A076302 Triangle T(n,k) = number of k-smooth divisors of n, read by rows.

1, 1, 2, 1, 1, 2, 1, 3, 3, 3, 1, 1, 1, 1, 2, 1, 2, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 2, 1, 4, 4, 4, 4, 4, 4, 4, 1, 1, 3, 3, 3, 3, 3, 3, 3, 1, 2, 2, 2, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Author

Reinhard Zumkeller, Mar 14 2003

Keywords

• nonn
• tabl

Examples

			Triangle begins:
                   1
                 1   2
               1   1   2
             1   3   3   3
           1   1   1   1   2
         1   2   4   4   4   4
       1   1   1   1   1   1   2
     1   4   4   4   4   4   4   4
   1   1   3   3   3   3   3   3   3
		

Links

• Eric Weisstein's World of Mathematics, Divisor Function.
• Eric Weisstein's World of Mathematics, Smooth Number.

Crossrefs

Cf. A000005, A001511, A072078, A355583.

Programs

Mathematica

T[n_, k_] := Times@@(IntegerExponent[n, #]+1& /@ Select[Range[2, k], PrimeQ]);
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2021 *)

Formula

T(n,n) = A000005(n);

T(n,2) = A001511(n) for n>1.

T(n,3) = A072078(n) for n>2.

T(n,5) = A355583(n) for n>4.

Limit_{m->oo} (1/m) * Sum_{n=k..m} T(n,k) = 1/Product_{p prime <= k} (1 - 1/p). - Amiram Eldar, Apr 17 2025