A378181 -id:A378181 - cp's OEIS frontend

A378632 Numbers that set records in A378181.

1, 4, 6, 12, 24, 48, 60, 96, 120, 240, 420, 480, 840, 1680, 3360, 6720, 13440, 18480, 26880, 36960, 53760, 73920, 147840, 295680, 480480, 591360, 960960, 1182720, 1921920, 3843840, 7687680, 15375360, 30750720, 61501440, 65345280, 123002880, 130690560, 246005760

Offset: 1

Michael De Vlieger, Dec 02 2024

Numbers n that set records for binomial(bigomega(n)+omega(n)-1, omega(n)), where bigomega = A001222 and omega = A001221.
a(n) is of the form 2^k * P(i), k >= 0, where primorial P = A002110.
Proper subset of A070175.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000079, A001221, A001222, A002110, A007947, A070175, A378181.

f[x_] := Block[{i, k, m, nn, p}, nn = Product[Prime[j], {j, x}]; Set[{k, i, p}, Range[0, 2]]; {1}~Join~Union@ Reap[Until[i > x, While[Set[m, 2^k*p] <= nn, Sow[m]; k++]; k = 0; i++; p *= Prime[i] ] ][[-1, 1]] ] (* generate A070175 *);
r = 0; Reap[Do[If[# > r, r = #; Sow[n]] &@ Binomial[#2 + #1 - 1, #1] & @@ {PrimeNu[n], PrimeOmega[n]}, {n, f[10]}] ][[-1, 1]]

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

Original entry on oeis.org

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

A378632 Numbers that set records in A378181.

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