A230373 Numbers n such that sigma(sigma*(n)) = sigma*(sigma(n)), where sigma*(n) is the sum of anti-divisors of n (A066417).

3, 265, 450, 1989, 18278, 31639, 55474, 71306, 96639, 197518, 267026, 1620723, 1888235, 3605481, 4448715, 10837215, 12128451, 22598820, 84681074, 96503379, 130118331, 152234714, 162138375, 189149834, 211239421, 343379954, 353833749, 404994939, 599244123, 804486314

Offset: 1

Author

Paolo P. Lava, Oct 23 2013

Keywords

• nonn
• hard

Examples

			Divisors of 450 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450 and sigma(450) = 1209; anti-divisors of 1209 are 2, 6, 26, 41, 59, 62, 78, 186, 806 and sigma*(1209) = 1266.
Anti-divisors of 450 are 4, 12, 17, 20, 29, 31, 36, 53, 60, 100, 180, 300 and sigma*(450) = 842; divisors of 842 are 1, 2, 421, 842 and sigma(842) = 1266.
Therefore 450 is part of the sequence because sigma(sigma*(450)) = sigma*(sigma(450)) = 1266.
		

Crossrefs

Cf. A000203, A066417.

Programs

Maple

with(numtheory);P:=proc(q) local a,b,c,k,j,n;
for n from 3 to q do c:=sigma(n);
k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
k:=0; j:=c; while j mod 2<>1 do k:=k+1; j:=j/2; od;
b:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2;
if sigma(a)=b then print(n); fi; od; end: P(10^6);

Extensions

a(12)-a(30) from Giovanni Resta, Oct 23 2013