login
A346730
Irregular triangle read by rows: T(n,k) is the number of n-bit numbers with k divisors.
5
1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 4, 0, 1, 0, 5, 1, 4, 1, 3, 0, 2, 0, 7, 1, 11, 0, 6, 0, 4, 1, 1, 0, 1, 0, 13, 1, 20, 1, 9, 1, 9, 1, 2, 0, 6, 0, 0, 0, 1, 0, 23, 1, 39, 0, 15, 0, 25, 2, 3, 0, 12, 0, 1, 1, 3, 0, 2, 0, 1, 0, 43, 2, 74, 0, 27, 0, 48, 3, 6, 0, 25, 0, 2, 2, 13, 0, 5, 0, 2, 0, 0, 0, 4
OFFSET
1,3
COMMENTS
The number of terms in row n is A346729(n).
LINKS
FORMULA
T(n,2) = A162145(n) for n > 1.
T(n,n) = A300509(n).
Sum_{k>=1} k * T(n,k) = A153876(n). - Alois P. Heinz, Aug 01 2021
EXAMPLE
There are four 3-bit numbers: 4 = 100_2, 5 = 101_2 = 5, 6 = 110_2, 7 = 111_2. 5 and 7 are both prime, so each has 2 divisors; 4 = 2^2 has 3 divisors (1, 2, and 4), and 6 = 2*3 has 4 divisors (1, 2, 3, and 6). Thus, among the 3-bit numbers, the counts of those having 1, 2, 3, and 4 divisors are 0, 2, 1, and 1, respectively, so the 3rd row of the table is 0, 2, 1, 1.
Triangle begins:
1;
0, 2;
0, 2, 1, 1;
0, 2, 1, 4, 0, 1;
0, 5, 1, 4, 1, 3, 0, 2;
0, 7, 1, 11, 0, 6, 0, 4, 1, 1, 0, 1;
0, 13, 1, 20, 1, 9, 1, 9, 1, 2, 0, 6, 0, 0, 0, 1;
MAPLE
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add
(x^numtheory[tau](i), i=2^(n-1)..2^n-1)):
seq(T(n), n=1..10); # Alois P. Heinz, Jul 31 2021
MATHEMATICA
Map[BinCounts[#, {0, Max[#] + 1, 1}] &, Table[DivisorSigma[0, 2^n + k], {n, 0, 8}, {k, 0, 2^n - 1}]] // Flatten (* Michael De Vlieger, Aug 29 2021 *)
CROSSREFS
Row sums give A011782.
Columns k=1-2 give: A063524, A162145 (for n>=2).
Sequence in context: A091182 A001822 A351380 * A249603 A231714 A366445
KEYWORD
nonn,base,tabf
AUTHOR
Jon E. Schoenfield, Jul 30 2021
STATUS
approved