A240667 - OEIS

%I #38 Dec 19 2024 06:15:41

%S 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,

%T 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,

%U 32,0,0,0,0,67,0,0,0,1,0,73,0,0,0,45,0,1,0,0

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

%C 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.

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

%H Antti Karttunen, <a href="/A240667/b240667.txt">Table of n, a(n) for n = 1..10000</a>

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%F a(A007369(n)) = 0.

%e There are no integers such that sigma(x) = 2, so a(2) = 0.

%e There is a single integer, x = 2, such that sigma(x) = 3, so a(3) = 2.

%e There are 2 integers, x = 6 and 11, such that sigma(x)=12, their gcd is 1, so a(12) = 1.

%p A240667 := n -> igcd(op(select(k->sigma(k)=n, [$1..n]))):

%p seq(A240667(n), n=1..82); # _Peter Luschny_, Apr 13 2014

%t a[n_] := GCD @@ Select[Range[n], DivisorSigma[1, #] == n&];

%t Array[a, 100] (* _Jean-François Alcover_, Jul 30 2018 *)

%o (PARI) sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));

%o a(n) = {v = sigv(n); if (#v == 0, 0, gcd(v));}

%o (PARI) a(n) = my(s = invsigma(n)); if(#s, gcd(s), 0); \\ _Amiram Eldar_, Dec 19 2024, using _Max Alekseyev_'s invphi.gp

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

%K nonn

%O 1,3

%A _Michel Marcus_, Apr 10 2014