A133943 Sum mu(k), where the sum is over the integers k which are the "isolated divisors" of n and mu(k) is the Moebius function (mu(k) = A008683(k)). A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0

Offset

1,132

Comments

A133943(n) = -A133944(n), for n >= 2.

Is every term either 0 or 1?

No, a(132)=2, a(870)=3, a(8844)=4, a(420)=-1, a(1190)=-2, a(1260)=-3, a(7140)=-4. - Ray Chandler, Jun 25 2008

And a(14280) = -5. - Antti Karttunen, Sep 02 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Maple

A133943 := proc(n) local divs, k, i, a ; divs := convert(numtheory[divisors](n), list); a := 0 ; for i from 1 to nops(divs) do k := op(i, divs) ; if not k-1 in divs and not k+1 in divs then a := a+numtheory[mobius](k); fi ; od: RETURN(a) ; end: seq(A133943(n), n=1..120) ; # R. J. Mathar, Oct 21 2007

Mathematica

a[n_] := MoebiusMu /@ Select[Divisors[n], (# == 1 || Mod[n, #-1] > 0) && Mod[n, #+1] > 0&] // Total;
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 08 2024 *)

Prog

(PARI) A133943(n) = sumdiv(n, d, (!!if((1==d), (n%2), (n%(d-1))&&(n%(d+1))))*moebius(d)); \\ Antti Karttunen, Sep 02 2018

Crossrefs

Cf. A133944.

Sequence in context: A348033 A327153 A374197 * A014084 A014159 A361016

Adjacent sequences: A133940 A133941 A133942 * A133944 A133945 A133946

Keyword

sign

Author

Leroy Quet, Sep 30 2007, Oct 27 2007

Extensions

More terms from R. J. Mathar, Oct 21 2007

Secondary offset added by Antti Karttunen, Sep 02 2018

Status

approved