A011782 -id:A011782 - cp's OEIS frontend

A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

0, 0, 0, 1, 3, 9, 21, 49, 106, 226, 470, 968, 1971, 3995, 8057, 16208, 32537, 65239, 130687, 261654, 523661, 1047784, 2096150, 4193049, 8387033, 16775258, 33551996, 67105854, 134214010, 268430891, 536865308, 1073734982, 2147475299, 4294957153, 8589922282

Offset: 0

Alford Arnold, Aug 29 2000

Previous name was: Counts members of A056808 by number of factors.
A056808 relates to least prime signatures (cf. A025487)
a(n) is also the number of compositions of n that are not partitions of n. - Omar E. Pol, Jan 31 2009, Oct 14 2013
a(n) is the number of compositions of n into positive parts containing pattern [1,2]. - Bob Selcoe, Jul 08 2014

			A011782 begins     1 1 2 4 8 16 32 64 128 256 ...;
A000041 begins     1 1 2 3 5  7 11 15  22  30 ...;
so sequence begins 0 0 0 1 3  9 21 49 106 226 ... .
For n = 3 the factorizations are 8=2*2*2, 12=2*2*3, 18=2*3*3 and 30=2*3*5.
a(5) = 9: {[1,1,1,2], [1,1,2,1], [1,1,3], [1,2,1,1], [1,2,2], [1,3,1], [1,4], [2,1,2], [2,3]}. - _Bob Selcoe_, Jul 08 2014

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A025487, A056808, A117989.
The version for patterns is A002051.
(1,2)-avoiding compositions are just partitions A000041.
The (1,1)-matching version is A261982.
The version for prime indices is A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns matched by compositions are counted by A335456.
Cf. A011782, A056986, A106356, A144300, A238279, A269134, A333755.

a:= n-> ceil(2^(n-1))-combinat[numbpart](n):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 30 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!GreaterEqual@@#&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)
a[n_] := If[n == 0, 0, 2^(n-1) - PartitionsP[n]];
a /@ Range[0, 37] (* Jean-François Alcover, May 23 2021 *)

a(n) = A011782(n) - A000041(n).
a(n) = 2*a(n-1) + A117989(n-1). - Bob Selcoe, Apr 11 2014
G.f.: (1 - x) / (1 - 2*x) - Product_{k>=1} 1 / (1 - x^k). - Ilya Gutkovskiy, Jan 30 2020

More terms from James Sellers, Aug 31 2000

New name from Joerg Arndt, Sep 02 2013

A158780 a(2n) = A131577(n), a(2n+1) = A011782(n).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152, 4194304

Offset: 0

Paul Curtz, Mar 26 2009

This construction combines the 2 basic sequences which equal their first differences in the same way as A138635 does for sequences which equal their 3rd differences and A137171 does for sequences which equal their fourth differences.
Essentially the same as A016116, A060546, and A131572. - R. J. Mathar, Apr 08 2009
Dropping a(0), this is the inverse binomial transform of A024537. - R. J. Mathar, Apr 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

[0,1] cat [2^Floor((n-2)/2): n in [2..50]]; // G. C. Greubel, Apr 19 2023

Table[(2^Floor[n/2] +Boole[n==1] -Boole[n==0])/2, {n,0,50}] (* or *) LinearRecurrence[{0,2}, {0,1,1,1}, 51] (* G. C. Greubel, Apr 19 2023 *)

a(n)=if(n>3,([0,1; 2,0]^n*[1;1])[1,1]/2,n>0) \\ Charles R Greathouse IV, Oct 18 2022

def A158780(n): return (2^(n//2) + int(n==1) - int(n==0))/2
[A158780(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

a(2n) + a(2n+1) = A000079(n).
G.f.: x*(1+x-x^2)/(1-2*x^2). - R. J. Mathar, Apr 08 2009
a(n) = (1/2)*(2^floor(n/2) + [n=1] - [n=0]). - G. C. Greubel, Apr 19 2023
E.g.f.: (2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) + 2*x - 2)/4. - Stefano Spezia, May 13 2023

Edited by R. J. Mathar, Apr 08 2009

A349452 Dirichlet inverse of A011782, 2^(n-1).

