A325790 -id:A325790 - cp's OEIS frontend

A188431 The number of n-full sets, F(n).

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 10, 13, 14, 17, 20, 25, 28, 34, 40, 46, 54, 62, 69, 80, 90, 102, 115, 131, 144, 167, 186, 213, 239, 273, 304, 349, 388, 441, 495, 563, 625, 710, 790, 890, 990, 1114, 1232, 1387, 1530, 1713, 1894, 2119, 2330, 2605, 2866, 3192, 3512, 3910, 4289, 4774, 5237, 5809, 6377, 7068, 7739

Offset: 0

Madjid Mirzavaziri, Mar 31 2011

nonn

Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. Then F(n) denotes the number of n-full sets.
Also the number of distinct and complete partitions of n, by definition, which are counted by A000009 and A126796. - George Beck, Nov 06 2017
An integer partition of n is complete (see also A325781) if every number from 0 to n is the sum of some submultiset of the parts. The Heinz numbers of these partitions are given by A325986. - Gus Wiseman, May 31 2019

			a(26) = 10, because there are 10 26-full sets: {1,2,4,5,6,8}, {1,2,3,5,7,8}, {1,2,3,5,6,9}, {1,2,3,4,7,9}, {1,2,3,4,6,10}, {1,2,3,4,5,11}, {1,2,4,8,11}, {1,2,4,7,12}, {1,2,4,6,13}, {1,2,3,7,13}.
G.f.: 1 = 1/(1+x) + 1*x/((1+x)*(1+x^2)) + 0*x^2/((1+x)*(1+x^2)*(1+x^3)) + 1*x^3/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...+ a(n)*x^n / Product_{k=1..n+1} (1+x^k) +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..1000 from Reinhard Zumkeller)
Mohammad Saleh Dinparvar, Python program
L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.
Arash Sal Moslehian, JavaScript p5 program for three different algorithms

Cf. A188429, A188430, A126796.

Cf. A002033, A103295, A108917, A276024, A325763, A325765, A325781, A325782, A325788, A325986, A325790, A325791.

import Data.MemoCombinators (memo2, integral, Memo)
a188431 n = a188431_list !! (n-1)
a188431_list = map
   (\x -> sum [fMemo x i | i <- [a188429 x .. a188430 x]]) [1..] where
   fMemo = memo2 integral integral f
   f _ 1 = 1
   f m i = sum [fMemo (m - i) j |
                j <- [a188429 (m - i) .. min (a188430 (m - i)) (i - 1)]]
-- Reinhard Zumkeller, Aug 06 2015

sums:= proc(s) local i, m;
          m:= max(s[]);
         `if`(m<1, {}, {m, seq([i, i+m][], i=sums(s minus {m}))})
       end:
a:= proc(n) local b;
      b:= proc(i,s) local si;
            if i=1 then `if`(sums(s)={$1..n}, 1, 0)
          else si:= s union {i};
               b(i-1, s)+ `if`(max(sums(si)[])>n, 0, b(i-1, si))
            fi
          end; b(n, {1})
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2n or i>n-i+1, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 20 2017

Sums[s_] := Sums[s] = With[{m = Max[s]}, If[m < 1, {}, Union @ Flatten @ Join[{m}, Table[{i, i + m}, {i, Sums[s ~Complement~ {m}]}]]]];
a[n_] := Module[{b}, b[i_, s_] := b[i, s] = Module[{si}, If[i == 1, If[Sums[s] == Range[n], 1, 0], si = s ~Union~ {i}; b[i-1, s] + If[Max[ Sums[si]] > n, 0, b[i - 1, si]]]]; b[n, {1}]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 80}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Union[Total/@Union[Subsets[#]]]==Range[0,n]&]],{n,30}] (* Gus Wiseman, May 31 2019 *)

/* As coefficients in g.f. */
{a(n)=local(A=[1]); for(i=1, n+1, A=concat(A,0); A[#A]=polcoeff(1 - sum(m=1,#A,A[m]*x^m/prod(k=1, m, 1+x^k +x*O(x^#A) )), #A) ); A[n+1]}
for(n=0, 50, print1(a(n),", ")) /* Paul D. Hanna, Mar 06 2012 */

F(n) = Sum_(i=L(n) .. U(n), F(n,i)), where F(n,i) = Sum_(j=L(n-i) .. min(U(n-i),i-1), F(n-i,j) ) and L(n), U(n) are defined in A188429 and A188430, respectively.
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1+x^k), with a(0)=1. - Paul D. Hanna, Mar 08 2012
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(3/4), where c = 0.03316508... - Vaclav Kotesovec, Oct 21 2019

