A364878 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.

1, 4, 2, 36, 12, 6, 900, 180, 60, 30, 44100, 6300, 1260, 420, 210, 5336100, 485100, 69300, 13860, 4620, 2310, 901800900, 69369300, 6306300, 900900, 180180, 60060, 30030, 260620460100, 15330615300, 1179278100, 107207100, 15315300, 3063060, 1021020, 510510

Offset: 0

Author

Jon E. Schoenfield, Aug 11 2023

Keywords

• nonn
• tabl

Comments

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.

From Michael De Vlieger, Aug 12 2023: (Start)

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

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

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

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

Examples

			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.
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.
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.
Table begins:
  n\k|         0         1        2       3       4      5      6
  ---+-----------------------------------------------------------
  0  |         1;
  1  |         4,        2;
  2  |        36,       12,       6;
  3  |       900,      180,      60,     30;
  4  |     44100,     6300,    1260,    420,    210;
  5  |   5336100,   485100,   69300,  13860,   4620,  2310;
  6  | 901800900, 69369300, 6306300, 900900, 180180, 60060, 30030;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0..150, flattened)

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

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

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

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