87, 89, 90, 91, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 122, 123, 125

Offset

1,1

Comments

Conjectures: 2n > a(n) or 2n < a(n) for infinitely many values of n and abs(a(n)-2n) < sqrt(n) for n > 45. a(n)=2n for n = 318, 338, 350, 488, 490, 492, 494,...

Numbers divisible by 16 which have no prime factors = 1 mod 4, together with numbers not divisible by 16 which have exactly one prime factor = 1 mod 4. This refutes the conjectures. - Charles R Greathouse IV, Apr 16 2012

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) seems to be asymptotic to 2n.

Mathematica

Select[Range[3, 125], Length[Reduce[x^4 - 1 == 0, x, Modulus -> #]] == 2*Length[Reduce[x^2 - 1 == 0, x, Modulus -> #]] &] (* Jayanta Basu, Jul 01 2013 *)

Prog

(PARI) is(n)=my(v=factor(n)[, 1]%4, s=sum(i=1, #v, v[i]==1), e=valuation(n, 2)); s==(e<4) \\ Charles R Greathouse IV, Apr 16 2012

Crossrefs

Cf. A060594, A073103.

Sequence in context: A224448 A046130 A241143 * A313378 A226386 A200995

Adjacent sequences: A073500 A073501 A073502 * A073504 A073505 A073506

Keyword

easy,nonn

Author

Benoit Cloitre, Aug 19 2002

Status

approved