A095026 Triangle read by rows, where T(n,k) is the number of occurrences of the digit k as least significant digit in the base-n multiplication table, and 0 <= k < n.

1, 3, 1, 5, 2, 2, 8, 2, 4, 2, 9, 4, 4, 4, 4, 15, 2, 6, 5, 6, 2, 13, 6, 6, 6, 6, 6, 6, 20, 4, 8, 4, 12, 4, 8, 4, 21, 6, 6, 12, 6, 6, 12, 6, 6, 27, 4, 12, 4, 12, 9, 12, 4, 12, 4, 21, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 40, 4, 8, 10, 16, 4, 20, 4, 16, 10, 8, 4, 25, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset

1,2

Comments

Assumes a suitable continuation of the representation of digits in bases 11, 12 (9,A,B,..).

From Natalia L. Skirrow, Feb 10 2026: (Start)

Number of elements of order n in R/Z X Z/kZ.

In a sense, A162728(n) = T(n,2^oo) (the limit of T(n,2^i) as i->oo). (End)

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).

David Book, The Multiplying Digits Problem.

Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, arXiv:1201.3139 [math.NT], 2012.

Natalia L. Skirrow, Notes on A095026

Formula

Sum_{k=0..n-1} T(n,k) = n^2.

From Natalia L. Skirrow, Feb 10 2026: (Start)

T(n,k) = Sum_{j=0..n-1} gcd(n,j) * e^(2*Pi*i * k*j/n).

Sum_{k=0..n-1} T(n,k) * e^(2*Pi*i * k/n) = n.

T(n,k) = Sum_{g | gcd(n,k)} g*phi(n/g), with phi = A000010.

T(n,k) = Sum_{d|n} d * mu(n/d) * tau(gcd(d,k)), with mu = A008683 and tau = A000005.

Dirichlet g.f. of kth column: zeta(s-1)/zeta(s) * Prod_{primes p|k} (1-p^((1-s)*val_p(k))) / (1-p^(1-s)).

Sum_{m=0..n} T(m,k) ~ n^2/(2*zeta(2)) * Product_{prime p|k} (1 - 1/p^(val_p(k)+1)) / (1-1/p). (End)

Example

Table of T(n,k):
n\k 0  1  2  3  4  5  6  7  8  9 10 11
--+-----------------------------------
 1| 1  .  .  .  .  .  .  .  .  .  .  .
 2| 3  1  .  .  .  .  .  .  .  .  .  .
 3| 5  2  2  .  .  .  .  .  .  .  .  .
 4| 8  2  4  2  .  .  .  .  .  .  .  .
 5| 9  4  4  4  4  .  .  .  .  .  .  .
 6|15  2  6  5  6  2  .  .  .  .  .  .
 7|13  6  6  6  6  6  6  .  .  .  .  .
 8|20  4  8  4 12  4  8  4  .  .  .  .
 9|21  6  6 12  6  6 12  6  6  .  .  .
10|27  4 12  4 12  9 12  4 12  4  .  .
11|21 10 10 10 10 10 10 10 10 10 10  .
12|40  4  8 10 16  4 20  4 16 10  8  4
a(2)=T(2,0)=3 because 3 of the 4 possible combinations of last digits in the product of two binary numbers produce 0 as last digit of the result. a(3)=T(2,1)=1 because only ...1 * ...1 gives a result with last digit=1.
T(10,k)={27,4,12,4,12,9,12,4,12,4} gives the probability in percent (n^2=100) to get {0,1,2,...,9} as last decimal digit in the decimal representation of the product of two arbitrary integers.

Mathematica

A095026[n_, k_] := DivisorSum[GCD[n, k], #*EulerPhi[n/#] &];
Table[A095026[n, k], {n, 13}, {k, 0, n-1}] (* Paolo Xausa, Feb 16 2026 *)

Crossrefs

Columns: A018804 (k=0), A000010 (k=1), A345082 (k=2).

Row sums: A000290.

Cf. A162728.

Sequence in context: A233940 A134033 A185051 * A184997 A385594 A094367

Adjacent sequences: A095023 A095024 A095025 * A095027 A095028 A095029

Keyword

nonn,tabl,base

Author

Hugo Pfoertner, Jun 02 2004

Extensions

More terms from David Wasserman, Jun 03 2004

Offset in k axis changed from 1 to 0 by Natalia L. Skirrow, Feb 10 2026

Status

approved