A304792 -id:A304792 - cp's OEIS frontend

A367402 Number of integer partitions of n whose semi-sums cover an interval of positive integers.

1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 26, 31, 38, 44, 58, 64, 81, 95, 116, 137, 166, 192, 233, 278, 330, 385, 459, 542, 636, 759, 879, 1038, 1211, 1418, 1656, 1942, 2242, 2618, 3029, 3535, 4060, 4735, 5429, 6299, 7231, 8346, 9556, 11031, 12593, 14482, 16525

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (3,2,1,1) has semi-sums {2,3,4,5}, which is an interval, so y is counted under a(7).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

For parts instead of sums we have A034296, ranks A073491.
For all subset-sums we have A126796, ranks A325781, strict A188431.
The complement for parts instead of sums is A239955, ranks A073492.
The complement for all sub-sums is A365924, ranks A365830, strict A365831.
The complement is counted by A367403.
The strict case is A367410, complement A367411.
A000009 counts partitions covering an initial interval, ranks A055932.
A086971 counts semi-sums of prime indices.
A261036 counts complete partitions by maximum.
A276024 counts positive subset-sums of partitions, strict A284640.
Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365661, A365918, A365921.

Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]==Union[d])&]], {n,0,15}]

A367403 Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 9, 13, 22, 30, 46, 63, 91, 118, 167, 216, 290, 374, 490, 626, 810, 1022, 1297, 1628, 2051, 2551, 3176, 3929, 4845, 5963, 7311, 8932, 10892, 13227, 16035, 19395, 23397, 28156, 33803, 40523, 48439, 57832, 68876, 81903, 97212, 115198

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The a(0) = 0 through a(9) = 13 partitions:
  .  .  .  .  .  (311)  (411)   (331)    (422)     (441)
                        (3111)  (421)    (431)     (522)
                                (511)    (521)     (531)
                                (4111)   (611)     (621)
                                (31111)  (3311)    (711)
                                         (4211)    (4311)
                                         (5111)    (5211)
                                         (41111)   (6111)
                                         (311111)  (33111)
                                                   (42111)
                                                   (51111)
                                                   (411111)
                                                   (3111111)

The complement for parts instead of sums is A034296, ranks A073491.
The complement for all sub-sums is A126796, ranks A325781, strict A188431.
For parts instead of sums we have A239955, ranks A073492.
For all subset-sums we have A365924, ranks A365830, strict A365831.
The complement is counted by A367402.
The strict case is A367411, complement A367410.
A000009 counts partitions covering an initial interval, ranks A055932.
A086971 counts semi-sums of prime indices.
A261036 counts complete partitions by maximum.
A276024 counts positive subset-sums of partitions, strict A284640.
Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365658.

Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,15}]

A367410 Number of strict integer partitions of n whose semi-sums cover an interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 6, 6, 7, 7, 8, 8, 11, 9, 11, 11, 12, 12, 15, 14, 15, 16, 16, 16, 19, 18, 19, 22, 21, 21, 24, 22, 25, 26, 26, 26, 30, 28, 29, 32, 31, 32, 37, 35, 36, 38, 39, 39, 43, 42, 43, 47, 46, 49, 51, 52, 51, 58

Offset: 0

Gus Wiseman, Nov 18 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is not counted under a(7).
The a(1) = 1 through a(9) = 6 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
                          (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
                                 (3,2,1)  (6,1)  (7,1)  (7,2)
                                                        (8,1)
                                                        (4,3,2)

For parts instead of sums we have A001227:
- non-strict A034296, ranks A073491
- complement A238007
- non-strict complement A239955, ranks A073492
The non-binary version is A188431:
- non-strict A126796, ranks A325781
- complement A365831
- non-strict complement A365924, ranks A365830
The non-strict version is A367402.
The non-strict complement is A367403.
The complement is counted by A367411.
A000009 counts partitions covering an initial interval, ranks A055932.
A046663 counts partitions w/o submultiset summing to k, strict A365663.
A365543 counts partitions w/ submultiset summing to k, strict A365661.
Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#,{2}]; If[d=={},{}, Range[Min@@d, Max@@d]]==Union[d])&]], {n,0,30}]

