OFFSET
1,5
COMMENTS
Smallest odd number a(n) such that product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is the smallest available for a(n) <= 2^x - a(n) < 2^x.
Product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is also called radical: rad((2^n)*a(n)*(2^n - a(n))).
For numbers 2^n - a(n) see A143701.
For minimal values of rad((2^n)*a(n)*(2^n - a(n))) see A143702.
Related to the abc conjecture. - M. F. Hasler, Nov 13 2008
MATHEMATICA
a = {{1, 1}}; aa = {1}; bb = {}; 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}]; aa (* Artur Jasinski with assistance of M. F. Hasler *)
PROG
(PARI) A143700(n) = {my(b=1, m=2^n-b); forstep(a=3, 2^(n-1), 2, A007947(a)*A007947(2^n-a)<m || next; m=A007947((2^n-a)*b=a)); b; } \\ M. F. Hasler, Nov 13 2008
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Artur Jasinski, Nov 10 2008
EXTENSIONS
a(28)-a(33) from M. F. Hasler, Nov 13 2008
STATUS
approved