A182936 Greatest common divisor of the proper divisors of n, 0 if there are none.

0, 0, 0, 2, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 7, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Author

Peter Luschny, Mar 22 2011

Keywords

• nonn

Comments

Here a proper divisor d of n is a divisor of n such that 1 < d < n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014963, A020500, A048671, A034387, A061397, A072107, A380118.

Programs

Maple

A182936 := n -> igcd(op(numtheory[divisors](n) minus {1,n}));
seq(A182936(i), i=1..79); # Peter Luschny, Mar 22 2011

Mathematica

Join[{0}, Table[GCD@@Most[Rest[Divisors[n]]],{n,2,110}]] (* Harvey P. Dale, May 04 2018 *)
(* From Peter Luschny, Jan 31 2025: (Start) *)
Join[{0}, Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0], {n,2,110}]]
(* or *)
Table[Cyclotomic[n, 1] - If[PrimeQ[n], n, 0], {n,1,110}] (* End *)

PARI

A182936(n) = { my(divs=divisors(n)); if(#divs<3,0,gcd(vector(numdiv(n)-2,k,divs[k+1]))); }; \\ Antti Karttunen, Sep 23 2017

Formula

a(n) = 0 if n is not composite, p if n is a proper power of prime p, and 1 otherwise. - Franklin T. Adams-Watters, Mar 22 2011

Conjecture: Sum_{k=1..n} a(k) = A072107(n) - A034387(n) - 1. - Vaclav Kotesovec, Jan 29 2025

From Peter Luschny, Jan 31 2025: (Start)

a(n) = A014963(n) - A061397(n) for n > 1. In other words, this sequence is the exponential von Mangoldt function restricted to proper divisors of n. See A380118. This implies the above conjecture.

a(n) = A020500(n) - A061397(n). (End)

Extensions

More terms from Antti Karttunen, Sep 23 2017