Original entry on oeis.org

1, -2, -4, -4, -16, -16, -64, -104, -240, -448, -1024, -1904, -4096, -7936, -16256, -32272, -65536, -129888, -262144, -522176, -1048064, -2093056, -4194304, -8379520, -16776960, -33538048, -67106880, -134184704, -268435456, -536801024, -1073741824, -2147352224, -4294959104, -8589672448, -17179867136, -34359197184

Offset: 1

Antti Karttunen, Nov 22 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A011782.

Cf. also A349449, A349450, A349451, A349453 and A349563, A349565, A349567, A349569.

a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * 2^(n/# - 1) &, # < n &]; Array[a, 36] (* Amiram Eldar, Nov 22 2021 *)

A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A011782(n/d) * a(d).

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} 2^(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022

A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89

Offset: 0

Omar E. Pol, Mar 30 2015

Partial sums give A256264.
First differs from A160552 at a(27).
Appears to be a canonical sequence partially related to the cellular automata of A139250, A147562, A162795, A169707, A255366, A256250. See also A256264 and A256260.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0;
1;
1,3;
1,3,5,7;
1,3,5,7,5,11,17,15;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31,5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
...
Right border gives A000225.
Apart from the initial 0 the row sums give A000302.
Rows converge to A256258.
.
Illustration of initial terms in the fourth quadrant of the square grid:
---------------------------------------------------------------------------
n   a(n)                 Compact diagram
---------------------------------------------------------------------------
0    0     _
1    1    |_|_ _
2    1      |_| |
3    3      |_ _|_ _ _ _
4    1          |_| | | |
5    3          |_ _| | |
6    5          |_ _ _| |
7    7          |_ _ _ _|_ _ _ _ _ _ _ _
8    1                  |_| | | |_ _  | |
9    3                  |_ _| | |_  | | |
10   5                  |_ _ _| | | | | |
11   7                  |_ _ _ _| | | | |
12   5                  | | |_ _ _| | | |
13  11                  | |_ _ _ _ _| | |
14  17                  |_ _ _ _ _ _ _| |
15  15                  |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
16   1                                  |_| | | |_ _  | |_ _ _ _ _ _  | |
17   3                                  |_ _| | |_  | | |_ _ _ _ _  | | |
18   5                                  |_ _ _| | | | | |_ _ _ _  | | | |
19   7                                  |_ _ _ _| | | | |_ _ _  | | | | |
20   5                                  | | |_ _ _| | | |_ _  | | | | | |
21  11                                  | |_ _ _ _ _| | |_  | | | | | | |
22  17                                  |_ _ _ _ _ _ _| | | | | | | | | |
23  15                                  |_ _ _ _ _ _ _ _| | | | | | | | |
24   5                                  | | | | | | |_ _ _| | | | | | | |
25  11                                  | | | | | |_ _ _ _ _| | | | | | |
26  17                                  | | | | |_ _ _ _ _ _ _| | | | | |
27  23                                  | | | |_ _ _ _ _ _ _ _ _| | | | |
28  29                                  | | |_ _ _ _ _ _ _ _ _ _ _| | | |
29  35                                  | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
30  41                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
31  31                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
A256264(n) gives the total number of cells after n-th stage.

Ivan Neretin, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000225, A000302, A011782, A038573, A006257, A016969, A139251, A160552, A256250, A256258, A256260, A256261, A256264, A256265.

Flatten@Join[{0}, NestList[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 6]] (* Ivan Neretin, Feb 14 2017 *)

Terms a(95) to a(98) fixed by Ivan Neretin, Feb 14 2017

A324939 Triangle T(n,k) read by rows in which n-th row lists in increasing order all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i); n>=0, 1<=k<=A011782(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 54, 60, 90, 150, 210, 32, 48, 72, 108, 120, 162, 180, 270, 300, 420, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 216, 240, 324, 360, 486, 540, 600, 810, 840, 900, 1260, 1350, 1500, 1890, 2100, 2250, 2940, 3150, 3750, 4410, 4620, 5250, 6930, 7350, 10290, 11550, 16170, 25410, 30030

Offset: 0

Alois P. Heinz, Sep 04 2019

All terms sorted give A055932.

All terms first sorted by number of factors give A057335.

			Triangle T(n,k) begins:
   1;
   2;
   4,  6;
   8, 12, 18,  30;
  16, 24, 36,  54,  60,  90, 150, 210;
  32, 48, 72, 108, 120, 162, 180, 270, 300, 420, 450, 630, 750, 1050, 1470, 2310;
  ...

