A334097 a(n) is the exponent of the eventual power of 2 reached 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.

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 6, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 8, 7, 7, 7, 8, 7, 8, 7, 8, 8, 7, 7, 7, 7, 8, 7, 7, 7, 8, 8, 8, 8, 7, 7, 8

Offset: 1

Author

Antti Karttunen, Apr 29 2020

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A007814, A052126, A078701, A209229, A331410, A334098, A334862.

Cf. also A064415 (analogous sequence when using the map k -> k - k/p).

Programs

Mathematica

Array[Log2@ NestWhile[# + #/FactorInteger[#][[-1, 1]] &, #, !IntegerQ@ Log2@ # &] &, 105] (* Michael De Vlieger, Apr 30 2020 *)

PARI

A334097(n) = if(!bitand(n,n-1),valuation(n,2),my(f=factor(n)[, 1]); A334097(n+(n/f[2-(n%2)])));

PARI

A334097(n) = if(!bitand(n,n-1),valuation(n,2),A334097(n+(n/vecmax(factor(n)[, 1]))));

PARI

A334097(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],f[k,2],f[k,2]*A334097(1+f[k,1]))); };

Formula

Totally additive sequence: a(2) = 1, a(p) = a(p+1) for odd primes p, a(m*n) = a(m)+a(n) for m, n > 1.

If A209229(n) == 1, a(n) = A007814(n), otherwise a(n) = a(n+A052126(n)), or equally, a(n) = a(n+(n/A078701(n))).

a(n) = A331410(n) + A334098(n) = A334862(n) + A064415(n).