A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 4, 5, 8, 10, 14, 16, 23, 27, 35, 42, 52, 61, 75, 89, 106, 126, 149, 173, 204, 237, 274, 319, 369, 424, 490, 560, 642, 734, 838, 952, 1085, 1231, 1394, 1579, 1784, 2011, 2269, 2554, 2872, 3225, 3619, 4054, 4540, 5077, 5671, 6332

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is counted under a(7).
The a(7) = 1 through a(13) = 10 partitions:
  (4,2,1)  (4,3,1)  (5,3,1)  (5,3,2)  (5,4,2)  (6,4,2)    (6,4,3)
           (5,2,1)  (6,2,1)  (5,4,1)  (6,3,2)  (6,5,1)    (6,5,2)
                             (6,3,1)  (6,4,1)  (7,3,2)    (7,4,2)
                             (7,2,1)  (7,3,1)  (7,4,1)    (7,5,1)
                                      (8,2,1)  (8,3,1)    (8,3,2)
                                               (9,2,1)    (8,4,1)
                                               (5,4,2,1)  (9,3,1)
                                               (6,3,2,1)  (10,2,1)
                                                          (6,4,2,1)
                                                          (7,3,2,1)

For parts instead of sums we have A238007:
- complement A001227
- non-strict complement A034296, ranks A073491
- non-strict  A239955, ranks A073492
The non-strict version is A367403.
The non-strict complement is A367402.
The complement is counted by A367410.
The non-binary version is A365831:
- non-strict complement A126796, ranks A325781
- complement A188431
- non-strict A365924, ranks A365830
A000009 counts partitions covering an initial interval, ranks A055932.
A046663 counts partitions w/o submultiset summing to k, strict A365663.
A365543 counts partitions w/ submultiset summing to k, strict A365661.
Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#, {2}];If[d=={},{}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,30}]

A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 12, 14, 14, 15, 15, 18, 17, 19, 18, 20, 20, 22, 21, 25, 23, 26, 25, 28, 26, 29, 27, 31, 29, 32, 31, 34, 33, 35, 34, 38, 35, 41, 37, 42, 40, 43, 41, 45, 42, 46, 44, 48, 45, 50, 46, 52, 49, 53

Offset: 0

Gus Wiseman, Oct 13 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			Representative partitions for each of the a(16) = 11 alternating products:
         (16) -> 16
     (14,1,1) -> 14
     (12,2,2) -> 12
     (10,3,3) -> 10
      (8,4,4) -> 8
  (9,3,2,1,1) -> 6
     (10,4,2) -> 5
     (12,3,1) -> 4
  (6,4,2,2,2) -> 3
     (10,5,1) -> 2
        (8,8) -> 1

The even-length version is A000035.
The non-reverse version is A028310.
The version for factorizations has special cases:
- no changes: A046951
- non-reverse: A046951
- non-integer: A038548
- odd-length: A046951 + A010052
- non-reverse non-integer: A347460
- non-integer odd-length: A347708
- non-reverse odd-length: A046951 + A010052
- non-reverse non-integer odd-length: A347708
The odd-length version is a(n) - A059841(n).
These partitions are counted by A347445, non-reverse A347446.
Counting non-integers gives A347462, non-reverse A347461.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A119620 counts partitions with alternating product 1, ranked by A028982.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A304792 counts distinct subset-sums of partitions.
A325534 counts separable partitions, complement A325535.
A345926 counts possible alternating sums of permutations of prime indices.
Cf. A000070, A002033, A002219, A108917, A122768, A325765, A344654, A344740, A347443, A347444, A347448, A347449.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[Union[revaltprod/@IntegerPartitions[n]],IntegerQ]],{n,0,30}]

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 0, 1, 0, 3, 1, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 1, 2, 0, 1, 0, 2, 1, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 1, 1, 0, 0, 5

