OFFSET
1,3
COMMENTS
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.
EXAMPLE
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.
From Bob Selcoe, Oct 12 2017: (Start)
Triangle starts:
n/i 1 2 3 4 5 6
1 1
2 1 3
3 2 5 15
4 8 14 35 105
5 48 88 154 385 1155
6 480 624 1144 2002 5005 15015
(End)
MATHEMATICA
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]
(* Second program: *)
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
T. D. Noe, Apr 01 2010
STATUS
approved