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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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 22 15:04 EDT 2021. Contains 347607 sequences. (Running on oeis4.)