OFFSET
1,3
COMMENTS
Cubic analog of A046073. - Steven Finch, Mar 01 2006
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
FORMULA
a(n) = phi(n) / A060839(n).
Multiplicative with a(3) = 2, a(3^k) = 2*3^(k-2) otherwise;
a(p^k) = (p-1)*p^(k-1)/3 if prime p == 1 mod 6; a(p^k) = (p-1)*p^(k-1) for all other primes p. - Robert Israel, Jan 04 2015
Sum_{k=1..n} a(k) ~ c * n^2/log(n)^(1/3), where c = (17/(36*Gamma(2/3))) * Product_{p = 3 or p prime == 2 (mod 3)} (1+1/*p)*(1-1/p)^(2/3) * Product_{p prime == 1 (mod 3)} (1+1/(3*p))*(1-1/p)^(2/3) = 0.33051128776333262024... (Finch and Sebah, 2006). - Amiram Eldar, Oct 18 2022
MAPLE
b:= proc(p, i)
if p = 3 then if i=1 then 2 else 2*3^(i-2) fi
elif p mod 6 = 1 then (p-1)*p^(i-1)/3
else (p-1)*p^(i-1)
fi
end proc:
seq(mul(b(f[1], f[2]), f = ifactors(n)[2]), n = 1 .. 1000); # Robert Israel, Jan 04 2015
MATHEMATICA
Map[Length, Table[Select[Range[n], CoprimeQ[#, n] && IntegerQ[PowerMod[#, 1/3, n]] &], {n, 1, 82}]] (* Geoffrey Critzer, Jan 07 2015 *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 6] == 1, 3, 1]; f[3, e_] := 2*3^(e-2); f[3, 1] = 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)
PROG
(PARI) a(n) = my(f = factor(n)); prod(j=1, #f~, p=f[j, 1]; k=f[j, 2]; if (p == 3, if (k==1, 2, 2*3^(k-2)), if ((p % 6) == 1, ((p-1)*p^(k-1))/3, (p-1)*p^(k-1)))); \\ Michel Marcus, Jan 05 2015
CROSSREFS
KEYWORD
mult,nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003
EXTENSIONS
More terms from Steven Finch, Mar 01 2006
STATUS
approved