A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030

Offset

0,2

Comments

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021

From David James Sycamore, Jan 09 2025: (Start)

Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.

Permutation of A005117, Squarefree version A379746. (End)

Links

Alois P. Heinz, Rows n = 0..14, flattened

Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.

Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.

Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

Formula

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Example

Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Maple

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Mathematica

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]], {n, 5}]

Crossrefs

A011782 gives row lengths.

A339360 gives row sums.

A008578 (shifted) is column k = 1.

A100484 is column k = 2.

A001748 is column k = 3.

A002110 is column k = 2^(n-1).

A070826 is column k = 2^(n-1) - 1.

A209862 takes prime indices to binary indices in these terms.

A246867 groups squarefree numbers by Heinz weight, with row sums A147655.

A261144 divides the n-th row by prime(n), with row sums A054640.

A339116 is the restriction to semiprimes, with row sums A339194.

A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.

A006881 lists squarefree semiprimes.

A072047 counts prime factors of squarefree numbers.

A319246 is the sum of prime indices of the n-th squarefree number.

A329631 lists prime indices of squarefree numbers, reversed: A319247.

A338899/A270650/A270652 give the prime indices of squarefree semiprimes.

Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.

Cf. A379746.

Sequence in context: A077320 A379548 A381112 * A344085 A381500 A019565

Adjacent sequences: A339192 A339193 A339194 * A339196 A339197 A339198

Keyword

nonn,tabf

Author

Gus Wiseman, Dec 02 2020

Extensions

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

Status

approved