More terms from Alois P. Heinz, Apr 03 2011

a(0)=1 prepended by Alois P. Heinz, May 20 2017

A326020 Number of complete subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 15, 27, 50, 95, 185, 365, 724, 1441, 2873, 5735, 11458, 22902, 45789, 91561, 183102, 366180, 732331, 1464626, 2929209, 5858367, 11716674, 23433277, 46866473, 93732852, 187465596, 374931067, 749861989, 1499723808, 2999447418

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,3,4}
                                      {1,2,3,5}    {1,2,3,5}
                                      {1,2,4,5}    {1,2,3,6}
                                      {1,2,3,4,5}  {1,2,4,5}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,4,5,6}
                                                   {1,2,3,4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..300
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326021, A326022, A326036.

Table[Length[Select[Subsets[Range[n]],Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,0,10}]

a(17)-a(34) from Charlie Neder, Jun 05 2019

A325788 Number of complete strict necklace compositions of n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 4, 4, 4, 0, 20, 6, 16, 12, 10, 0, 84, 40, 74, 42, 66, 38, 22, 254, 238, 188, 356, 242, 272, 150, 148, 1140, 1058, 1208, 1546, 1288

Offset: 1

Gus Wiseman, May 22 2019

A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part (counted by A032153). A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.

			The a(1) = 1 through a(16) = 6 complete strict necklace compositions (empty columns not shown):
  (1)  (12)  (123)  (124)  (1234)  (1253)  (1245)  (1264)  (12345)  (12634)
             (132)  (142)  (1324)  (1325)  (1326)  (1327)  (12354)  (13624)
                           (1423)  (1352)  (1542)  (1462)  (12435)  (14263)
                           (1432)  (1523)  (1623)  (1723)  (12453)  (14326)
                                                           (12543)  (14362)
                                                           (13254)  (16234)
                                                           (13425)
                                                           (13452)
                                                           (13524)
                                                           (13542)
                                                           (14235)
                                                           (14253)
                                                           (14325)
                                                           (14523)
                                                           (14532)
                                                           (15234)
                                                           (15243)
                                                           (15324)
                                                           (15342)
                                                           (15432)

Cf. A000740, A002033, A008965, A032153, A103295, A126796, A188431, A325684, A325785, A325786, A325787, A325790, A325791.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,30}]

A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.

Original entry on oeis.org

1, 1, 1, 2, 4, 20, 82, 252, 1074, 7912, 39552, 152680, 776094, 5550310, 30026848, 108376910

Offset: 0

Gus Wiseman, May 23 2019

A necklace permutation is a permutation that is either empty or whose first part is the minimum. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. The only circular subsequence of maximum length is the sequence itself, not any rotation of it. For example, the circular subsequences of (1,3,2) are: (), (1), (2), (3), (1,3), (2,1), (3,2), (1,3,2).

			The a(1) = 1 through a(5) = 20 permutations:
  (1)  (1,2)  (1,2,3)  (1,2,3,4)  (1,2,3,4,5)
              (1,3,2)  (1,3,2,4)  (1,2,3,5,4)
                       (1,4,2,3)  (1,2,4,3,5)
                       (1,4,3,2)  (1,2,4,5,3)
                                  (1,2,5,4,3)
                                  (1,3,2,5,4)
                                  (1,3,4,2,5)
                                  (1,3,4,5,2)
                                  (1,3,5,2,4)
                                  (1,3,5,4,2)
                                  (1,4,2,3,5)
                                  (1,4,2,5,3)
                                  (1,4,3,2,5)
                                  (1,4,5,2,3)
                                  (1,4,5,3,2)
                                  (1,5,2,3,4)
                                  (1,5,2,4,3)
                                  (1,5,3,2,4)
                                  (1,5,3,4,2)
                                  (1,5,4,3,2)

Cf. A002033, A008965, A032153, A103295, A126796, A304793, A325684, A325781, A325786, A325788, A325789, A325790.

subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Permutations[Range[n]],#=={}||First[#]==1&&Range[n*(n+1)/2]==Union[Total/@subalt[#]]&]],{n,0,5}]

a(11)-a(15) from Bert Dobbelaere, Nov 01 2020

A325986 Heinz numbers of complete strict integer partitions.

