cp's OEIS Frontend

A250208 Ratio of the primitive part of 2^n-1 to the product of primitive prime factors of 2^n-1.

%I A250208 #50 Nov 12 2024 09:06:10
%S A250208 1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,3,1,5,7,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A250208 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A250208 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A250208 Ratio of the primitive part of 2^n-1 to the product of primitive prime factors of 2^n-1.
%C A250208 As with A178764, it can be shown that all terms are either 1 or prime.
%C A250208 a(2*3^n)   =  3 (n>=1).
%C A250208 a(4*5^n)   =  5 (n>=1).
%C A250208 a(3*7^n)   =  7 (n>=1).
%C A250208 a(10*11^n) = 11 (n>=1).
%C A250208 a(12*13^n) = 13 (n>=1).
%C A250208 a(8*17^n)  = 17 (n>=1).
%C A250208 a(18*19^n) = 19 (n>=1).
%C A250208 ...
%C A250208 a(A014664(k)*prime(k)^n) = prime(k).
%C A250208 For other n (while Phi_n(2) is squarefree), a(n) = 1.
%C A250208 a(n) != 1 for n = {6, 18, 20, 21, 54, 100, 110, 136, 147, 155, 156, 162, ...}.
%C A250208 At least, a(A049093(n)) = 1. (In fact, since Phi_n(2) is not completely factored for n = 991, 1207, 1213, 1217, 1219, 1229, 1231, 1237, 1243, 1249, ..., so it is unknown whether they are squarefree or not, but it is likely that Phi_n(2) is squarefree for all n except 364 and 1755 (because it is likely 1093 and 3511 are the only two Wieferich primes), so a(991), a(1207), a(1213), ..., are likely to be 1.)
%H A250208 Eric Chen, <a href="/A250208/b250208.txt">Table of n, a(n) for n = 1..990</a>
%H A250208 Eric Chen, <a href="/A250208/a250208_1.txt">Table of n, a(n) for n = 1..1280 status (with unknown terms but all are conjectured to be 1)</a>
%F A250208 a(n) = A019320(n) / A064078(n) while Phi_n(2) is squarefree.
%F A250208 a(n) = GCD(Phi_n(2), n) while Phi_n(2) is squarefree.
%F A250208 Notice: a(364) = 1093, a(1755) = 3511. (See A001220.)
%e A250208 a(11) = 1 since Phi_11(2) = (2^11-1)/(2-1) = 2047, and the primitive prime factors of 2^11-1 are 23 and 89, so a(11) = 2047/(23*89) = 1.
%e A250208 a(18) = 3 since Phi_18(2) = 2^6 - 2^3 + 1 = 57, and the only primitive prime factor of 2^18-1 is 19, so a(18) = 57/19 = 3.
%t A250208 a250208[n_] = If[n == 364, 1093, If[n == 1755, 3511, GCD[Cyclotomic[n, 2], n]]]; Table[a250208[n], {n, 0, 200}]
%o A250208 (PARI) a(n) = if (n==364, 1093, if (n==1755, 3511, gcd(polcyclo(n, 2), n)));
%o A250208 (PARI) isprimitive(p, n) = {for (r=1, n-1, if (((2^r-1) % p) == 0, return (0)); ); return (1); }
%o A250208 ppf(n) = {my(pf = factor(2^n-1)[,1]); prod(k=1,#pf, if (isprimitive(pf[k], n), pf[k], 1));}
%o A250208 a(n) = if (issquarefree(m=polcyclo(n,2)), gcd(m, n), m/ppf(n)); \\ _Michel Marcus_, Mar 06 2015
%Y A250208 Cf. A178764.
%Y A250208 Cf. A001220, A237043, A093106, A049094, A049093, A014491, A072226, A161508.
%K A250208 nonn
%O A250208 1,6
%A A250208 _Eric Chen_, Mar 02 2015