A332018 a(n) = A038502(A000265(n)) if n is even or n == 0 (mod 3), a(n) = A038502(A000265(5*n + 1)) otherwise.

1, 1, 1, 1, 13, 1, 1, 1, 1, 5, 7, 1, 11, 7, 5, 1, 43, 1, 1, 5, 7, 11, 29, 1, 7, 13, 1, 7, 73, 5, 13, 1, 11, 17, 11, 1, 31, 19, 13, 5, 103, 7, 1, 11, 5, 23, 59, 1, 41, 25, 17, 13, 133, 1, 23, 7, 19, 29, 37, 5, 17, 31, 7, 1, 163, 11, 7, 17, 23, 35, 89, 1, 61, 37

Offset: 1

Author

Davis Smith, Feb 04 2020

Keywords

• nonn
• easy

Comments

a(n) is the greatest divisor of n coprime to 6 if n is not coprime to 6, otherwise a(n) is the greatest divisor of 5*n + 1 coprime to 6.

This is the '5x+1' map with the successive dividing steps removed. The 'Px+1' map with those steps removed: If x is divisible by any prime < P, then divide out those primes; otherwise multiply x by P, add 1, and then divide out the primes < P.

There is a conjecture which states that for any value of n > 0 there is a k such that a^{k}(n) = 1 or a^{k}(n) enters one of a finite number of periodic cycles, where a^{0}(n) = n and a^{k + 1}(n) = a(a^{k}(n)).

Links

• J. Lesieutre, On a Generalization of the Collatz Conjecture, Research Science Institute, 2007.
• T. Oliveira e Silva, Computational verification of the 5x+1 and 7x+1 conjectures.

Crossrefs

Cf. A000265, A038502, A133419.

Programs

Magma

[Gcd(n,6) ne 1 select n/(Gcd(n, 2^n)*Gcd(n, 3^n)) else (5*n + 1)/(Gcd(5*n + 1, 2^(5*n + 1))*Gcd(5*n + 1, 3^(5*n + 1))):n in [1..75]]; // Marius A. Burtea, Feb 06 2020

Maple

A332018 := proc(n) option remember;
if n mod 2 = 0 or n mod 3 = 0 then n/(2^padic[ordp](n, 2)*3^padic[ordp](n, 3))
else (5*n+1)/(2^padic[ordp](5*n+1, 2)*3^padic[ordp](5*n+1, 3)) fi end:
seq(A332018(n), n=1..80);

Mathematica

b[n_]:=Denominator[2^n/n]; c[n_]:=Denominator[3^n/n]; Table[If[EvenQ[n]||(Mod[n, 3] == 0), c[b[n]], c[b[5*n + 1]]], {n, 1, 80}]

PARI

A332018(n)=my(val(x)=x/(2^valuation(x,2)*3^valuation(x,3))); val(if(n%2&&n%3,5*n+1,n))

Formula

a(n) = A038502(A000265(A133419(n))).

a(n) = n/(gcd(n, 2^n)*gcd(n, 3^n)) if n is not coprime to 6, a(n) = (5*n + 1)/(gcd(5*n + 1, 2^(5*n + 1))*gcd(5*n + 1, 3^(5*n + 1))) otherwise.