A317623 Number of distinct values of X*(3*X-1) mod n.

1, 1, 3, 2, 3, 3, 4, 4, 9, 3, 6, 6, 7, 4, 9, 8, 9, 9, 10, 6, 12, 6, 12, 12, 11, 7, 27, 8, 15, 9, 16, 16, 18, 9, 12, 18, 19, 10, 21, 12, 21, 12, 22, 12, 27, 12, 24, 24, 22, 11, 27, 14, 27, 27, 18, 16, 30, 15, 30, 18, 31, 16, 36, 32, 21, 18, 34, 18, 36, 12, 36

Offset: 1

Author

Andrew Howroyd, Aug 01 2018

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000224, A290731, A290732.

Programs

Mathematica

f[2, e_] := 2^(e-1); f[3, e_] := 3^e; f[p_, e_] := 1 + Floor[p^(e+1)/(2*p+2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 13 2020 *)

PARI

a(n)={my(v=vector(n)); for(i=0, n-1, v[i*(3*i-1)%n + 1]=1); vecsum(v)}

PARI

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p<=3, if(p==2, 2^(e-1), 3^e), 1 + p^(e+1)\(2*p+2)))}

Formula

Multiplicative with a(2^e) = 2^(e-1), a(3^e) = 3^e, a(p^e) = 1 + floor( p^(e+1)/(2*p+2) ) for prime p >= 5.