A378183 -id:A378183 - cp's OEIS frontend

A378180 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) < bigomega(n), where rad = A007947 and bigomega = A001222.

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 5, 1, 1, 2, 3, 4, 6, 9, 1, 1, 2, 7, 1, 3, 5, 1, 2, 4, 8, 1, 1, 2, 3, 4, 6, 9, 1, 1, 2, 4, 5, 10, 25, 1, 3, 7, 1, 2, 11, 1, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 5, 1, 2, 13, 1, 3, 9, 1, 2, 4, 7, 14, 49, 1, 1, 2, 3, 4, 5, 6, 9, 10, 15, 25

Offset: 2

Michael De Vlieger, Nov 19 2024

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).
Row n contains numbers m such that rad(m) | n, where the number of prime factors of m with repetition is less than that of n.
Row 1 of this sequence is {}, hence offset of this sequence is set to 2.
For n = p^k (in A246655), row n contains p^j, j = 0..k-1.
For prime p, row p = {1}.
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, and since row n of A162306 contains n itself.

			Select rows n, showing nondivisors k parenthetically (i.e., k not in row n of A027750), and numbers k > n in brackets (i.e., k neither in row n of A162306 nor in row n of A027750):
   n    row n of this sequence:
  -------------------------------------------
   2:   1;
   3:   1;
   4:   1, 2;
   6:   1, 2, 3;
   8:   1, 2, 4;
   9:   1, 3;
  10:   1, 2, 5;
  12:   1, 2, 3,  4,   6,  (9);
  18:   1, 2, 3, (4),  6,   9;
  20:   1, 2, 4,  5,  10, [25];
  24:   1, 2, 3,  4,   6,   8, (9), 12, (18), [27];
  28:   1, 2, 4,  7,  14, [49];
  30:   1, 2, 3, (4),  5,   6, (9), 10,  15,  (25);
  36:   1, 2, 3,  4,   6,   8,  9,  12,  18,  (27);

Michael De Vlieger, Table of n, a(n) for n = 2..10325

Cf. A000961, A001221, A001222, A007947, A010846, A024619, A027750, A162306, A376567, A376248, A377070, A378181, A378183.

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}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) < bigomega(n) } = S \ T, where S is row n of A376248, and T is row n of A377070.
A378181(n) = binomial(bigomega(n) + omega(n) - 1, omega(n)) = Length of row n, where omega = A001221.
A378183(n) = rad(n)^binomial(omega(n) + bigomega(n) - 1, bigomega(n)-2) = A377073(n)/A377379(n) = product of row n.

cp's OEIS Frontend

A378180 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) < bigomega(n), where rad = A007947 and bigomega = A001222.

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