A329814 The smallest base b where the sum of the digits for the number n in the base b is the smallest, with 1 < b < n and a(1) = a(2) = 1.

1, 1, 2, 2, 2, 2, 6, 2, 3, 2, 10, 2, 12, 7, 14, 2, 2, 2, 18, 2, 20, 11, 22, 2, 5, 5, 3, 3, 28, 3, 30, 2, 2, 2, 34, 6, 6, 19, 38, 2, 40, 6, 42, 22, 44, 23, 46, 2, 7, 5, 50, 26, 52, 3, 54, 7, 56, 29, 58, 30, 60, 31, 62, 2, 2, 2, 66, 2, 68, 35, 70, 2, 72, 37, 74

Offset: 1

Author

Haris Ziko, Nov 21 2019

Keywords

• nonn
• base
• look

Comments

The smallest sum of digits corresponding to a(n) is equal to 2-A075802(n), i.e., it is 1 when n is 1 or a perfect power and 2 otherwise. - Giovanni Resta, Nov 22 2019

a(n)=n-1 if and only if n is in A088905 but not in A001597. a(n)<= n/2 if n is even. - Robert Israel, Dec 05 2019

Examples

			For n = 5:
  n in base 2 = [1, 0, 1] -> digitSum(5, 2) = 2.
  n in base 3 = [1, 2] -> digitSum(5, 3) = 3.
  n in base 4 = [1, 1] -> digitSum(5, 4) = 2.
  Base 2 has the smallest sum of the digits for n = 5 ->
    therefore a(5) = 2.
For n = 7:
  n in base 2 = [1, 1, 1] -> digitSum(7, 2) = 3.
  n in base 3 = [2, 1] -> digitSum(7, 3) = 3.
  n in base 4 = [1, 3] -> digitSum(7, 4) = 4.
  n in base 5 = [1, 2] -> digitSum(7, 5) = 3.
  n in base 6 = [1, 1] -> digitSum(7, 6) = 2.
  Base 6 has the smallest sum of the digits for n = 7 ->
    therefore a(7) = 6.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597, A075802, A088905.

Programs

Maple

f:= proc(n) local F, t,d,bmin,s,r,b;
   F:= ifactors(n)[2];
   d:= igcd(seq(t[2],t=F));
   if d > 1 then return mul(t[1]^(t[2]/d),t=F) fi;
   F:= ifactors(n-1)[2];
   d:= igcd(seq(t[2],t=F));
   if d=1 then bmin:= n-1 else bmin:= mul(t[1]^(t[2]/d),t=F) fi;
   for s in numtheory:-divisors(n) do
     r:= n/s-1;
     F:= ifactors(s)[2];
     d:= igcd(seq(t[2],t=F));
     b:= mul(t[1]^(t[2]/d),t=F);
     if b < bmin and r = b^padic:-ordp(r,b) then bmin:= b fi
   od;
   bmin;
end proc:
map(f, [$1..100]); # Robert Israel, Dec 05 2019

Mathematica

a[n_] := Block[{b=1, r=n, t}, Do[t = Plus @@ IntegerDigits[n, i]; If[t < r, r=t; b=i], {i, 2, n-1}]; b]; Array[a, 75] (* Giovanni Resta, Nov 22 2019 *)

PARI

a(n)={my(best_b=1, best_dig_sum=n); if(n>1, for(b=2, n-1, dig_sum=sumdigits(n, b); if(best_dig_sum>dig_sum, best_dig_sum=dig_sum; best_b=b))); best_b};

Extensions

More terms from Giovanni Resta, Nov 22 2019