A376248 - OEIS

%I #20 Oct 13 2024 11:21:30

%S 1,1,2,1,3,1,2,4,1,5,1,2,3,4,6,9,1,7,1,2,4,8,1,3,9,1,2,4,5,10,25,1,11,

%T 1,2,3,4,6,8,9,12,18,27,1,13,1,2,4,7,14,49,1,3,5,9,15,25,1,2,4,8,16,1,

%U 17,1,2,3,4,6,8,9,12,18,27,1,19,1,2,4,5,8,10,20,25,50,125

%N Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

%C Analogous to A162306 regarding m such that rad(m) | n, but instead of taking m <= n, we take m such that bigomega(m) <= bigomega(n).

%C 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).

%C For prime power n = p^k, k >= 0 (i.e., n in A000961), row p^k of this sequence is the same as row p^k of A027750 and A162306. Therefore, for prime p, row p of this sequence is the same as row p of A027750 and A162306: {1, p}.

%C For n in A024619, row n of this sequence does not match row n of A162306, since the former contains gpf(n)^bigomega(n) = A006530(n)^A001222(n), which is larger than n.

%H Michael De Vlieger, <a href="/A376248/b376248.txt">Table of n, a(n) for n = 1..17475</a> (rows n = 1..1000, flattened)

%H Michael De Vlieger, <a href="/A376248/a376248.png">Log log scatterplot of rows n = 1..2^16 of this sequence, flattened</a>, (3153752 terms).

%F Row n of this sequence is { m : rad(m) | n, bigomega(m) <= bigomega(n) }.

%F A376567(n) = binomial(bigomega(n) + omega(n)) = Length of row n, where omega = A001221.

%e Triangle begins:

%e    n    row n of this sequence:

%e   -------------------------------------------

%e    1:   1;

%e    2:   1,  2;

%e    3:   1,  3;

%e    4:   1,  2   4;

%e    5:   1,  5;

%e    6:   1,  2,  3,  4,  6,  9;

%e    7:   1,  7;

%e    8:   1,  2,  4,  8;

%e    9:   1,  3,  9;

%e   10:   1,  2,  4,  5, 10, 25;

%e   11:   1, 11;

%e   12:   1,  2,  3,  4,  6,  8, 9, 12, 18, 27;

%e         ...

%e Row n = 10 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 5^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m and the parenthetic 8 are in row 10 of A162306:

%e    1   2   4  (8)

%e    5  10

%e   25*

%e Row n = 12 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 3^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m are in row 12 of A162306:

%e    1   2   4   8

%e    3   6  12

%e    9  18*

%e   27*

%t Table[Clear[p]; MapIndexed[Set[p[First[#2]], #1] &, FactorInteger[n][[All, 1]]]; k = PrimeOmega[n]; w = PrimeNu[n]; Union@ Map[Times @@ MapIndexed[p[First[#2]]^#1 &, #] &, Select[Tuples[Range[0, k], w], Total[#] <= k &] ], {n, 120}]

%Y Cf. A000961, A001221, A001222, A006530, A007947, A010846, A024619, A027750, A162306, A376567.

%K nonn,tabf

%O 1,3

%A _Michael De Vlieger_, Oct 09 2024