A263774 Sequence is defined by the condition that Sum_{d|n} a(d)^(n/d) = mu(n)^2, where mu(n) is the Möbius function.

1, 0, 0, -1, 0, 0, 0, -2, -1, 0, 0, 0, 0, 0, 0, -6, 0, -2, 0, 0, 0, 0, 0, 6, -1, 0, 0, 0, 0, 0, 0, -54, 0, 0, 0, -5, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 114, -1, -2, 0, 0, 0, 6, 0, 126, 0, 0, 0, 0, 0, 0, 0, -4470, 0, 0, 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 0, 0, 0

Offset

1,8

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(1) = 1.

If p>2, a(p) = 0, a(p^2) = -1, a(p^n) = 0 for n>2.

a(p1*p2*..*pn) = 0, a(2*p1*...*pn) = 0, a(4*p1*...*pn) = 0.

If p1,...,pn are odd it appears that:

a(p1^2*p2^2*...*pn^2) = (-1)^n,

a(p1^k1*p2^k2*...*pn^kn) = 0, if one of k1,...,kn > 2,

a(2*p1^k1*p2^k2*...*pn^kn) > 0 if one of k1,...,kn > 1.

a(2^n) = A264610(n).

Example

For a prime p, a(p)^1 + a(1)^p = mu(p)^2=1 => a(p) = 0.
For n=6, a(1)^6 + a(2)^3 + a(3)^2 + a(6)^1 = mu(6) = 1, so 1 - 0 + 0 + a(6) = 1, so 1 + a(6) = 1, so a(6) = 0.

Maple

a := proc (n) option remember; numtheory:-mobius(n)^2-add(procname(n/d)^d, d = `minus`(numtheory:-divisors(n), {1})) end proc; a(1) := 1; La := seq(a(i), i = 1 .. 100)

Mathematica

a[n_]:=If[n<2, 1, MoebiusMu[n]^2 - Sum[If[d==1, 0, a[n/d]^d], {d, Divisors[n]}]]; Table[a[n], {n, 100}] (* Indranil Ghosh, Mar 26 2017 *)

Prog

(PARI) a(n) = if (n==1, 1, moebius(n)^2- sumdiv(n, d, if (d==1, 0, a(n/d)^d)));
(Haskell)
a263774 1 = 1
a263774 n = foldl (-) (a008966 n) $ zipWith (^) (map a' $ reverse ds) ds
            where a' x = if x == n then 0 else a263774 x
                  ds = a027750_row n
-- Reinhard Zumkeller, Dec 06 2015

Crossrefs

Cf. A008683, A264610.

Cf. A008966, A027750.

Sequence in context: A074079 A037858 A037876 * A161519 A286561 A068101

Adjacent sequences: A263771 A263772 A263773 * A263775 A263776 A263777

Keyword

sign

Author

Gevorg Hmayakyan, Nov 28 2015

Status

approved