A321882 -id:A321882 - cp's OEIS frontend

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

Rémy Sigrist, Nov 21 2018

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))
Index entries for sequences related to carryless arithmetic

See A321882 for the additive variant.

Cf. A061909, A131577.

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 *)

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

a(n) = 2 iff n belongs to A131577.
a(n * a(n)) <= a(n).
a(A061909(n)) <= 10 for any n > 0.

A362366 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) is the least base >= 2 where the sum n + k can be computed without carry.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 5, 3, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 3, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 5, 5, 4, 4, 5, 5, 2, 2, 2, 3, 2, 6, 4, 4, 4, 6, 2, 3, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 5, 5, 3, 2, 5, 5, 5, 2, 3, 5, 5, 2

Offset: 0

Rémy Sigrist, Apr 17 2023

			Array A(n, k) begins:
  n\k | 0  1  2  3  4  5  6  7  8  9  10  11  12
  ----+-----------------------------------------
    0 | 2  2  2  2  2  2  2  2  2  2   2   2   2
    1 | 2  3  2  3  2  4  2  3  2  3   2   5   2
    2 | 2  2  5  3  2  2  3  5  2  2   5   5   2
    3 | 2  3  3  3  2  3  5  6  2  3   3   3   2
    4 | 2  2  2  2  3  4  4  4  2  2   2   2   3
    5 | 2  4  2  3  4  4  4  5  2  3   2   5   3
    6 | 2  2  3  5  4  4  5  5  2  2   3   3   5
    7 | 2  3  5  6  4  5  5  5  2  3   3   5   5
    8 | 2  2  2  2  2  2  2  2  6  3   5   5   6
    9 | 2  3  2  3  2  3  2  3  3  3   3   3   3
   10 | 2  2  5  3  2  2  3  3  5  3   3   5   3
   11 | 2  5  5  3  2  5  3  5  5  3   5   5   3
   12 | 2  2  2  2  3  3  5  5  6  3   3   3   3

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Rémy Sigrist, Colored representation of the array for n, k <= 1024 (the color is function of A(n, k), black pixels denote 2's)

Cf. A321882, A362367.

A(n, k) = { for (b = 2, oo, if (sumdigits(n+k, b) == sumdigits(n, b) + sumdigits(k, b), return (b););); }

A(n, k) <= max(2, n + k + 1).
A(n, k) = A(k, n).
A(n, 0) = 2.
A(n, n) = A321882(n).

A321909 a(n) is the least base b > 1 in which the additive persistence of n is <= 1.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 5, 2, 3, 3, 5, 3, 6, 6, 5, 2, 4, 3, 6, 4, 4, 7, 7, 4, 5, 5, 3, 3, 7, 3, 5, 2, 4, 8, 5, 3, 6, 6, 6, 5, 8, 6, 6, 6, 6, 9, 9, 4, 6, 5, 5, 5, 7, 3, 5, 5, 7, 7, 7, 5, 10, 10, 7, 2, 4, 4, 8, 4, 4, 7, 7, 4, 6, 6, 5, 5, 7, 6, 6, 4, 3, 3, 8, 3, 6

Offset: 0

Rémy Sigrist, Nov 21 2018

Equivalently, a(n) is the least base b > 1 in which the sum of digits of n is < b.
The sequence is well defined as, for any n > 0, the additive persistence of n is 0 in base n + 1.
This sequence is unbounded.

			For n = 42:
- in base 2, 42 has additive persistence 3: "101010" -> "11" -> "10" -> "1",
- in base 3, 42 has additive persistence 2: "1120" -> "11" -> "2",
- in base 4, 42 has additive persistence 2: "222" -> "12" -> "3",
- in base 5, 42 has additive persistence 2: "132" -> "11" -> "2",
- in base 6, 42 has additive persistence 1: "110" -> "2",
- hence a(42) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..1000000 (where the color is function of the initial digit of n in base a(n))

See A321882 for a similar sequence.

Cf. A031286, A131577.

Array[Block[{b = 2}, While[Total@ IntegerDigits[#, b] >= b, b++]; b] &, 86, 0] (* Michael De Vlieger, Nov 25 2018 *)

a(n) = for (b=2, oo, if (sumdigits(n, b) < b, return (b)))

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

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

cp's OEIS Frontend

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

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A362366 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) is the least base >= 2 where the sum n + k can be computed without carry.

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A321909 a(n) is the least base b > 1 in which the additive persistence of n is <= 1.

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