cp's OEIS Frontend

A336836 Number of iterations of x -> A003961(x) needed before A003961(x) < 2x, when starting from x=n, or -1 if such a number is never reached.

%I A336836 #22 Jun 13 2021 03:24:04
%S A336836 0,0,0,2,0,3,0,4,1,2,0,4,0,1,2,4,0,5,0,4,1,0,0,6,0,0,3,4,0,4,0,6,0,0,
%T A336836 1,6,0,0,1,4,0,4,0,4,3,0,0,6,1,2,0,4,0,5,0,4,1,0,0,6,0,0,3,6,0,4,0,4,
%U A336836 1,4,0,9,0,0,4,4,0,4,0,6,3,0,0,6,0,0,0,6,0,5,1,4,0,0,0,9,0,3,3,4,0,3,0,4,3
%N A336836 Number of iterations of x -> A003961(x) needed before A003961(x) < 2x, when starting from x=n, or -1 if such a number is never reached.
%C A336836 Starting from n, the number of prime shifts needed before a term of A246281 is reached.
%C A336836 It holds that a(n) >= A336835(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
%C A336836 Note that in contrast to abundancy used in A336835, the condition [A003961(x) > 2x] (= A252742) is not monotonic when iterating with A003961. For example, we have A003961(9) = 25 > 2*9, A003961(25) = 49 < 2*25, and then again A003961(49) = 121 > 2*49.
%C A336836 Question: Is the escape clause necessary in the definition?
%H A336836 Antti Karttunen, <a href="/A336836/b336836.txt">Table of n, a(n) for n = 1..65537</a>
%H A336836 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%o A336836 (PARI)
%o A336836 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A336836 A336836(n) = for(i=0,oo,my(n2 = n+n); n = A003961(n); if(n < n2, return(i)));
%Y A336836 Cf. A246281 (positions of zeros, numbers k for which A003961(k) < 2*k).
%Y A336836 Cf. A005940, A252742, A252743, A286385.
%Y A336836 Cf. also A246271, A252459, A336835 for similar iterations.
%K A336836 nonn
%O A336836 1,4
%A A336836 _Antti Karttunen_, Aug 07 2020