A130493 -id:A130493 - cp's OEIS frontend

A200748 Smallest number requiring n terms to be expressed as a sum of factorials.

0, 1, 3, 5, 11, 17, 23, 47, 71, 95, 119, 239, 359, 479, 599, 719, 1439, 2159, 2879, 3599, 4319, 5039, 10079, 15119, 20159, 25199, 30239, 35279, 40319, 80639, 120959, 161279, 201599, 241919, 282239, 322559, 362879, 725759, 1088639, 1451519, 1814399, 2177279

Offset: 0

Franklin T. Adams-Watters, Nov 28 2011

nonn

Indices of record values in A034968.
Records for sum of digits when numbers are written in factorial base.
Numbers, n, whose factorial base representation digitally dominates every number less than or equal to n; we say n digitally dominates m if each digit of n in its factorial base representation is greater than or equal to the corresponding digit of m in its factorial base representation. - Joanne Beckford, Sep 01 2017

Michael De Vlieger, Table of n, a(n) for n = 0..11324 (corresponding to rows 1 <= n <= 150 of A051683, see Formula)

Cf. A051683, A034968.

Partial sums of A130493.

With[{b = MixedRadix[Reverse@ Range[2, 12]]}, Function[s, {0}~Join~Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]@ Array[Total@ IntegerDigits[#, b] &, 10^5]] (* or *)
Prepend[-1 + Rest@ Flatten[Table[n!*k, {n, 9}, {k, n}]], 0] (* Michael De Vlieger, Sep 03 2017, after Jean-François Alcover at A051683 *)

k=0;m=1;s=0;vector(45,n,s+=m!;if(k++==m,k=0;m++);s)

a(n) = A051683(n+1) - 1.

A130478 Triangle T(n,k) = n! / A130477(n,k).

Original entry on oeis.org

1, 2, 2, 6, 3, 2, 24, 8, 3, 2, 120, 30, 8, 3, 2, 720, 144, 30, 8, 3, 2, 5040, 840, 144, 30, 8, 3, 2, 40320, 5760, 840, 144, 30, 8, 3, 2, 362880, 45360, 5760, 840, 144, 30, 8, 3, 2, 3628800, 403200, 45360, 5760, 840, 144, 30, 8, 3, 2

Offset: 1

Gary W. Adamson, May 31 2007

Sums of reciprocals of rows is 1. - Henry Bottomley, Nov 05 2009

			First few rows of the triangle:
     1;
     2,   2;
     6,   3,   2;
    24,   8,   3,  2;
   120,  30,   8,  3, 2;
   720, 144,  30,  8, 3, 2;
  5040, 840, 144, 30, 8, 3, 2;
  ...
Row 4 = (24, 8, 3, 2), terms such that (24, 8, 3, 2) dot (1, 3, 8, 12) = (24, 24, 24, 24), where (1, 3, 8, 12) = row 4 of A130477 and (24, 24, 24, 24) = row 4 of A130493.
Row 5 = (120, 30, 8, 3, 2) = 5! + (4!+3!) + (3!+2!) + (2!+1!) + (1!+1).
Row 5 = 120 followed by the first reversed 4 terms of A001048; i.e., 120 followed by 30, 8, 3, 2.

Cf. A130493 (row sums), A001048, A130493, A130477.

T(n,k) = n! / A130477(n,k); such that by rows as vector terms, (n-th row of A130477) dot (n-th row of A130478) = n-th row of A130493 = n! repeated n times.
Triangle by rows = n! followed by the first (n-1) reversed terms of A001048: (2, 3, 8, 30, 144, 840, ...).
Left border = (1, 2, 6, 24, 120, ...); while all other columns = A001048: (2, 3, 8, 30, ...).
n-th row of the triangle = n terms of: (n!; (n-1)!+(n-2)!; (n-2)!+(n-3)!; ...; 1! + 0!).

Corrected and extended by Henry Bottomley, Nov 05 2009

A343041 a(0) = 0 and for any n > 0, a(n) = A343040(a(n-1), n).

Original entry on oeis.org

0, 1, 3, 3, 5, 5, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 23, 23, 23, 23, 23, 23, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71

Offset: 0

Rémy Sigrist, Apr 03 2021

This sequence has similarities with A087052.
If we remove duplicate terms, then we obtain A200748.
The value A200748(k) appears A130493(k) times for any k > 0.

			The first terms, in decimal and in factorial base, are:
  n   a(n)  fact(n)  fact(a(n))
  --  ----  -------  ----------
   0     0        0           0
   1     1        1           1
   2     3       10          11
   3     3       11          11
   4     5       20          21
   5     5       21          21
   6    11      100         121
   7    11      101         121
   8    11      110         121
   9    11      111         121
  10    11      120         121
  11    11      121         121
  12    17      200         221
  13    17      201         221
  14    17      210         221

Rémy Sigrist, Table of n, a(n) for n = 0..5039
Rémy Sigrist, PARI program for A343041
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to factorial base representation

Cf. A087052, A130493, A200748, A343040, A343045.

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cp's OEIS Frontend

A200748 Smallest number requiring n terms to be expressed as a sum of factorials.

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A130478 Triangle T(n,k) = n! / A130477(n,k).

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A343041 a(0) = 0 and for any n > 0, a(n) = A343040(a(n-1), n).

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