cp's OEIS Frontend

A334098 a(n) = A334097(n) - A331410(n), where former is the exponent of the eventual power of 2 reached, and the latter is the number of iterations needed to get there, when starting from n and using the map k -> k + k/p, where p can be any odd prime factor of k, for example, the largest.

%I A334098 #14 Apr 30 2020 22:01:26
%S A334098 0,1,1,2,1,2,2,3,2,2,2,3,2,3,2,4,2,3,2,3,3,3,3,4,2,3,3,4,2,3,4,5,3,3,
%T A334098 3,4,2,3,3,4,3,4,3,4,3,4,4,5,4,3,3,4,3,4,3,5,3,3,3,4,4,5,4,6,3,4,3,4,
%U A334098 4,4,4,5,2,3,3,4,4,4,4,5,4,4,4,5,3,4,3,5,3,4,4,5,5,5,3,6,4,5,4,4,3,4,4,5,4
%N A334098 a(n) = A334097(n) - A331410(n), where former is the exponent of the eventual power of 2 reached, and the latter is the number of iterations needed to get there, when starting from n and using the map k -> k + k/p, where p can be any odd prime factor of k, for example, the largest.
%C A334098 Question: Are there any negative terms?
%H A334098 Antti Karttunen, <a href="/A334098/b334098.txt">Table of n, a(n) for n = 1..65537</a>
%F A334098 a(n) = A334097(n) - A331410(n).
%F A334098 Totally additive sequence: a(m*n) = a(m)+a(n), for all m, n.
%t A334098 Array[Log2@ Last[#] - (Length[#] - 1) &@ NestWhileList[# + #/FactorInteger[#][[-1, 1]] &, #, ! IntegerQ@ Log2@ # &] &, 105] (* _Michael De Vlieger_, Apr 30 2020 *)
%o A334098 (PARI) A334098(n) = { my(k=0); while(bitand(n,n-1), k++; my(f=factor(n)[, 1]); n += (n/f[2-(n%2)])); (valuation(n,2)-k); };
%Y A334098 Cf. A331410, A334097.
%K A334098 nonn
%O A334098 1,4
%A A334098 _Antti Karttunen_, Apr 29 2020