cp's OEIS Frontend

A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

%I A336499 #6 Aug 06 2020 09:27:00
%S A336499 1,1,1,1,1,2,0,1,2,1,2,1,1,3,1,3,2,0,1,3,2,5,3,3,2,1,1,4,2,7,4,4,3,2,
%T A336499 0,1,4,2,7,4,5,7,7,6,3,2,0,1,4,2,8,8,9,10,11,11,7,8,5,2,0,1,4,3,11,8,
%U A336499 11,16,16,15,15,15,13,9,6,3,1,1,5,3,14,10,13,21,21,20,19,21,18,13,9,5,2,0
%N A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.
%C A336499 Row lengths are A022559(n) + 1.
%e A336499 Triangle begins:
%e A336499   1
%e A336499   1
%e A336499   1  1
%e A336499   1  2  0
%e A336499   1  2  1  2  1
%e A336499   1  3  1  3  2  0
%e A336499   1  3  2  5  3  3  2  1
%e A336499   1  4  2  7  4  4  3  2  0
%e A336499   1  4  2  7  4  5  7  7  6  3  2  0
%e A336499   1  4  2  8  8  9 10 11 11  7  8  5  2  0
%e A336499   1  4  3 11  8 11 16 16 15 15 15 13  9  6  3  1
%e A336499   1  5  3 14 10 13 21 21 20 19 21 18 13  9  5  2  0
%e A336499   1  5  3 14 10 14 25 23 27 24 30 28 28 25 20 16 11  5  2  0
%e A336499 Row n = 7 counts the following divisors:
%e A336499   1  2  4  8   16  48   144  720   {}
%e A336499      3  9  12  24  72   360  1008
%e A336499      5     18  40  80   504
%e A336499      7     20  56  112
%e A336499            28
%e A336499            45
%e A336499            63
%t A336499 Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]
%Y A336499 A000720 is column k = 1.
%Y A336499 A022559 gives row lengths minus one.
%Y A336499 A056172 appears to be column k = 2.
%Y A336499 A336414 gives row sums.
%Y A336499 A336420 is the version for superprimorials.
%Y A336499 A336498 is the version counting all divisors.
%Y A336499 A336865 is the generalization to non-factorials.
%Y A336499 A336866 lists indices of rows with a final 1.
%Y A336499 A336867 lists indices of rows with a final 0.
%Y A336499 A336868 gives the final terms in each row.
%Y A336499 A000110 counts divisors of superprimorials with distinct prime exponents.
%Y A336499 A008302 counts divisors of superprimorials by number of prime factors.
%Y A336499 A130091 lists numbers with distinct prime exponents.
%Y A336499 A181796 counts divisors with distinct prime exponents.
%Y A336499 A327498 gives the maximum divisor of n with distinct prime exponents.
%Y A336499 Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.
%Y A336499 Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.
%K A336499 nonn,tabf
%O A336499 0,6
%A A336499 _Gus Wiseman_, Aug 03 2020