Alois P. Heinz, Rows n = 0..16, flattened
Wikipedia, Gödel's encoding

Column k=1 gives A000079.
Last elements of rows give A002110.
Row sums give A325054.
Row lengths give A011782.
Cf. A000040, A055932, A057335, A087443, A215366.

b:= n-> `if`(n=0, [[]], [seq(map(x-> [j, x[]], b(n-j))[], j=1..n)]):
T:= n-> sort(map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n)))[]:
seq(T(n), n=0..7);

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

Original entry on oeis.org

1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 127, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137

Offset: 1

Omar E. Pol, Apr 04 2015

Row sums give A002001.
The sum of all terms of first n rows gives A000302(n-1).
The rows of triangle A256263 converge to this sequence.
Rows converge to A016969.
First 11 terms agree with A151548.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
5,7;
5,11,17,15;
5,11,17,23,29,35,41,31;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,131,137,143,149,155,161,167,173,179,185,127;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1    1      |_| | | |_ _  | |_ _ _ _ _ _  | |
2    3      |_ _| | |_  | | |_ _ _ _ _  | | |
3    5      |_ _ _| | | | | |_ _ _ _  | | | |
4    7      |_ _ _ _| | | | |_ _ _  | | | | |
5    5      | | |_ _ _| | | |_ _  | | | | | |
6   11      | |_ _ _ _ _| | |_  | | | | | | |
7   17      |_ _ _ _ _ _ _| | | | | | | | | |
8   15      |_ _ _ _ _ _ _ _| | | | | | | | |
9    5      | | | | | | |_ _ _| | | | | | | |
10  11      | | | | | |_ _ _ _ _| | | | | | |
11  17      | | | | |_ _ _ _ _ _ _| | | | | |
12  23      | | | |_ _ _ _ _ _ _ _ _| | | | |
13  29      | | |_ _ _ _ _ _ _ _ _ _ _| | | |
14  35      | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
15  41      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
16  31      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
It appears that A241717 can be represented by a similar diagram.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A000225, A000302, A002001, A011782, A016969, A141548, A241717, A256260, A256261, A256263, A256264.

Nest[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 7] (* Ivan Neretin, Feb 14 2017 *)

A332977 Triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m satisfying Omega(m) + pi(gpf(m)) - [m<>1] = n; n>=0, 1<=k<=A011782(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 18, 25, 27, 11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125, 13, 22, 28, 33, 40, 42, 48, 55, 60, 63, 64, 70, 72, 77, 90, 98, 100, 105, 108, 121, 135, 147, 150, 162, 175, 225, 243, 245, 250, 343, 375, 625

Offset: 0

Alois P. Heinz, Mar 04 2020

Integer m > 0 is listed in row n if the index of the largest prime factor of m (or 0 for empty prime factor set) plus the cardinality of the other prime factors of m (counted with multiplicity) equals n.
Row n+k-1 contains prime(n)^k (for all n, k >= 1).
The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1, 2, 3, 4, 5, 6, 10, ... and inverse permutation A332990.
This is a variant with sorted rows of A005940 (offset differs) or A163511.

			Triangle T(n,k) begins:
   1;
   2;
   3,  4;
   5,  6,  8,  9;
   7, 10, 12, 15, 16, 18, 25, 27;
  11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125;
  ...

Alois P. Heinz, Rows n = 0..16, flattened
Wikipedia, Iverson bracket
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization

Columns k=1-2 give: A008578(n+1), A100484(n-1) for n>1.
Last elements of rows give A332979.
Row sums give A252737.
Product of row elements give A252738.
Row lengths give A011782.
Cf. A000027, A000040, A000720 (pi), A001222 (Omega), A006530 (GPF), A060576 ([n<>1]), A061395 (pi(gpf(n))), A215366, A332990.
Cf. A005940, A163511.

b:= proc(n, i) option remember; `if`(n=0, [1], sort([seq(map(x-> x*
      ithprime(j), b(n-`if`(i=0, j, 1), j))[], j=1..`if`(i=0, n, i))]))
    end:
T:= n-> b(n, 0)[]:
seq(T(n), n=0..7);

b[n_, i_] := b[n, i] = If[n == 0, {1}, Sort[Flatten[Table[#*
    Prime[j]& /@ b[n-If[i == 0, j, 1], j], {j, 1, If[i == 0, n, i]}]]]];
T[n_] := b[n, 0];
T /@ Range[0, 7] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)

A349563 Dirichlet convolution of right-shifted Catalan numbers with A349452 (Dirichlet inverse of A011782, 2^(n-1)).

