A103295 -id:A103295 - cp's OEIS frontend

A325986 Heinz numbers of complete strict integer partitions.

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).

A325765 Number of integer partitions of n with a unique consecutive subsequence summing to every positive integer from 1 to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3

Offset: 0

Gus Wiseman, May 20 2019

After a(0) = 1, same as A032741(n + 1) (number of proper divisors of n + 1).

The Heinz numbers of these partitions are given by A325764.

			The a(1) = 1 through a(13) = 3 partitions:
  (1)  (11)  (21)   (1111)  (221)    (111111)  (2221)     (3311)
             (111)          (311)              (4111)     (11111111)
                            (11111)            (1111111)
.
  (22221)      (1111111111)  (33311)        (111111111111)  (2222221)
  (51111)                    (44111)                        (7111111)
  (111111111)                (222221)                       (1111111111111)
                             (611111)
                             (11111111111)

Cf. A000041, A002033, A103295, A103300, A143823, A169942, A325676, A325683, A325768, A325769, A325770.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]]&&UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,20}]

A325786 Number of complete necklace compositions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 7, 12, 19, 41, 71, 141, 255, 509, 924, 1882, 3395, 6838, 12715, 25233, 47049

Offset: 1

Gus Wiseman, May 22 2019

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. A circular subsequence is a sequence of consecutive terms where the first and last 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(8) = 19 necklace compositions:
  (1)  (11)  (12)   (112)   (113)    (123)     (124)      (1124)
             (111)  (1111)  (122)    (132)     (142)      (1133)
                            (1112)   (1113)    (1114)     (1142)
                            (11111)  (1122)    (1123)     (1214)
                                     (1212)    (1132)     (1223)
                                     (11112)   (1213)     (1322)
                                     (111111)  (1222)     (11114)
                                               (11113)    (11123)
                                               (11122)    (11132)
                                               (11212)    (11213)
                                               (111112)   (11222)
                                               (1111111)  (11312)
                                                          (12122)
                                                          (111113)
                                                          (111122)
                                                          (111212)
                                                          (112112)
                                                          (1111112)
                                                          (11111111)

Cf. A000740, A002033, A008965, A103295, A108917, A126796, A276024, A325549, A325682, A325781, A325788, A325789, 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/@IntegerPartitions[n],neckQ[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,15}]

A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 7, 3, 2, 1, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Dec 14 2021

We use the notation [n] = {0, 1, ..., n-1}. If S is a subset of [n] then we define the distset of S (set of distances of S) as {|x - y|: x, y in S}. We call a subset S of the natural numbers self-measuring if and only if S = distset(S).

			Triangle starts:
[ 0] [1]
[ 1] [1, 1]
[ 2] [1, 1,  1]
[ 3] [1, 1,  2, 1]
[ 4] [1, 1,  3, 1, 1]
[ 5] [1, 1,  4, 2, 1, 1]
[ 6] [1, 1,  5, 2, 1, 1, 1]
[ 7] [1, 1,  6, 3, 2, 1, 1, 1]
[ 8] [1, 1,  7, 3, 2, 1, 1, 1, 1]
[ 9] [1, 1,  8, 4, 2, 2, 1, 1, 1, 1]
[10] [1, 1,  9, 4, 3, 2, 1, 1, 1, 1, 1]
[11] [1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1]
[12] [1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1]
.
The first  column is 1,1,...  because {} = distset({}) and |{}| = 0.
The second column is 1,1,... because {0} = distset({0}) and |{0}| = 1.
The third  column is n-1  because {0, j} = distset({0, j}) and |{0, j}| = 2 for j = 1..n - 1.
The main diagonal is 1,1,... because [n] = distset([n]) and |[n]| = n (these are the complete rulers A103295).

Winston de Greef, Table of n, a(n) of the first 150 rows, flattened (n = 0..11324)
Peter Luschny, Illustrating self-measuring subsets of {0, 1, 2, 3}.

Cf. A350102 (row sums), A349976, A103295, A003988, A010766, A123706.

T := (n, k) -> ifelse(k < 2, 1, floor((n - 1) / (k - 1))):
seq(print(seq(T(n, k), k = 0..n)), n = 0..12);

distSet[s_] := Union[Map[Abs[Differences[#][[1]]] &, Union[Sort /@ Tuples[s, 2]]]]; T[n_, k_] := Count[Subsets[Range[0, n - 1]], ?((ds = distSet[#]) == # && Length[ds] == k &)]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Amiram Eldar, Dec 16 2021 *)

T(n, k) = if(k<=1, 1, (n - 1) \ (k - 1)) \\ Winston de Greef, Jan 31 2024

# generating and counting (slow)
def isSelfMeasuring(R):
    S, L = Set([]), len(R)
    R = Set([r - 1 for r in R])
    for i in range(L):
        for j in (0..i):
            S = S.union(Set([abs(R[i] - R[i - j])]))
    return R == S
def A349976row(n):
    counter = [0] * (n + 1)
    for S in Subsets(n):
        if isSelfMeasuring(S): counter[len(S)] += 1
    return counter
for n in range(10): print(A349976row(n))

T(n, k) = floor((n - 1) / (k - 1)) for k >= 2.

