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A174909 Triangle T(n,i) whose n-th row gives the number of numbers in any prime(n)# consecutive numbers whose smallest prime factor is prime(n-i+1).

%I A174909 #18 Oct 30 2017 23:20:58
%S A174909 1,1,3,2,5,15,8,14,35,105,48,88,154,385,1155,480,624,1144,2002,5005,
%T A174909 15015,5760,8160,10608,19448,34034,85085,255255,92160,109440,155040,
%U A174909 201552,369512,646646,1616615,4849845,1658880,2119680,2517120,3565920
%N A174909 Triangle T(n,i) whose n-th row gives the number of numbers in any prime(n)# consecutive numbers whose smallest prime factor is prime(n-i+1).
%C A174909 Here prime(n)# denotes the product of the first n primes. Row n begins with A005867(n-1). The other n-1 terms in row n are prime(n) times the previous row. The sum of the terms in row n is cototient(prime(n)#), which is A053144(n), and which equals prime(n)#-A005867(n). This sequence is a generalization of a comment in A005867 by Dennis Martin.
%e A174909 For n=3, we have prime(n)=5 and any range of 2*3*5=30 consecutive numbers has 2 numbers whose smallest prime factor is 5, 5 numbers whose smallest prime factor is 3, and 15 numbers whose smallest prime factor is 2.
%e A174909 From _Bob Selcoe_, Oct 12 2017: (Start)
%e A174909 Triangle starts:
%e A174909 n/i  1   2   3    4    5     6
%e A174909 1    1
%e A174909 2    1   3
%e A174909 3    2   5   15
%e A174909 4    8  14   35  105
%e A174909 5   48  88  154  385  1155
%e A174909 6  480 624 1144 2002  5005 15015
%e A174909 (End)
%t A174909 t={{1}}; q=2; Do[p=Prime[n]; t=AppendTo[t, Join[{(q-1)*t[[ -1,1]]}, p*t[[ -1]]]]; q=p, {n,2,9}]; Flatten[t]
%t A174909 (* Second program: *)
%t A174909 Block[{nn = 8, s}, s = Array[FactorInteger[#][[1, 1]] &, Product[Prime@i, {i, nn}]]; Table[With[{P = Product[Prime@ k, {k, n}]}, Count[Take[s, P], _?(# == Prime[n - i + 1] &)]], {n, nn}, {i, n}]] (* _Michael De Vlieger_, Oct 14 2017 *)
%Y A174909 Cf. A002110, A005867 (first column), A020639, A053144, A070826 (main diagonal).
%Y A174909 Cf. A293558 (transpose).
%K A174909 nonn,tabl
%O A174909 1,3
%A A174909 _T. D. Noe_, Apr 01 2010