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

Author

Rémy Sigrist, Nov 21 2018

Keywords

• nonn
• base

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

Crossrefs

See A321882 for the additive variant.

Cf. A061909, A131577.

Programs

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 *)

PARI

a(n) = for (b=2, oo, my (d=if(n==0, [0], digits(n,b))); if (vecmax(d)^2
				

Formula

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

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

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