Original entry on oeis.org

1, 2, 6, 30, 42, 210, 330, 390, 462, 510, 546, 714, 798, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7854, 8778, 8970, 9282, 9570, 9690, 10230, 10374, 10626, 11310, 11730, 12090, 12210, 12558, 13398, 13566, 14322, 14430

Offset: 1

Gus Wiseman, May 30 2019

nonn

Strict partitions are counted by A000009, while complete partitions are counted by A126796.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition of n is complete (A126796, A325781) if every number from 0 to n is the sum of some submultiset of the parts.
The enumeration of these partitions by sum is given by A188431.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     42: {1,2,4}
    210: {1,2,3,4}
    330: {1,2,3,5}
    390: {1,2,3,6}
    462: {1,2,4,5}
    510: {1,2,3,7}
    546: {1,2,4,6}
    714: {1,2,4,7}
    798: {1,2,4,8}
   2310: {1,2,3,4,5}
   2730: {1,2,3,4,6}
   3570: {1,2,3,4,7}
   3990: {1,2,3,4,8}
   4290: {1,2,3,5,6}
   4830: {1,2,3,4,9}
   5610: {1,2,3,5,7}

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A304793.

Cf. A325780, A325781, A325782, A325788, A325790, A325988.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Select[Range[1000],SquareFreeQ[#]&&Union[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

Intersection of A005117 (strict partitions) and A325781 (complete partitions).

A326021 Number of complete subsets of {1..n} with maximum n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 23, 45, 90, 180, 359, 717, 1432, 2862, 5723, 11444, 22887, 45772, 91541, 183078, 366151, 732295, 1464583, 2929158, 5858307, 11716603, 23433196, 46866379, 93732744, 187465471, 374930922, 749861819, 1499723610

Offset: 1

Gus Wiseman, Jun 04 2019

nonn

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

			The a(1) = 1 through a(7) = 12 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,4}    {1,2,3,5}    {1,2,3,6}      {1,2,3,7}
                       {1,2,3,4}  {1,2,4,5}    {1,2,4,6}      {1,2,4,7}
                                  {1,2,3,4,5}  {1,2,3,4,6}    {1,2,3,4,7}
                                               {1,2,3,5,6}    {1,2,3,5,7}
                                               {1,2,4,5,6}    {1,2,3,6,7}
                                               {1,2,3,4,5,6}  {1,2,4,5,7}
                                                              {1,2,4,6,7}
                                                              {1,2,3,4,5,7}
                                                              {1,2,3,4,6,7}
                                                              {1,2,3,5,6,7}
                                                              {1,2,4,5,6,7}
                                                              {1,2,3,4,5,6,7}

Charlie Neder, Table of n, a(n) for n = 1..300
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326022, A326036.

Table[Length[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,10}]

a(18)-a(34) from Charlie Neder, Jun 05 2019

A326022 Number of minimal complete subsets of {1..n} with maximum n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 4, 8, 8, 8, 10, 14, 25, 40, 49, 62

Offset: 1

Gus Wiseman, Jun 04 2019

A set of positive integers summing to m is complete if every nonnegative integer up to m is the sum of some subset. For example, (1,2,3,6,13) is a complete set because we have:
= (empty sum)
= 1
= 2
= 3
= 1 + 3
= 2 + 3
= 6
= 6 + 1
= 6 + 2
= 6 + 3
= 1 + 3 + 6
= 2 + 3 + 6
= 1 + 2 + 3 + 6
and the remaining numbers 13-25 are obtained by adding 13 to each of these.

			The a(3) = 1 through a(9) = 8 subsets:
  {1,2,3}  {1,2,4}  {1,2,3,5}  {1,2,3,6}  {1,2,3,7}  {1,2,4,8}    {1,2,3,4,9}
                    {1,2,4,5}  {1,2,4,6}  {1,2,4,7}  {1,2,3,5,8}  {1,2,3,5,9}
                                                     {1,2,3,6,8}  {1,2,3,6,9}
                                                     {1,2,3,7,8}  {1,2,3,7,9}
                                                                  {1,2,4,5,9}
                                                                  {1,2,4,6,9}
                                                                  {1,2,4,7,9}
                                                                  {1,2,4,8,9}

Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326021, A326036.

fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]]],{n,10}]

cp's OEIS Frontend

A188431 The number of n-full sets, F(n).

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A326020 Number of complete subsets of {1..n}.

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A325788 Number of complete strict necklace compositions of n.

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A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.

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A325986 Heinz numbers of complete strict integer partitions.

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A326021 Number of complete subsets of {1..n} with maximum n.

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A326022 Number of minimal complete subsets of {1..n} with maximum n.

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