A341651 a(n) is the integer k in the interval [2,n] at which log(n)*log(k) is closest to an integer.

2, 3, 2, 2, 3, 5, 7, 4, 9, 8, 5, 7, 14, 3, 6, 17, 2, 15, 20, 10, 7, 13, 17, 12, 16, 21, 11, 8, 19, 25, 32, 31, 30, 22, 7, 4, 3, 3, 34, 33, 19, 11, 31, 18, 23, 38, 48, 47, 10, 35, 27, 34, 43, 20, 12, 32, 15, 31, 39, 7, 38, 23, 29, 11, 58, 57, 35, 17, 27, 54, 42

Offset

2,1

Comments

a(1398) = 658; log(1398)*log(658) = 46.999999997186..., a result remarkably close to an integer (see A341577).

Values of j and k (2 <= k <= j) at which the absolute difference between log(j)*log(k) and the nearest integer reaches a record low as j runs through the positive integers are in A341652 and A341653, respectively.

Links

Table of n, a(n) for n=2..72.

Example

For n=2, the only integer k in the interval [2,n] is 2, so a(2) = 2.
For n=3, log(3)*log(2) = 0.76150..., which differs from the nearest integer (1) by 0.23849..., but log(3)*log(3) = 1.20694..., which differs from the nearest integer (again, 1) by only 0.20694..., which is less than 0.23849..., so a(3) = 3.
For n=12, we have
   k  P=log(12)*log(k)   |P - round(P)|
  --  ----------------  ----------------
   2  1.72240603825...  0.27759396174...
   3  2.72994898165...  0.27005101834...
   4  3.44481207651...  0.44481207651...
   5  3.99930297102...  0.00069702897... (minimum)
   6  4.45235501990...  0.45235501990...
   7  4.83540506927...  0.16459493072...
   8  5.16721811476...  0.16721811476...
   9  5.45989796330...  0.45989796330...
  10  5.72170900928...  0.27829099071...
  11  5.95854590887...  0.04145409112...
  12  6.17476105816...  0.17476105816...
so a(12) = 5.

Crossrefs

Cf. A341577, A341652, A341653.

Sequence in context: A053812 A177865 A307835 * A017828 A140087 A174329

Adjacent sequences: A341648 A341649 A341650 * A341652 A341653 A341654

Keyword

nonn

Author

Jon E. Schoenfield, Feb 17 2021

Status

approved