

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).


2



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
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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



