A364878 - OEIS

%I #21 Aug 13 2023 02:46:49

%S 1,4,2,36,12,6,900,180,60,30,44100,6300,1260,420,210,5336100,485100,

%T 69300,13860,4620,2310,901800900,69369300,6306300,900900,180180,60060,

%U 30030,260620460100,15330615300,1179278100,107207100,15315300,3063060,1021020,510510

%N Triangle read by rows: T(n,k), 0 <= k <= n, is the smallest number that has n distinct prime factors, k of which are unique.

%C If a number's prime factorization is written as the product of the number's prime factors without using exponents (e.g., 1260 = 2*2*3*3*5*7), a prime factor is unique (i.e., one of a kind) if it appears only once; prime factors that have one or more duplicates are not unique. E.g., the distinct prime factors of 1260 are 2, 3, 5, and 7, but the only unique prime factors of 1260 are 5 and 7.

%C From _Michael De Vlieger_, Aug 12 2023: (Start)

%C T(n,k) is in A025487 by definition.

%C The number 2 is the only prime term and 4 the only composite prime power term.

%C T(n,k), k < n, n > 2 is neither squarefree nor a prime power and is in A126706.

%C No prime power divisor p^e | T(n,k) is such that e > 2. (End)

%H Michael De Vlieger, <a href="/A364878/b364878.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150, flattened)

%F T(n,k) = Product_{j=1..n} prime(j)^m where m = 1 if j > n - k, m = 2 otherwise.

%F T(n,k) = A002110(n)*A002110(n-k).

%F T(n,n) = A002110(n).

%F T(n,0) = A002110(n)^2.

%e T(2,0) = 36: 36 = 2*2*3*3, so 36 has 2 distinct prime factors (2 and 3) but no unique prime factors (each prime factor has a duplicate), and 36 is the smallest number with this property.

%e T(2,2) = 6: 6 = 2*3, so 6 has 2 distinct prime factors (2 and 3), and each of those is a unique prime factor (having no duplicates), and 6 is the smallest number with this property.

%e T(3,2) = 60: 60 = 2*2*3*5, so 60 has 3 distinct prime factors (2, 3, and 5), but only 2 unique prime factors (3 and 5, since the factor 2 is duplicated), and 60 is the smallest number having this property.

%e Table begins:

%e   n\k|         0         1        2       3       4      5      6

%e   ---+-----------------------------------------------------------

%e   0  |         1;

%e   1  |         4,        2;

%e   2  |        36,       12,       6;

%e   3  |       900,      180,      60,     30;

%e   4  |     44100,     6300,    1260,    420,    210;

%e   5  |   5336100,   485100,   69300,  13860,   4620,  2310;

%e   6  | 901800900, 69369300, 6306300, 900900, 180180, 60060, 30030;

%e   ...

%t T[n_,k_]:=Module[{primes=Array[Prime,n],primeProducts},primeProducts=Table[If[j>n-k,primes[[j]],primes[[j]]^2],{j,1,n}];Times@@primeProducts];lst[rows_]:=Table[T[n,k],{n,0,rows},{k,0,n}]//Flatten;lst[7] (* _Robert P. P. McKone_, Aug 12 2023 *)

%Y Cf. A002110, A025487, A061742, A088860, A126706, A228593.

%K nonn,tabl

%O 0,2

%A _Jon E. Schoenfield_, Aug 11 2023