A393591 Greatest integer b such that the sum of the integer digits of n in bases 2..b strictly increases.

2, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 5, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 5, 2, 3, 3, 4, 4, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 2, 2, 3, 4

Offset

1,1

Comments

Even if we interpret A000042 as base 1, it cannot be incorporated into this sequence, since the count of ones in A000042 would always exceed the count of ones in base 2.

Links

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

Example

For the examples, define f(n, b) as the sum of the digits in the representation of n in base b.
a(1) is 2 because f(1, 3) is not larger than f(1, 2);
a(2) is 3 because f(2, 2) is less than f(2, 3) but f(2, 3) is not less than f(2, 4);
a(14) is 5 because f(14, 2) < f(14, 3) < f(14, 4) < f(14, 5) but f(14, 5) is not less than f(14, 6), etc.

Maple

f:= proc(n) local b, s, sb;
  s:= convert(convert(n, base, 2), `+`);
  for b from 3 do
    sb:= convert(convert(n, base, b), `+`);
    if sb <= s then return b-1 fi;
    s:= sb
  od;
end proc:
map(f, [$1..200]); # Robert Israel, Feb 22 2026

Mathematica

a[n_] := Block[{b = 3, c = Total[ IntegerDigits[n, 2]]}, While[d = Total[ IntegerDigits[n, b]]; c < d, c = d; b++]; b - 1]; Array[a, 105]

Prog

(PARI) isok(n, b)= my(v=vector(b-1, bb, sumdigits(n, bb+1))); v == Set(v);
a(n) = my(b=2); while (isok(n, b), b++); b-1; \\ Michel Marcus, Feb 27 2026

Crossrefs

Cf. A393592.

Sequence in context: A200323 A075370 A332297 * A030350 A026240 A124474

Adjacent sequences: A393588 A393589 A393590 * A393592 A393593 A393594

Keyword

base,nonn

Author

Robert G. Wilson v, Feb 22 2026

Status

approved