A319478 a(n) is the least base b > 1 such that the product n * n can be computed without carry by long multiplication.

2, 2, 2, 3, 2, 4, 5, 5, 2, 3, 3, 5, 3, 6, 7, 7, 2, 4, 9, 9, 4, 4, 10, 11, 11, 5, 5, 3, 3, 13, 3, 3, 2, 11, 11, 5, 3, 3, 6, 13, 13, 13, 6, 6, 6, 15, 15, 15, 6, 7, 5, 5, 17, 17, 18, 5, 7, 7, 7, 19, 19, 20, 20, 7, 2, 4, 8, 22, 4, 4, 17, 23, 6, 6, 8, 24, 19, 19, 6

Offset

0,1

Comments

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.

Links

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))

Index entries for sequences related to carryless arithmetic

Formula

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

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

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

Mathematica

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 *)

Prog

(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))))

Crossrefs

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

Keyword

nonn,base

Author

Rémy Sigrist, Nov 21 2018

Status

approved