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 A377070 #14 Oct 29 2024 10:27:42
%S A377070 1,2,3,4,5,4,6,9,7,8,9,4,10,25,11,8,12,18,27,13,4,14,49,9,15,25,16,17,
%T A377070 8,12,18,27,19,8,20,50,125,9,21,49,4,22,121,23,16,24,36,54,81,25,4,26,
%U A377070 169,27,8,28,98,343,29,8,12,18,20,27,30,45,50,75,125,31
%N A377070 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) = bigomega(n), where rad = A007947 and bigomega = A001222.
%C A377070 Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k = bigomega(n), that is, numbers m such that rad(m) | n and m has the same number of prime factors with repetition than does n.
%H A377070 Michael De Vlieger, <a href="/A377070/b377070.txt">Table of n, a(n) for n = 1..12021</a>, (rows n = 1..1500, flattened)
%H A377070 Michael De Vlieger, <a href="/A377070/a377070.png">Diagrams of select a(n)</a> illustrating rank omega(n)-1 simplexes formed by the arrangement of terms in row n by prime power decomposition.
%H A377070 Michael De Vlieger, <a href="/A377070/a377070_1.png">Log log scatterplot of a(n)</a>, rows n = 1..65536 (1278755 terms).
%F A377070 Row n of this sequence is { m : rad(m) | n, bigomega(m) = bigomega(n) }.
%F A377070 For prime p, row p of this sequence is {p}, generally for prime power p^k, row p^k of this sequence is {p^k}.
%F A377070 For n in A024619, row n of this sequence has more than 1 term.
%F A377070 A377071(n) = length of row n of this sequence.
%e A377070 Triangle begins:
%e A377070 n row n of this sequence:
%e A377070 -------------------------------------------
%e A377070 1: {1}
%e A377070 2: {2}
%e A377070 3: {3}
%e A377070 4: {4}
%e A377070 5: {5}
%e A377070 6: {4, 6, 9}
%e A377070 7: {7}
%e A377070 8: {8}
%e A377070 9: {9}
%e A377070 10: {4, 10, 25}
%e A377070 ... (Select rows appear below)
%e A377070 12: {8, 12, 18, 27}
%e A377070 14: {4, 14, 49}
%e A377070 15: {9, 15, 25}
%e A377070 18: {8, 12, 18, 27}
%e A377070 20: {8, 20, 50, 125}
%e A377070 24: {16, 24, 36, 54, 81}
%e A377070 30: {8, 12, 18, 20, 27, 30, 45, 50, 75, 125}
%e A377070 42: {8, 12, 18, 27, 28, 42, 63, 98, 147, 343}
%e A377070 60: {16, 24, 36, 40, 54, 60, 81, 90, 100, 135, 150, 225, 250, 375, 625}.
%e A377070 .
%e A377070 Diagrams of the rank omega(n)-1 simplexes created by row n of this sequence for select n, ordering k in row n by prime decomposition. The number k = n appears in brackets:
%e A377070 Rank 3:
%e A377070 n = 30: n = 42:
%e A377070 8 8
%e A377070 / \ / \
%e A377070 12 -- 20 12 -- 28
%e A377070 / \ / \ / \ / \
%e A377070 18 --[30]-- 50 18 --[42]-- 98
%e A377070 / \ / \ / \ / \ / \ / \
%e A377070 27 -- 45 -- 75 -- 125 27 -- 63 --147 -- 343
%e A377070 .
%e A377070 n = 60: 16
%e A377070 / \
%e A377070 24 -- 40
%e A377070 / \ / \
%e A377070 36 --[60]-- 50
%e A377070 / \ / \ / \
%e A377070 54 -- 90 -- 75 --125
%e A377070 / \ / \ / \ / \
%e A377070 81 --150 --135 --375 --625
%e A377070 .
%e A377070 Rank 4:
%e A377070 n = 210:
%e A377070 16
%e A377070 40
%e A377070 24 56
%e A377070 100
%e A377070 60 140
%e A377070 36 84 196
%e A377070 250
%e A377070 150 350
%e A377070 90 [210] 490
%e A377070 54 126 294 686
%e A377070 625
%e A377070 375 875
%e A377070 225 525 1225
%e A377070 135 315 735 1715
%e A377070 81 189 441 1029 2401
%t A377070 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A377070 Table[k = PrimeOmega[n]; Select[Range[n^PrimeNu[n]], Divisible[n, rad[#]] && PrimeOmega[#] == k &], {n, 30}]
%Y A377070 Cf. A001221, A001222, A007947, A024619, A376248, A377071.
%K A377070 nonn,easy,tabf
%O A377070 1,2
%A A377070 _Michael De Vlieger_, Oct 25 2024