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

%I A076302 #16 Apr 17 2025 08:09:14
%S A076302 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 A076302 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 A076302 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 A076302 Triangle T(n,k) = number of k-smooth divisors of n, read by rows.
%H A076302 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.
%H A076302 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.
%F A076302 T(n,n) = A000005(n);
%F A076302 T(n,2) = A001511(n) for n>1.
%F A076302 T(n,3) = A072078(n) for n>2.
%F A076302 T(n,5) = A355583(n) for n>4.
%F A076302 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
%e A076302 Triangle begins:
%e A076302                    1
%e A076302                  1   2
%e A076302                1   1   2
%e A076302              1   3   3   3
%e A076302            1   1   1   1   2
%e A076302          1   2   4   4   4   4
%e A076302        1   1   1   1   1   1   2
%e A076302      1   4   4   4   4   4   4   4
%e A076302    1   1   3   3   3   3   3   3   3
%t A076302 T[n_, k_] := Times@@(IntegerExponent[n, #]+1& /@ Select[Range[2, k], PrimeQ]);
%t A076302 Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 15 2021 *)
%Y A076302 Cf. A000005, A001511, A072078, A355583.
%K A076302 nonn,tabl
%O A076302 1,3
%A A076302 _Reinhard Zumkeller_, Mar 14 2003