Offset: 1

Gus Wiseman, Oct 14 2021

nonn

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any length gives A347460, reverse A038548.
Allowing any odd length gives A347708.
A001055 counts factorizations (strict A045778, ordered A074206).
A122179 counts length-3 factorizations.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions, positive A276024.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 3, 2, 6, 6, 16, 12, 20, 26, 59, 45, 79, 94, 186, 142, 231, 244, 442, 470, 616, 746, 1340, 1053, 1548, 1852, 2780, 2826, 3874, 4320, 6617, 6286, 7924, 9178, 13180, 13634, 17494, 20356, 28220, 29176, 37188, 41932, 56037

Offset: 0

Gus Wiseman, Sep 16 2023

Are n = 1, 2, 4 the only n such that none of these partitions has 1?

Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?

			The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
The a(1) = 1 through a(7) = 16 partitions:
(2)  (2,2)  (4,2)    (4,2,2)    (4,3,3)      (6,4,2)        (6,5,3)
            (5,1)    (2,2,2,2)  (4,4,2)      (6,5,1)        (8,4,2)
            (2,2,2)             (6,2,2)      (4,4,2,2)      (8,5,1)
                                (8,1,1)      (6,2,2,2)      (9,3,2)
                                (4,2,2,2)    (4,2,2,2,2)    (9,4,1)
                                (2,2,2,2,2)  (2,2,2,2,2,2)  (10,3,1)
                                                            (11,2,1)
                                                            (4,4,4,2)
                                                            (5,3,3,3)
                                                            (6,4,2,2)
                                                            (8,2,2,2)
                                                            (11,1,1,1)
                                                            (4,4,2,2,2)
                                                            (6,2,2,2,2)
                                                            (4,2,2,2,2,2)
                                                            (2,2,2,2,2,2,2)

For n instead of 2n we have A126796.
Central column n = 2k of A365658.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002219 counts partitions of 2n with a submultiset summing to n.
A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.
A122768 counts distinct nonempty submultisets of partitions.
A299701 counts sums of submultisets of prime indices, of partitions A304792.
A364272 counts sum-full strict partitions, sum-free A364349.
A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.
Cf. A000041, A008967, A095944, A364911, A365376, A365377.

msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]],{n,0,10}]

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A365660(n):
    c = 0
    for p in partitions(n<<1):
        q, s = list(Counter(p).elements()), set()
        for l in range(1,len(q)+1):
            for k in multiset_combinations(q,l):
                s.add(sum(k))
                if len(s) > n:
                    break
            else:
                continue
            break
        if len(s)==n:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(21)-a(38) from Chai Wah Wu, Sep 20 2023

a(39)-a(43) from Chai Wah Wu, Sep 21 2023

A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 4, 6, 11, 15, 25, 35, 53, 72, 108

Offset: 1

Gus Wiseman, Sep 30 2023

A non-subset-sum of a multiset of positive integers summing to n is an element of {1..n} that is not the sum of any submultiset. A multiset is incomplete if it has at least one non-subset-sum.

			The non-subset-sums of y = {2,2,3} are {1,6}, with maximum 6, so y is counted under a(6).
The a(1) = 1 through a(6) = 15 multisets:
  {2}  {3}    {4}      {5}        {6}          {7}
       {1,3}  {1,4}    {1,5}      {1,6}        {1,7}
              {2,2}    {2,3}      {2,4}        {2,5}
              {1,1,4}  {1,1,5}    {3,3}        {3,4}
                       {1,2,5}    {1,1,6}      {1,1,7}
                       {1,1,1,5}  {1,2,6}      {1,2,7}
                                  {1,3,3}      {1,3,4}
                                  {2,2,2}      {2,2,3}
                                  {1,1,1,6}    {1,1,1,7}
                                  {1,1,2,6}    {1,1,2,7}
                                  {1,1,1,1,6}  {1,1,3,7}
                                               {1,2,2,7}
                                               {1,1,1,1,7}
                                               {1,1,1,2,7}
                                               {1,1,1,1,1,7}

