A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 1, 9, 13, 8, 0, 0, 1, 0, 19, 0, 0, 0, 1, 0, 0, 0, 12, 0, 29, 1, 1, 0, 0, 0, 22, 0, 37, 18, 27, 0, 1, 0, 43, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 49, 0, 0, 1, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 1, 0, 73, 0, 0, 0, 45, 0, 1, 0, 0

Offset: 1

Author

Michel Marcus, Apr 10 2014

Keywords

• nonn

Comments

From n = 1 to 5, the least integers such that a(x) = n, depending on if singletons (see A007370 and A211656) are accepted or not, are 1, 3, 4, 7, 6 or 12, 126, 124, 210, 22152.

Is it possible to find an integer n such that a(n) = 6? Answer: n = A241625(6) = 6187272.

Examples

			There are no integers such that sigma(x) = 2, so a(2) = 0.
There is a single integer, x = 2, such that sigma(x) = 3, so a(3) = 2.
There are 2 integers, x = 6 and 11, such that sigma(x)=12, their gcd is 1, so a(12) = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000203, A007369, A007370, A211656, A241479 (a variant), A241625.

Programs

Maple

A240667 := n -> igcd(op(select(k->sigma(k)=n, [$1..n]))):
seq(A240667(n), n=1..82); # Peter Luschny, Apr 13 2014

Mathematica

a[n_] := GCD @@ Select[Range[n], DivisorSigma[1, #] == n&];
Array[a, 100] (* Jean-François Alcover, Jul 30 2018 *)

PARI

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
a(n) = {v = sigv(n); if (#v == 0, 0, gcd(v));}

PARI

a(n) = my(s = invsigma(n)); if(#s, gcd(s), 0); \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

Formula

a(A007369(n)) = 0.