cp's OEIS Frontend

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

%I A339195 #38 Jan 09 2025 13:21:36
%S A339195 1,2,3,6,5,10,15,30,7,14,21,35,42,70,105,210,11,22,33,55,66,77,110,
%T A339195 154,165,231,330,385,462,770,1155,2310,13,26,39,65,78,91,130,143,182,
%U A339195 195,273,286,390,429,455,546,715,858,910,1001,1365,1430,2002,2145,2730,3003,4290,5005,6006,10010,15015,30030
%N A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.
%C A339195 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).
%C A339195 A019565 in its triangle form, with each row's terms in increasing order. - _Peter Munn_, Feb 26 2021
%C A339195 From _David James Sycamore_, Jan 09 2025: (Start)
%C A339195 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.
%C A339195 Permutation of A005117, Squarefree version A379746. (End)
%H A339195 Alois P. Heinz, <a href="/A339195/b339195.txt">Rows n = 0..14, flattened</a>
%H A339195 Michael De Vlieger, <a href="/A339195/a339195.png">Plot p | a(n) at (x,y) = (n,pi(p))</a>, n = 0..2047, 12X vertical exaggeration.
%H A339195 Michael De Vlieger, <a href="/A339195/a339195_1.png">Fan style binary tree showing a(n)</a>, n = 0..2047, with a color function related to the order of a(n) in A019565.
%H A339195 Michael De Vlieger, <a href="/A339195/a339195_2.png">Fan style binary tree showing a(n)</a>, 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.
%F A339195 For n > 1, T(n,k) = prime(n) * A261144(n-1,k).
%F A339195 a(n) = A019565(A379770(n)). - _Michael De Vlieger_, Jan 08 2025
%e A339195 Triangle begins:
%e A339195    1
%e A339195    2
%e A339195    3   6
%e A339195    5  10  15  30
%e A339195    7  14  21  35  42  70  105  210
%p A339195 T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
%p A339195       x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
%p A339195     end:
%p A339195 seq(T(n), n=0..6);  # _Alois P. Heinz_, Jan 08 2025
%t A339195 Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]
%Y A339195 A011782 gives row lengths.
%Y A339195 A339360 gives row sums.
%Y A339195 A008578 (shifted) is column k = 1.
%Y A339195 A100484 is column k = 2.
%Y A339195 A001748 is column k = 3.
%Y A339195 A002110 is column k = 2^(n-1).
%Y A339195 A070826 is column k = 2^(n-1) - 1.
%Y A339195 A209862 takes prime indices to binary indices in these terms.
%Y A339195 A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
%Y A339195 A261144 divides the n-th row by prime(n), with row sums A054640.
%Y A339195 A339116 is the restriction to semiprimes, with row sums A339194.
%Y A339195 A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
%Y A339195 A006881 lists squarefree semiprimes.
%Y A339195 A072047 counts prime factors of squarefree numbers.
%Y A339195 A319246 is the sum of prime indices of the n-th squarefree number.
%Y A339195 A329631 lists prime indices of squarefree numbers, reversed: A319247.
%Y A339195 A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
%Y A339195 Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
%Y A339195 Cf. A379746.
%K A339195 nonn,tabf
%O A339195 0,2
%A A339195 _Gus Wiseman_, Dec 02 2020
%E A339195 Row n=0 (term 1) prepended by _Alois P. Heinz_, Jan 08 2025