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

%I A240667 #38 Dec 19 2024 06:15:41
%S A240667 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 A240667 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 A240667 32,0,0,0,0,67,0,0,0,1,0,73,0,0,0,45,0,1,0,0
%N A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.
%C A240667 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 A240667 Is it possible to find an integer n such that a(n) = 6? Answer: n = A241625(6) = 6187272.
%H A240667 Antti Karttunen, <a href="/A240667/b240667.txt">Table of n, a(n) for n = 1..10000</a>
%H A240667 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%F A240667 a(A007369(n)) = 0.
%e A240667 There are no integers such that sigma(x) = 2, so a(2) = 0.
%e A240667 There is a single integer, x = 2, such that sigma(x) = 3, so a(3) = 2.
%e A240667 There are 2 integers, x = 6 and 11, such that sigma(x)=12, their gcd is 1, so a(12) = 1.
%p A240667 A240667 := n -> igcd(op(select(k->sigma(k)=n, [$1..n]))):
%p A240667 seq(A240667(n), n=1..82); # _Peter Luschny_, Apr 13 2014
%t A240667 a[n_] := GCD @@ Select[Range[n], DivisorSigma[1, #] == n&];
%t A240667 Array[a, 100] (* _Jean-François Alcover_, Jul 30 2018 *)
%o A240667 (PARI) sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
%o A240667 a(n) = {v = sigv(n); if (#v == 0, 0, gcd(v));}
%o A240667 (PARI) a(n) = my(s = invsigma(n)); if(#s, gcd(s), 0); \\ _Amiram Eldar_, Dec 19 2024, using _Max Alekseyev_'s invphi.gp
%Y A240667 Cf. A000203, A007369, A007370, A211656, A241479 (a variant), A241625.
%K A240667 nonn
%O A240667 1,3
%A A240667 _Michel Marcus_, Apr 10 2014