A076302 - OEIS

%I #11 Feb 16 2025 08:32:47

%S 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,

%T 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,

%U 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

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.

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

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

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

%e Triangle begins:

%e                    1

%e                  1   2

%e                1   1   2

%e              1   3   3   3

%e            1   1   1   1   2

%e          1   2   4   4   4   4

%e        1   1   1   1   1   1   2

%e      1   4   4   4   4   4   4   4

%e    1   1   3   3   3   3   3   3   3

%t T[n_, k_] := Times@@(IntegerExponent[n, #]+1& /@ Select[Range[2, k], PrimeQ]);

%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 15 2021 *)

%Y Cf. A000005, A001511, A072078.

%K nonn,tabl,changed

%O 1,3

%A _Reinhard Zumkeller_, Mar 14 2003