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.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Aug 07 2020

Keywords

• nonn

Comments

Starting from n, the number of prime shifts needed before a term of A246281 is reached.

It holds that a(n) >= A336835(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).

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.

Question: Is the escape clause necessary in the definition?

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A246281 (positions of zeros, numbers k for which A003961(k) < 2*k).

Cf. A005940, A252742, A252743, A286385.

Cf. also A246271, A252459, A336835 for similar iterations.

Programs

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336836(n) = for(i=0,oo,my(n2 = n+n); n = A003961(n); if(n < n2, return(i)));