For least instead of greatest we have A126796, ranks A325781, strict A188431.
These multisets have ranks A365830.
Counts appearances of n in the rank statistic A365920.
Column sums of A365921.
These multisets counted by sum are A365924, strict A365831.
The strict case is A366129.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365918 counts non-subset-sums of partitions.
A365923 counts partitions by non-subset sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365658, A365919, A365925.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n],Max@@nmz[#]==n&]],{n,5}]

A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 7, 11, 11, 15, 18, 23, 28, 36, 40, 50, 59, 70, 83, 101, 118, 141, 166, 195, 227, 268, 306, 358, 414, 478, 549, 640, 730, 846, 968, 1113, 1271, 1462, 1657, 1897, 2154, 2451

Offset: 1

Gus Wiseman, Oct 07 2023

A non-subset-sum of a set summing to n is a positive integer up to n that is not the sum of any subset. For example, the non-subset-sums of {1,3,4} are {2,6}.

			The a(1) = 1 through a(8) = 11 sets:
  {2}  {3}    {4}    {5}      {6}      {7}      {8}        {9}
       {1,3}  {1,4}  {2,3}    {2,4}    {2,5}    {2,6}      {2,7}
                     {1,5}    {1,6}    {3,4}    {3,5}      {3,6}
                     {1,2,5}  {1,2,6}  {1,7}    {1,8}      {4,5}
                                       {1,3,4}  {1,3,5}    {2,3,4}
                                       {1,2,7}  {1,2,8}    {1,9}
                                                {1,2,3,8}  {1,3,6}
                                                           {1,4,5}
                                                           {1,2,9}
                                                           {1,2,3,9}
                                                           {1,2,4,9}

For least instead of greatest: A188431, non-strict A126796 (ranks A325781).
The version counting multisets instead of sets is A366127.
These sets counted by sum are A365924, strict A365831.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A365923 counts partitions by number of non-subset-sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365543, A365658, A365661, A365918, A365920, A365921, A365925.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n], UnsameQ@@#&&Max@@nmz[#]==n&]],{n,15}]

a(31)-a(42) from Erich Friedman, Nov 13 2024

A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

Original entry on oeis.org

1, 12, 24, 48, 60, 192, 144, 120, 180, 336, 240, 630, 420, 360, 900, 1344, 960, 1008, 720, 840, 2340, 1980, 1260, 1440, 3120, 2640, 1680, 4032, 2880, 6840, 3600, 4620, 3780, 2520, 6480, 11700, 8820, 6300, 7200, 10560, 6720, 12240, 9360, 7920, 5040, 10920, 9240

Offset: 1

Gus Wiseman, Nov 09 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.An integer n is a subset-sum (A299701, A304792) of a multiset y if there exists a submultiset of y with sum n.

			The divisors of 60 are {1,2,3,4,5,6,10,12,15,20,30,60}, and the distinct subset-sums of its prime indices {1,1,2,3} are {0,1,2,3,4,5,6,7}, so the difference is 12 - 8 = 4. Since 60 is the first number with this difference, we have a(4) = 60.
The terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    24: {1,1,1,2}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   144: {1,1,1,1,2,2}
   180: {1,1,2,2,3}
   192: {1,1,1,1,1,1,2}
   240: {1,1,1,1,2,3}
   336: {1,1,1,1,2,4}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}
   840: {1,1,1,2,3,4}
   900: {1,1,2,2,3,3}
   960: {1,1,1,1,1,1,2,3}

The first part (divisors) is A000005.
The second part (subset-sums of prime indices) is A299701, positive A304793.
These are the positions of first appearances in the difference A325801.
The binary version is A367093, firsts of  A086971 - A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A220264, A304792, A365920, A366740, A367097.

nn=1000;
w=Table[DivisorSigma[0,n]-Length[Union[Total/@Subsets[prix[n]]]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[0,i]]];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

A000005(a(n)) - A299701(a(n)) = n.

cp's OEIS Frontend

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