T(n, k) = 1 if k = 0 or k = 1 or n >= k >= floor((n + 1)/2).

A103296 Number of complete rulers with n segments.

Original entry on oeis.org

1, 1, 3, 10, 38, 175, 885, 5101, 32080, 219569, 1616882, 12747354, 106948772, 950494868

Offset: 0

Peter Luschny, Feb 28 2005

For definitions, references and links related to complete rulers see A103294.

a(10) > 1616740 (contributions from rows of A103294 up to 39). - Hugo Pfoertner, Dec 16 2021

			a(2)=3 counts the complete rulers with 2 segments, {[0,1,2],[0,1,3],[0,2,3]}.

Index entries for sequences related to perfect rulers.

Cf. A103301 (perfect rulers with n segments), A103299 (optimal rulers with n segments).

Cf. A103294, A103295 (complete rulers of length n).

! Link to FORTRAN program given in A103295.

a(n) = Sum_{i=n..A004137(n+1)} T(i, n) where T is the A103294 triangle.

a(9) from Hugo Pfoertner, Mar 17 2005
a(10)-a(11) from Fausto A. C. Cariboni, Mar 03 2022
a(12)-a(13) from Fausto A. C. Cariboni, Mar 08 2022

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}]

A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 54, 56, 64, 100, 128, 162, 176, 256, 392, 416, 486, 500, 512, 1024, 1088, 1458, 1936, 2048, 2432, 2500, 2744, 4096, 4374, 5408, 5888, 8192, 12500, 13122, 14848, 16384, 18496, 19208, 21296, 31744, 32768, 39366, 46208, 62500, 65536

Offset: 1

Gus Wiseman, May 20 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325765.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   176: {1,1,1,1,5}
   256: {1,1,1,1,1,1,1,1}
   392: {1,1,1,4,4}
   416: {1,1,1,1,1,6}
   486: {1,2,2,2,2,2}
   500: {1,1,3,3,3}
   512: {1,1,1,1,1,1,1,1,1}

Cf. A002033, A056239, A103295, A103300, A112798, A143823, A169942, A325676, A325685, A325763, A325765, A325769, A325770.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{_,s__,_}:>{s}]]&&Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#],{_,s__,_}:>Plus[s]]]&]

A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 10, 10, 24, 24, 43, 42, 88, 72, 136, 122, 242, 213, 392, 320, 630, 490, 916, 742, 1432, 1160, 1955, 1604, 2826, 2310, 3850, 2888, 5416, 4426, 7332, 5814, 10046, 7983, 12946, 10236, 17780, 14100, 22674, 17582, 30232, 23674, 37522, 29426, 49832

Offset: 0

Gus Wiseman, Jun 02 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

The contiguous case is A325676.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,1,1)  (4,1)        (4,2)
                                  (1,1,3)      (5,1)
                                  (1,2,2)      (1,1,4)
                                  (2,2,1)      (2,2,2)
                                  (3,1,1)      (4,1,1)
                                  (1,1,1,1,1)  (1,1,1,1,1,1)

These compositions are ranked by A334967.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702, while the strict case is counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A003022, A029931, A103295, A143823, A325680, A333224.

b:= proc(n, s) option remember; `if`(n=0, 1, add((h->
      `if`(nops(h)=nops(map(l-> add(i, i=l), h)),
       b(n-j, h), 0))({s[], map(l-> [l[], j], s)[]}), j=1..n))
    end:
a:= n-> b(n, {[]}):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 03 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15}]

a(18)-a(47) from Alois P. Heinz, Jun 03 2020

A376423 Nonnegative numbers m such that the run lengths in binary expansion of m, say (r_1, ..., r_k), correspond to a complete ruler: the sums r_i + ... r_j with i <= j <= k cover an initial interval of the positive integers.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 10, 11, 13, 18, 19, 20, 21, 22, 23, 25, 26, 29, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 49, 50, 53, 54, 58, 68, 69, 70, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 98, 101, 102, 105, 106, 109

Offset: 1

Rémy Sigrist, Sep 22 2024

There are A103295(k) terms with k binary digits (ignoring leading zeros).

			The binary expansion of 35 is "100011", the corresponding run lengths are (1, 3, 2); the sums 1, 2, 3, 1+3, 3+2, 1+3+2 cover the positive integers between 1 and 6, hence 35 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A101211, A103295, A376314.

toruns(n) = { my (r = []); while (n, my (v = valuation(n+n%2, 2)); n \= 2^v; r = concat(v, r)); r }
is(n) = { my (r = toruns(n)); #setbinop((i, j) -> vecsum(r[i..j]), [1..#r])==vecsum(r); }

cp's OEIS Frontend

A325986 Heinz numbers of complete strict integer partitions.

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A325765 Number of integer partitions of n with a unique consecutive subsequence summing to every positive integer from 1 to n.

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A325786 Number of complete necklace compositions of n.

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A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

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A103296 Number of complete rulers with n segments.

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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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A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.

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