0,1

Apparently, a(n) is also the least base b > 1 where the square of the digital sum of n equals the digital sum of the square of n.

The sequence is well defined as, for any n > 0, n * n can be computed without carry in base n^2 + 1.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..50000 (where the color is function of the initial digit of n in base a(n))

a(n) = 2 iff n belongs to A131577.

a(n * a(n)) <= a(n).

a(A061909(n)) <= 10 for any n > 0.

Array[Block[{b = 2}, While[AnyTrue[With[{d = IntegerDigits[#, b]}, Function[{s, t}, Total@ Map[PadLeft[#, t] &, s]] @@ {#, Max[Length /@ #]} &@ MapIndexed[Join[d #, ConstantArray[0, First@ #2 - 1]] &, Reverse@ d]], # >= b &], b++]; b] &, 79, 0] (* Michael De Vlieger, Nov 25 2018 *)

(PARI) a(n) = for (b=2, oo, my (d=if(n==0, [0], digits(n, b))); if (vecmax(d)^2<b, my (s=0, ok=1); forstep (i=#d, 1, -1, s \= b; my (t=d[i]*n); if (sumdigits(s+t, b)==sumdigits(s, b)+sumdigits(t, b), s += t, ok = 0; break)); if (ok, return (b))))

See A321882 for the additive variant.

Cf. A061909, A131577.

Sequence in context: A318058 A345901 A305434 * A341408 A104239 A283751

Adjacent sequences: A319475 A319476 A319477 * A319479 A319480 A319481

nonn,base

Rémy Sigrist, Nov 21 2018

approved