A143700 a(n) is the least odd number m minimizing A007947(m*(2^n-m)).

1, 1, 1, 1, 5, 1, 3, 13, 169, 25, 243, 375, 11, 49, 7, 3, 18225, 71875, 4913, 1701, 144027, 1825, 3483, 2197, 9156027, 131989, 1103, 5103, 38525, 458703, 1523, 3483891, 19283525

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

Links

Table of n, a(n) for n=1..33.

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

Cf. A007947, A085152, A085153, A147298-A147307, A147638-A147643.

Sequence in context: A365457 A365456 A134894 * A036790 A201847 A185559

Adjacent sequences: A143697 A143698 A143699 * A143701 A143702 A143703

Keyword

nonn,more

Author

Artur Jasinski, Nov 10 2008

Extensions

a(28)-a(33) from M. F. Hasler, Nov 13 2008

Status

approved