A064985 -id:A064985 - cp's OEIS frontend

A064986 Number of partitions of n into factorial parts (0! not allowed).

1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 12, 12, 15, 15, 18, 18, 22, 22, 26, 26, 30, 30, 36, 36, 42, 42, 48, 48, 56, 56, 64, 64, 72, 72, 82, 82, 92, 92, 102, 102, 114, 114, 126, 126, 138, 138, 153, 153, 168, 168, 183, 183, 201, 201, 219, 219, 237, 237, 258, 258, 279, 279

Offset: 0

Naohiro Nomoto, Oct 30 2001

a(2*n+1) = a(2*n) = A117930(n). [Reinhard Zumkeller, Dec 04 2011]

			a(3) = 2 because we can write 3 = 2!+1! = 1!+1!+1!.
a(10) = 9 because 10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 = 1 + 1 + 1 + 1 + 2 + 2 + 2 = 1 + 1 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 6 = 1 + 1 + 2 + 6 = 2 + 2 + 6.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..250 from Reinhard Zumkeller)
Youkow Homma, Jun Hwan Ryu and Benjamin Tong, Sequence non-squashing partitions, Slides from a talk, Jul 24 2014.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
Index entries for sequences related to factorial numbers

Cf. A000142, A064985, A115944, A197182.

Bisection gives A090632.

a064986 = p (tail a000142_list) where
   p _          0             = 1
   p fs'@(f:fs) m | m < f     = 0
                  | otherwise = p fs' (m - f) + p fs m
-- Reinhard Zumkeller, Dec 04 2011

b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i!>n, 0, b[n-i!, i]]];
c[n_] := Module[{i}, For[i = 1, i!<2n, i++]; b[2n, i]];
a[n_] := If[OddQ[n], c[(n-1)/2], c[n/2]];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz in A117930 *)
Table[Length@IntegerPartitions[n, All, Factorial[Range[6]]], {n, 0, 63}] (* Robert Price, Jun 04 2020 *)

N=66; x='x+O('x^N); m=1; while(N>=m!, m++); Vec(1/prod(k=1, m-1, 1-x^k!)) \\ Seiichi Manyama, Oct 13 2019

N=66; x='x+O('x^N); m=1; while(N>=m!, m++); Vec(1+sum(i=1, m-1, x^i!/prod(j=1, i, 1-x^j!))) \\ Seiichi Manyama, Oct 13 2019

G.f.: 1/Product_{i>=1} (1-x^(i!)).
G.f.: 1 + Sum_{n>0} x^(n!) / Product_{k=1..n} (1 - x^(k!)). - Seiichi Manyama, Oct 12 2019
G.f.: 1 + x/(1-x) + x^2/((1-x)*(1-x^2)) + x^6/((1-x)*(1-x^2)*(1-x^6)) + ... . - Seiichi Manyama, Oct 12 2019

More terms from Vladeta Jovovic and Don Reble, Nov 02 2001

A133041 Sum of n and partition number of n.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 17, 22, 30, 39, 52, 67, 89, 114, 149, 191, 247, 314, 403, 509, 647, 813, 1024, 1278, 1599, 1983, 2462, 3037, 3746, 4594, 5634, 6873, 8381, 10176, 12344, 14918, 18013, 21674, 26053, 31224, 37378, 44624, 53216

Offset: 0

Omar E. Pol, Oct 29 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A001365, A014688, A064985, A090631, A102379.
Cf. A000041 (the partition numbers).
Partial sums of A232697.

Table[n+PartitionsP[n],{n,0,50}] (* Harvey P. Dale, Jul 17 2014 *)

a(n)=numbpart(n)+n \\ Charles R Greathouse IV, Jan 06 2016

a(n) = n + partition(n) = n + A000041(n).

a(n) = A207779(A000041(n)), n >= 1. - Omar E. Pol, Jul 17 2014

A328325 Expansion of Product_{k>=0} 1/(1 - x^(k^k)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 113, 122, 131, 140, 150, 161, 172, 183, 195, 208, 221, 234, 248, 263, 278, 293, 309, 326, 343, 360, 378, 397, 416, 435, 455, 476, 497, 519, 542, 566, 590, 615, 641, 668, 695

Offset: 0

Seiichi Manyama, Oct 12 2019

nonn

Partial sums of A328301.

			G.f.: 1/(1-x) + x/(1-x)^2 + x^4/((1-x)^2*(1-x^4)) + x^27/((1-x)^2*(1-x^4)*(1-x^27)) + ... .

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000312, A064985, A086858, A328301.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(i^i))
    end:
a:= proc(n) option remember; `if`(n<2, n+1, a(n-1)+
      b(n, floor((t-> t/LambertW(t))(log(n)))))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 12 2019

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + With[{p = i^i}, If[p > n, 0, b[n-p, i]]]];
a[n_] := a[n] = If[n < 2, n+1, a[n-1] + b[n, Floor[PowerExpand[Log[n]/ ProductLog[Log[n]]]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 14 2022, after Alois P. Heinz *)

N=99; x='x+O('x^N); m=1; while(N>=m^m, m++); Vec(1/prod(k=0, m-1, 1-x^k^k))

a(n) = Sum_{k=0..n} A328301(k).

G.f.: 1/(1-x) + Sum_{n>0} x^(n^n) / Product_{k=0..n} (1 - x^(k^k)).

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A064986 Number of partitions of n into factorial parts (0! not allowed).

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A133041 Sum of n and partition number of n.

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A328325 Expansion of Product_{k>=0} 1/(1 - x^(k^k)).

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