cp's OEIS Frontend

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

%I A332018 #23 Jun 20 2025 08:12:40
%S A332018 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,
%T A332018 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,
%U A332018 7,19,29,37,5,17,31,7,1,163,11,7,17,23,35,89,1,61,37
%N A332018 a(n) = A038502(A000265(n)) if n is even or n == 0 (mod 3), a(n) = A038502(A000265(5*n + 1)) otherwise.
%C A332018 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.
%C A332018 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.
%C A332018 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)).
%H A332018 J. Lesieutre, <a href="http://web.mit.edu/rsi/www/pdfs/papers/2004/2004-lesjohn.pdf">On a Generalization of the Collatz Conjecture</a>, Research Science Institute, 2007.
%H A332018 T. Oliveira e Silva, <a href="http://sweet.ua.pt/tos/px+1.html">Computational verification of the 5x+1 and 7x+1 conjectures</a>.
%F A332018 a(n) = A038502(A000265(A133419(n))).
%F A332018 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.
%p A332018 A332018 := proc(n) option remember;
%p A332018 if n mod 2 = 0 or n mod 3 = 0 then n/(2^padic[ordp](n, 2)*3^padic[ordp](n, 3))
%p A332018 else (5*n+1)/(2^padic[ordp](5*n+1, 2)*3^padic[ordp](5*n+1, 3)) fi end:
%p A332018 seq(A332018(n), n=1..80);
%t A332018 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}]
%o A332018 (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))
%o A332018 (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
%Y A332018 Cf. A000265, A038502, A133419.
%K A332018 nonn,easy
%O A332018 1,5
%A A332018 _Davis Smith_, Feb 04 2020