|
|
A161213
|
|
a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 18.
|
|
5
|
|
|
1, 131071, 64570081, 8589869056, 190734863281, 8463265086751, 38771752331201, 562945658454016, 2779530261754401, 24999809265103951, 50544702849929377, 554648540725313536, 720867993281778161, 5081852349802846271
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
a(n) is the number of lattices L in Z^17 such that the quotient group Z^17 / L is C_n. - Álvar Ibeas, Nov 26 2015
|
|
LINKS
|
|
|
FORMULA
|
a(n) = J_17(n)/A000010(n), where J_17 is the 17th Jordan totient function.
Multiplicative with a(p^e) = p^(16e-16) * (p^17-1) / (p-1).
For squarefree n, a(n) = A000203(n^16). (End)
Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} (1 + (p^16-1)/((p-1)*p^17)) = 0.1143286202... .
Sum_{k>=1} 1/a(k) = zeta(16)*zeta(17) * Product_{p prime} (1 - 2/p^17 + 1/p^33) = 1.000007645061593... . (End)
|
|
MAPLE
|
add(numtheory[mobius](n/d)*d^17, d=numtheory[divisors](n)) ;
%/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
|
|
MATHEMATICA
|
A161213[n_]:=DivisorSum[n, MoebiusMu[n/#]*#^(18-1)/EulerPhi[n]&]; Array[A161213, 20]
f[p_, e_] := p^(16*e - 16) * (p^17-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
|
|
PROG
|
(PARI) A161213(n)=sumdiv(n, d, moebius(n/d)*d^17)/eulerphi(n);
(PARI) vector(100, n, sumdiv(n^16, d, if(ispower(d, 17), moebius(sqrtnint(d, 16))*sigma(n^16/d), 0))) \\ Altug Alkan, Nov 26 2015
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^17 - 1)*f[i, 1]^(16*f[i, 2] - 16)/(f[i, 1] - 1)); } \\ Amiram Eldar, Nov 08 2022
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,mult
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|