A301593 -id:A301593 - cp's OEIS frontend

A089359 Primes which can be partitioned into distinct factorials. 0! and 1! are not considered distinct.

2, 3, 7, 31, 127, 151, 727, 751, 5167, 5791, 5881, 40351, 40471, 41047, 41161, 45361, 45481, 362911, 363751, 368047, 368647, 368791, 403327, 403951, 408241, 408271, 408361, 409081, 3628927, 3629671, 3633991, 3634591, 3669241, 3669847, 3669961

Offset: 1

Amarnath Murthy, Nov 07 2003

nonn

			From _Seiichi Manyama_, Mar 24 2018: (Start)
n | a(n) |
--+------+------------------
1 |    2 | 2!
2 |    3 | 2! + 1!
3 |    7 | 3! + 1!
4 |   31 | 4! + 3! + 1!
5 |  127 | 5! + 3! + 1!
6 |  151 | 5! + 4! + 3! + 1! (End)

Alois P. Heinz, Table of n, a(n) for n = 1..16812 (first 1000 terms from Seiichi Manyama)

Cf. A059590, A088332, A300947, A301593.

from sympy import isprime
def facbase(k, f):
    return sum(f[i] for i, bi in enumerate(bin(k)[2:][::-1]) if bi == "1")
def auptoN(N): # terms up to N factorial-base digits; 20 generates b-file
    f = [factorial(i) for i in range(1, N+1)]
    return list(filter(isprime, (facbase(k, f) for k in range(2**N))))
print(auptoN(10)) # Michael S. Branicky, Oct 15 2022

More terms from Vladeta Jovovic, Nov 08 2003

A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

Original entry on oeis.org

3, 7, 8, 25, 26, 30, 121, 122, 126, 144, 721, 722, 726, 744, 840, 5041, 5042, 5046, 5064, 5160, 5760, 40321, 40322, 40326, 40344, 40440, 41040, 45360, 362881, 362882, 362886, 362904, 363000, 363600, 367920, 403200, 3628801, 3628802, 3628806, 3628824, 3628920, 3629520, 3633840, 3669120, 3991680

Offset: 1

Seiichi Manyama, Mar 23 2018

nonn

Numbers of the form i! + j! where i > j > 0. - Altug Alkan, Mar 23 2018

Primes in this sequence are A088332(n) for n > 1.

			    + |   1    2    6   24
  ----+--------------------
    1 |
    2 |   3;
    6 |   7,   8;
   24 |  25,  26,  30;
  120 | 121, 122, 126, 144;

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000142, A001048, A038507, A059590, A088332, A301593.

Union[Total/@Subsets[Range[10]!,{2}]] (* Harvey P. Dale, Aug 25 2020 *)

A301652 Triangle read by rows: row n gives the digits of n in factorial base in reversed order.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 2, 1, 2, 2, 0, 0, 3, 1, 0, 3, 0, 1, 3, 1, 1, 3, 0, 2, 3, 1, 2, 3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 0, 0, 1, 1

Offset: 0

Seiichi Manyama, Mar 25 2018

Row n gives exponents for successive primes 2, 3, 5, 7, 11, etc., in the prime factorization of A276076(n). - Antti Karttunen, Mar 11 2024

			   n | 1  2  6
  ---+---------
   0 | 0;
   1 | 1;
   2 | 0, 1;
   3 | 1, 1;
   4 | 0, 2;
   5 | 1, 2;
   6 | 0, 0, 1;
   7 | 1, 0, 1;
   8 | 0, 1, 1;
   9 | 1, 1, 1;
  10 | 0, 2, 1;
  11 | 1, 2, 1;
  12 | 0, 0, 2;
  13 | 1, 0, 2;
  14 | 0, 1, 2;
  15 | 1, 1, 2;
  16 | 0, 2, 2;
  17 | 1, 2, 2;
  18 | 0, 0, 3;
  19 | 1, 0, 3;

Seiichi Manyama, Rows n = 0..2000, flattened
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Triangle A108731 with rows reversed.

Cf. A007623, A034968 (row sums), A208575 (row products), A227153 (products of nonzero terms on row n), A276076, A301593.

row[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, AppendTo[s, r]; m++]; s]; row[0] = {0}; Array[row, 31, 0] // Flatten (* Amiram Eldar, Mar 11 2024 *)

terms=25; print([0]+[x for sublist in [[floor(n/factorial(i))%(i+1) for i in [k for k in [1..n] if factorial(k)<=n]] for n in [1..terms]] for x in sublist]) # Tom Edgar, Aug 15 2018

T(n,k) = floor(n/k!) mod k+1. - Tom Edgar, Aug 15 2018

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A089359 Primes which can be partitioned into distinct factorials. 0! and 1! are not considered distinct.

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A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

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A301652 Triangle read by rows: row n gives the digits of n in factorial base in reversed order.

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Mathematica

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