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A376479 Array read by antidiagonals: T(n,k) is the index of prime(k)^n in the numbers with n prime factors, counted with multiplicity.

%I A376479 #6 Sep 27 2024 23:07:51
%S A376479 1,2,1,3,3,1,4,9,5,1,5,17,30,8,1,6,40,82,90,14,1,7,56,328,385,269,23,
%T A376479 1,8,90,551,2556,1688,788,39,1,9,114,1243,5138,18452,7089,2249,64,1,
%U A376479 10,164,1763,15590,44329,126096,28893,6340,103,1,11,253,3112,24646,179313,361249,827901,115180,17526
%N A376479 Array read by antidiagonals: T(n,k) is the index of prime(k)^n in the numbers with n prime factors, counted with multiplicity.
%C A376479 T(n,k) is the number of numbers j with n prime factors, counted with multiplicity, such that j <= prime(k)^n.
%e A376479 T(2,3) = 9 because the third prime is 5 and 5^2 = 25 is the 9th semiprime.
%p A376479 T:= Matrix(12,12):
%p A376479 with(priqueue);
%p A376479 for m from 1 to 12 do
%p A376479   initialize(pq);
%p A376479   insert([-2^m, [2$m]],pq);
%p A376479   k:= 0:
%p A376479   for count from 1 do
%p A376479     t:= extract(pq);
%p A376479     w:= t[2];
%p A376479     if nops(convert(w,set))=1 then
%p A376479       k:= k+1;
%p A376479       T[m,k]:= count;
%p A376479       if m+k = 13 then break fi;
%p A376479     fi;
%p A376479     p:= nextprime(w[-1]);
%p A376479     for i from m to 1 by -1 while w[i] = w[m] do
%p A376479       insert([t[1]*(p/w[-1])^(m+1-i),[op(w[1..i-1]),p$(m+1-i)]],pq);
%p A376479 od od od:
%p A376479 seq(seq(T[i,s-i],i=1..s-1),s=2..13)
%Y A376479 Cf. A001222, A078843 (second column), A078844 (third column), A078845 (fourth column), A078846 (fifth column), A128301 (second row), A128302 (third row), A128304 (fourth row).
%K A376479 nonn,tabl
%O A376479 1,2
%A A376479 _Robert Israel_, Sep 24 2024