A046073 Number of squares in multiplicative group modulo n.

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 6, 3, 2, 2, 8, 3, 9, 2, 3, 5, 11, 1, 10, 6, 9, 3, 14, 2, 15, 4, 5, 8, 6, 3, 18, 9, 6, 2, 20, 3, 21, 5, 6, 11, 23, 2, 21, 10, 8, 6, 26, 9, 10, 3, 9, 14, 29, 2, 30, 15, 9, 8, 12, 5, 33, 8, 11, 6, 35, 3, 36, 18, 10, 9, 15, 6, 39, 4, 27, 20, 41, 3, 16, 21

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy
• mult

Comments

a(n) is the number of different diagonal elements in Cayley table for multiplicative group modulo n. But the fact that the same number of different elements are on the diagonal of the Cayley table does not mean in every case that these groups are isomorphic. - Artur Jasinski, Jul 03 2010

The number of quadratic residues modulo n that are coprime to n. These residues are listed in A096103. - Peter Munn, Mar 10 2021

References

• Daniel Shanks, Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 95, 1993.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Steven R. Finch and Pascal Sebah, Square and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
• Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
• Eric Weisstein's World of Mathematics, Quadratic Residue.

Crossrefs

Cf. A046072, A007735, A060594, A000010, A087692, A000224.

Row lengths of A096103.

Positions of ones: A018253.

Programs

Maple

F:= n -> nops({seq}(`if`(igcd(t,n)=1,t^2 mod n,NULL), t=1..floor(n/2))):
1, seq(F(n), n=2..100); # Robert Israel, Jan 04 2015
# 2nd program
A046073 := proc(n)
    local a,p,e,pf;
    a := 1;
    for pf in ifactors(n)[2] do
        p := op(1,pf) ;
        e := op(2,pf) ;
        if p = 2 then
            a := a*p^max(e-3,0) ;
        else
            a := a*(p-1)/2*p^(e-1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 03 2016

Mathematica

Table[EulerPhi[n]/Sum[Boole[Mod[k^2, n] == 1] + Boole[n == 1], {k, n}], {n, 86}] (* or *)
Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, p == 2, 2^Max[e - 3, 0], True, (p - 1) p^(e - 1)/2]], {n, 86}] (* Michael De Vlieger, Jul 18 2017 *)

PARI

A060594(n) = if(n<=2, 1, 2^#znstar(n)[3]); \\ This function from Joerg Arndt, Jan 06 2015
A046073(n) = eulerphi(n)/A060594(n); \\ Antti Karttunen, Jul 17 2017, after Sharon Sela's Mar 09 2002 formula.

PARI

A046073(n)=if(n>4,(n=znstar(n))[1]>>#n[3],1) \\ Avoids duplicate computation of phi(n). - M. F. Hasler, Nov 27 2017, typo fixed Mar 11 2021

Python

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod([2**max(e - 3, 0) if p==2 else (p - 1)*p**(e - 1)//2 for p, e in factorint(n).items()])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 17 2017

Scheme

(define (A046073 n) (cond ((= 1 n) n) ((even? n) (* (A000079 (max (- (A007814 n) 3) 0)) (A046073 (A028234 n)))) (else (* (/ 1 2) (- (A020639 n) 1) (/ (A028233 n) (A020639 n)) (A046073 (A028234 n)))))) ;; Antti Karttunen, Jul 17 2017, after the given multiplicative formula.

Formula

a(n) * A060594(n) = A000010(n) = phi(n) (This gives a formula for a(n) using the one in A060594(n) ). - Sharon Sela (sharonsela(AT)hotmail.com), Mar 09 2002

Multiplicative with a(2^e) = 2^max(e-3,0), a(p^e) = (p-1)*p^(e-1)/2 for p an odd prime.

Sum_{k=1..n} a(k) ~ c * n^2/sqrt(log(n)), where c = (43/(80*sqrt(Pi))) * Product_{p prime} (1+1/(2*p))*sqrt(1-1/p) = 0.24627260085060864229... (Finch and Sebah, 2006). - Amiram Eldar, Oct 18 2022

Extensions

Edited and verified by Franklin T. Adams-Watters, Nov 07 2006