A254596 Convert number n to each of the bases from 2 to 10. a(n) is the minimum base that maximizes the sum of digits.

2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 7, 8, 8, 8, 5, 9, 9, 7, 10, 10, 8, 8, 7, 7, 7, 9, 10, 10, 10, 10, 7, 7, 9, 9, 9, 8, 8, 8, 7, 10, 9, 9, 9, 9, 8, 8, 10, 10, 10, 10, 9, 9, 9, 8, 10, 10, 10, 10, 10, 10, 9, 9, 10, 10, 10, 10, 10, 10, 10

Offset

1,1

Comments

For n <= 10^6 the frequencies are:

2 -> 1 (highest number: 1)

3 -> 1 (highest number: 2)

4 -> 7 (highest number: 832123)

5 -> 8607 (highest number: 993124)

6 -> 29338 (highest number: 995105)

7 -> 110167 (highest number: 991612)

8 -> 153583 (highest number: 999423)

9 -> 251998 (highest number: 1000000)

10 -> 446298 (highest number: 999999)

2 only occurs at a(1) and 3 only occurs at a(2), but all the other numbers, 4 through 10, occur with increasing frequency. 4 occurs at n: 3, 730, 731, 631530, 631531, 832122, 832123, ..., . - Robert G. Wilson v, Feb 05 2015

From Robert G. Wilson v, Feb 11 2015: (Start)

The following table is a count of the frequencies of the bases, b, 2 through 10, and for exponents, e, 1 through 9, of 10:

\b..2..3...4........5.........6.........7..........8..........9.........10

e\

1 1 1 1 1 2 1 1 1 1

2 1 1 1 2 4 15 14 19 43

3 1 1 3 19 61 127 171 233 384

4 1 1 3 161 419 1152 1525 2445 4293

5 1 1 3 1168 3706 12152 16503 24138 42328

6 1 1 7 8607 29338 110167 153583 251998 446298

7 1 1 7 66461 313656 918927 1559502 2616635 4524810

8 1 1 11 421478 2262670 8075616 16460917 25832934 46946372

9 1 1 26 3910736 19645806 77440365 167615009 254855326 476532730

(End)

Links

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Example

11 in base 2 is 1011 -> sum of digits: 3;
11 in base 3 is 102 -> sum of digits: 3;
11 in base 4 is 23 -> sum of digits: 5;
11 in base 5 is 21 -> sum of digits: 3;
11 in base 6 is 15 -> sum of digits: 6;
11 in base 7 is 14 -> sum of digits: 5;
11 in base 8 is 13 -> sum of digits: 4;
11 in base 9 is 12 -> sum of digits: 3;
11 in base 10 is 11 -> sum of digits: 2.
Therefore a(11) = 6.

Maple

with(numtheory): P:=proc(q) local a, b, c, d, j, k, n;
for n from 1 to q do c:=0;
for k from 2 to 10 do a:=convert(n, base, k);
b:=add(a[j], j=1..nops(a)); if b>c then c:=b; d:=k;
fi; od; print(d); od; end: P(10^4);

Mathematica

f[n_] := Block[{a = Plus @@@ IntegerDigits[ n, {2, 3, 4, 5, 6, 7, 8, 9, 10}]}, 1 + Position[a, Max@ a][[1, 1]]]; Array[f, 79] (* Robert G. Wilson v, Feb 04 2015 *)

Crossrefs

Cf. A254597.

Sequence in context: A365392 A305810 A171060 * A286603 A291751 A275987

Adjacent sequences: A254593 A254594 A254595 * A254597 A254598 A254599

Keyword

nonn,base

Author

Paolo P. Lava, Feb 02 2015

Status

approved