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

1,4

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

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A329027 A235987 A104597 * A340503 A072662 A030010

Adjacent sequences: A182933 A182934 A182935 * A182937 A182938 A182939

Keyword

nonn

Author

Peter Luschny, Mar 22 2011

Extensions

More terms from Antti Karttunen, Sep 23 2017

Status

approved