A143701 a(n) is the least odd number 2^n - m minimizing A007947(m*(2^n - m)).

1, 3, 7, 15, 27, 63, 125, 243, 343, 999, 1805, 3721, 8181, 16335, 32761, 65533, 112847, 190269, 519375, 1046875, 1953125, 4192479, 8385125, 16775019, 24398405, 66976875, 134216625

Offset: 1

Author

Artur Jasinski, Nov 10 2008

Keywords

• nonn
• more

Comments

a(n) is the smallest odd number such that the product of distinct prime divisors of (2^n)*a(n)*(2^n-a(n)) is the smallest for the range a(n) <= 2^x - a(n) < 2^x.

The product of distinct prime divisors of m*(2^n-m) is also called the radical of that number: rad(m*(2^n-m)).

Crossrefs

Cf. A007947, A085152, A085153.

Cf. A147298, A147299, A147300, A147302, A147303, A147305, A147306, A147307.

Cf. A147638, A147639, A147640, A147641, A147642, A147643.

Programs

Mathematica

aa = {1}; bb = {1}; rr = {}; Do[logmax = 0; k = 2^x; w = Floor[(k - 1)/2]; Do[m = FactorInteger[n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log > logmax, bmin = k - n; amax = n; logmax = log; r = rad], {n, 1, w, 2}]; Print[{x, amax}]; AppendTo[aa, amax]; AppendTo[bb, bmin]; AppendTo[rr, r]; AppendTo[a, {x, logmax}], {x, 2, 15}]; bb (* Artur Jasinski with assistance of M. F. Hasler *)

Formula

a(n) = 2^n - A143700(n).

Extensions

a(1) added by Jinyuan Wang, Aug 11 2020