cp's OEIS Frontend

A055021 Smallest number k such that n iterations of sigma() are required for the result to be >= 2k.

%I A055021 #24 Mar 18 2024 13:49:15
%S A055021 6,2,9,81
%N A055021 Smallest number k such that n iterations of sigma() are required for the result to be >= 2k.
%C A055021 These are the first terms of A023196, A107912, A107913, A107914. - _Jud McCranie_, May 28 2005
%C A055021 a(5) > 4*10^9, if it exists. - _Jud McCranie_, May 28 2005
%C A055021 There are no more terms: sigma(2*k) is never prime if k is not a power of 2, so an even number needs at most two steps; sigma(k) is odd iff k is a square or twice a square. So A107914 (four recursive steps) contains only odd squares. Assume p prime so sigma(p^2) = p^2 + p + 1 = m^2 never meets the condition with p + 2k = m that (p + 2k)^2 = m^2. This implies the impossibility of a solution for numbers of the form p^(2i) and numbers of the form p^(2i)q^(2i). - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jun 06 2005
%C A055021 If k is a power of 2 then sigma(sigma(2*k)) = sigma(4*k - 1) >= 4*k and so the number of iterations is exactly 2. - _David A. Corneth_, Mar 18 2024
%e A055021 sigma(sigma(sigma(9))) = 24 >= 2*9, so a(3)=9.
%o A055021 (PARI) isok(k, n) = my(kk=k); for (i=1, n, k = sigma(k); if ((i<n) && (k>=2*kk), return(0))); k >= 2*kk;
%o A055021 a(n) = my(k=2); while (!isok(k, n), k++); k; \\ _Michel Marcus_, Mar 18 2024
%o A055021 (PARI)
%o A055021 seq() = {
%o A055021 	my(todo = Set([1,2,3,4]), res = vector(4));
%o A055021 	for(i = 2, oo,
%o A055021 		t = 1;
%o A055021 		s = sigma(i);
%o A055021 		while(s < 2*i,				
%o A055021 			s = sigma(s);
%o A055021 			t++
%o A055021 		);
%o A055021 		if(res[t] == 0,
%o A055021 			res[t] = i;
%o A055021 			todo = setminus(todo, Set(t));
%o A055021 			if(#todo == 0,
%o A055021 				return(res)
%o A055021 			)
%o A055021 		);	
%o A055021 	)
%o A055021 } \\ _David A. Corneth_, Mar 18 2024
%Y A055021 Cf. A000203, A055020, A107912, A107913, A107914, A060800.
%K A055021 nonn,fini,full
%O A055021 1,1
%A A055021 _Jud McCranie_, May 31 2000