This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381500 #17 Mar 07 2025 11:34:17
%S A381500 1,2,3,6,5,10,15,30,7,14,21,35,42,70,105,210,11,22,33,55,77,66,110,
%T A381500 165,154,231,385,330,462,770,1155,2310,13,26,39,65,91,143,78,130,195,
%U A381500 182,273,455,286,429,715,1001,390,546,910,1365,858,1430,2145,2002,3003
%N A381500 a(n) = A019565(A187769(n)).
%C A381500 The squarefree numbers, ordered first by largest prime factor (dividing the sequence into rows), then by number of prime factors, then lexicographically by their prime factors (written in descending order).
%C A381500 We index (a(n)) from offset 0, matching the choice for A019565 and similar sequences.
%H A381500 Michael De Vlieger, <a href="/A381500/b381500.txt">Table of n, a(n) for n = 0..16383</a>
%H A381500 Michael De Vlieger, <a href="/A381500/a381500.png">Log log scatterplot of a(n)</a>, n = 0..2^16-1.
%H A381500 Michael De Vlieger, <a href="/A381500/a381500_1.png">Plot p | a(n) at (x,y) = (n, pi(p))</a>, n = 0..2047, 12X vertical exaggeration.
%H A381500 Michael De Vlieger, <a href="/A381500/a381500_2.png">Fan style binary tree showing a(n)</a>, n = 0..2048, with a color function showing smallest and greatest terms in each row in green and red, respectively.
%F A381500 a(n) = A019565(A187769(n)).
%F A381500 As an irregular triangle T(n,k), where row 0 = {1}:
%F A381500 For n > 1, omega(T(n,1)) = 1, omega(T(n, 2^(n-1))) = n, thus row n is divided into n segments S such that with S, omega(T(n,k)) = m, where m = 1..n. (See A187769 for the lengths of segments associated with Pascal's triangle A007318.)
%F A381500 S(-1,-1) = (1).
%F A381500 For n >= 0:
%F A381500 S(n-1, n) = (); S(n, -1) = ();
%F A381500 for 0 <= m <= n, S(n,m) = ( A253550'(S(n-1, m)), A119416'(S(n-1, m-1)) ), where Axxx'((i_1, i_2, ..., i_j)) denotes Axxx(i_1), Axxx(i_2), ..., Axxx(i_j).
%F A381500 a(A163866(n)) = A098012(n).
%e A381500 Table begins:
%e A381500 Row 0: 1;
%e A381500 Row 1: 2;
%e A381500 Row 2: 3, 6;
%e A381500 Row 3: 5, 10, 15, 30;
%e A381500 Row 4: 7, 14, 21, 35, 42, 70, 105, 210;
%e A381500 Row 5: 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310;
%e A381500 ...
%e A381500 Table of a(n) for n = 0..31, demonstrating relationship of this sequence with s = A187769:
%e A381500 <-factors <-factors
%e A381500 n a(n) 2 3 5 7 s(n) | n a(n) 2 3 5 7 11 s(n)
%e A381500 -------------------------|----------------------------
%e A381500 0 1 . 0 | 16 11 . . . . x 16
%e A381500 1 2 x 1 | 17 22 x . . . x 17
%e A381500 2 3 . x 2 | 18 33 . x . . x 18
%e A381500 3 6 x x 3 | 19 55 . . x . x 20
%e A381500 4 5 . . x 4 | 20 77 . . . x x 24
%e A381500 5 10 x . x 5 | 21 66 x x . . x 19
%e A381500 6 15 . x x 6 | 22 110 x . x . x 21
%e A381500 7 30 x x x 7 | 23 165 . x x . x 22
%e A381500 8 7 . . . x 8 | 24 154 x . . x x 25
%e A381500 9 14 x . . x 9 | 25 231 . x . x x 26
%e A381500 10 21 . x . x 10 | 26 385 . . x x x 28
%e A381500 11 35 . . x x 12 | 27 330 x x x . x 23
%e A381500 12 42 x x . x 11 | 28 462 x x . x x 27
%e A381500 13 70 x . x x 13 | 29 770 x . x x x 29
%e A381500 14 105 . x x x 14 | 30 1155 . x x x x 30
%e A381500 15 210 x x x x 15 | 31 2310 x x x x x 31
%e A381500 -------------------------|----------------------------
%e A381500 1 2 4 8 s(n) | 1 2 4 8 16 s(n)
%e A381500 bits-> bits->
%t A381500 a187769 = {{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten; a019565[x_] := Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[x, 2]; Map[a019565, a187769]
%Y A381500 Cf. A019565, A119416, A163866, A187769, A253550, A294648, A344085.
%Y A381500 A002110, A098012 are subsequences.
%K A381500 nonn,base,easy
%O A381500 0,2
%A A381500 _Michael De Vlieger_, _Peter Munn_, and _Antti Karttunen_, Feb 27 2025