A110797 -id:A110797 - cp's OEIS frontend

A083872 Triangle read by rows in which row n lists first appearance of m such that m divides n!.

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 20, 30, 40, 60, 120, 9, 16, 18, 36, 45, 48, 72, 80, 90, 144, 180, 240, 360, 720, 7, 14, 21, 28, 35, 42, 56, 63, 70, 84, 105, 112, 126, 140, 168, 210, 252, 280, 315, 336, 420, 504, 560, 630, 840, 1008, 1260, 1680, 2520, 5040, 32, 64

Offset: 1

Jon Perry, Jun 18 2003

Differs from A110797 starting at a(17)=9.
From Rémy Sigrist, Sep 17 2017: (Start)
Each number k > 0 appears exactly once in the triangle, on row A002034(k).
The n-th row of the triangle:
- contains A038024(n) terms,
- starts with A046021(n),
- ends with n! = A000142(n).
(End)

			1!:1
2!:1,2 -> 2 as 1 has already appeared
3!:1,2,3,6 -> 3,6
4!:1,2,3,4,6,8,12,24 -> 4,8,12,24

Rémy Sigrist, Rows n = 1..17, flattened

Cf. A000142, A002034, A038024, A046021.

lst = {1}; Do[lst = Join[lst, Complement[Divisors[n!], Divisors[(n - 1)!]]], {n, 2, 9}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)

Extended by Ray Chandler, Aug 23 2005

A123664 a(1) = 1, a(2) = 2; then all new products of subsets of pre-existing terms which include the most recent, then the first integer not present and so on.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 24, 48, 72, 144, 5, 10, 15, 20, 30, 40, 60, 80, 90, 120, 160, 180, 240, 320, 360, 480, 720, 960, 1080, 1440, 1920, 2160, 2880, 3840, 4320, 5760, 6480, 7680, 8640, 11520, 12960, 15360, 17280, 23040, 25920, 34560, 46080, 51840, 69120, 77760

Offset: 1

Joel B. Lewis, Nov 15 2006

nonn

Similar to A096113. However, each product must include the most recently added singleton. Thus after adding 4, the terms 18 and 36 are not added because they have no representation as a product of earlier terms, including 4. A110797 is similar, but only allows products of pairs (not of subsets).

			After a(1) = 1, a(2) = 2, all products are present, so we add the first integer not included, namely 3. Then we add all products of any subset of {1, 2, 3} which include 3 and are not already present, in this case just 6. Then we add the next integer not already present, 4. Then we add all products of any subset of {1, 2, 3, 6, 4} which include 4 and are not already present, 8 (=2*4), 12 (=3*4), 24 (=2*3*4=6*4), 48 (=2*6*4), 72 (=3*6*4) and 144 (=2*3*6*4). Then we add 5, the next integer not already present. And so on.

Cf. A096113.

M[2]={1,2} M[n_]:= Join[M[n-1], Complement[Union[M[n-1][[ -1]] * Exp[Map[Total, Log[Subsets[Delete[Delete[M[n-1],1],-1]]]]]], M[n-1]],{n}] M[6]

cp's OEIS Frontend

A083872 Triangle read by rows in which row n lists first appearance of m such that m divides n!.

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