Original entry on oeis.org

1, -1, -2, -1, -2, 18, 68, 311, 1182, 4370, 15772, 56754, 203916, 734636, 2658096, 9661591, 35292134, 129511602, 477376556, 1766730706, 6563071700, 24464139348, 91478369336, 343051112482, 1289887370140, 4861912443284, 18367285959072, 69533415236716, 263747683314904, 1002241674463968, 3814985428350480, 14544633872450487

Offset: 1

Antti Karttunen, Nov 22 2021

sign

Dirichlet convolution with A034729 gives A034731.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000108, A011782, A349452, A349564 (Dirichlet inverse).

Cf. also A034729, A034731, A349565, A349567, A349569.

s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * 2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, CatalanNumber[# - 1] * s[n/#] &]; Array[a, 32] (* Amiram Eldar, Nov 22 2021 *)

A000108(n) = (binomial(2*n, n)/(n+1));
A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349563(n) = sumdiv(n,d,A000108(d-1)*A349452(n/d));

a(n) = Sum_{d|n} A000108(d-1) * A349452(n/d).

A349565 Dirichlet convolution of Fibonacci numbers with A349452 (Dirichlet inverse of A011782, 2^(n-1)).

Original entry on oeis.org

1, -1, -2, -3, -11, -16, -51, -93, -214, -419, -935, -1812, -3863, -7649, -15698, -31443, -63939, -127676, -257963, -516037, -1037298, -2076547, -4165647, -8335716, -16702015, -33421217, -66911078, -133875827, -267921227, -535987784, -1072395555, -2145208557, -4291436930, -8584038291, -17170640199, -34344407256

Offset: 1

Antti Karttunen, Nov 22 2021

sign

Dirichlet convolution of this sequence with A034738 produces A034748.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000045, A011782, A349452, A349566 (Dirichlet inverse).

Cf. also A034738, A034748, A349563, A349567, A349569.

s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * 2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, Fibonacci[#] * s[n/#] &]; Array[a, 36] (* Amiram Eldar, Nov 22 2021 *)

A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349565(n) = sumdiv(n,d,fibonacci(d)*A349452(n/d));

a(n) = Sum_{d|n} A000045(d) * A349452(n/d).

A349566 Dirichlet convolution of A011782 (2^(n-1)) with A349451 (Dirichlet inverse of Fibonacci numbers).

Original entry on oeis.org

1, 1, 2, 4, 11, 20, 51, 100, 218, 441, 935, 1862, 3863, 7751, 15742, 31648, 63939, 128180, 257963, 516974, 1037502, 2078417, 4165647, 8339900, 16702136, 33428943, 66911942, 133891584, 267921227, 536021340, 1072395555, 2145272320, 4291440670, 8584166169, 17170641321, 34344672290, 68695318919, 137399603159, 274814652766

Offset: 1

Antti Karttunen, Nov 22 2021

nonn

Dirichlet convolution of this sequence with A034748 produces A034738.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000045, A011782, A349451, A349565 (Dirichlet inverse).

s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * Fibonacci[n/#] &, # < n &]; a[n_] := DivisorSum[n, 2^(# - 1) * s[n/#] &]; Array[a, 40] (* Amiram Eldar, Nov 22 2021 *)

memoA349451 = Map();
A349451(n) = if(1==n,1,my(v); if(mapisdefined(memoA349451,n,&v), v, v = -sumdiv(n,d,if(dA349451(d),0)); mapput(memoA349451,n,v); (v)));
A349566(n) = sumdiv(n,d,(2^(d-1)) * A349451(n/d));

a(n) = Sum_{d|n} 2^(d-1) * A349451(n/d).

cp's OEIS Frontend

A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A158780 a(2n) = A131577(n), a(2n+1) = A011782(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A349452 Dirichlet inverse of A011782, 2^(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A324939 Triangle T(n,k) read by rows in which n-th row lists in increasing order all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i); n>=0, 1<=k<=A011782(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A332977 Triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m satisfying Omega(m) + pi(gpf(m)) - [m<>1] = n; n>=0, 1<=k<=A011782(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica