The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A250208 Ratio of the primitive part of 2^n-1 to the product of primitive prime factors of 2^n-1. 1
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS As with A178764, it can be shown that all terms are either 1 or prime. a(2*3^n) = 3 (n>=1). a(4*5^n) = 5 (n>=1). a(3*7^n) = 7 (n>=1). a(10*11^n) = 11 (n>=1). a(12*13^n) = 13 (n>=1). a(8*17^n) = 17 (n>=1). a(18*19^n) = 19 (n>=1). ... a(A014664(k)*prime(k)^n) = prime(k). For other n (while Phi_n(2) is squarefree), a(n) = 1. a(n) != 1 for n = {6, 18, 20, 21, 54, 100, 110, 136, 147, 155, 156, 162, ...}. 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.) LINKS Eric Chen, Table of n, a(n) for n = 1..990 Eric Chen, Table of n, a(n) for n = 1..1280 status (with unknown terms but all are conjectured to be 1 FORMULA a(n) = A019320(n) / A064078(n) while Phi_n(2) is squarefree. a(n) = GCD(Phi_n(2), n) while Phi_n(2) is squarefree. Notice: a(364) = 1093, a(1755) = 3511. (See A001220.) EXAMPLE 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. 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. MATHEMATICA a250208[n_] = If[n == 364, 1093, If[n == 1755, 3511, GCD[Cyclotomic[n, 2], n]]]; Table[a250208[n], {n, 0, 200}] PROG (PARI) a(n) = if (n==364, 1093, if (n==1755, 3511, gcd(polcyclo(n, 2), n))); (PARI) isprimitive(p, n) = {for (r=1, n-1, if (((2^r-1) % p) == 0, return (0)); ); return (1); } ppf(n) = {my(pf = factor(2^n-1)[, 1]); prod(k=1, #pf, if (isprimitive(pf[k], n), pf[k], 1)); } a(n) = if (issquarefree(m=polcyclo(n, 2)), gcd(m, n), m/ppf(n)); \\ Michel Marcus, Mar 06 2015 CROSSREFS Cf. A178764. Cf. A001220, A237043, A093106, A049094, A049093, A014491, A072226, A161508. Sequence in context: A354997 A327537 A120263 * A325825 A030580 A030579 Adjacent sequences: A250205 A250206 A250207 * A250209 A250210 A250211 KEYWORD nonn AUTHOR Eric Chen, Mar 02 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 25 03:15 EDT 2024. Contains 371964 sequences. (Running on oeis4.)