A238628 -id:A238628 - cp's OEIS frontend

A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

0, 0, 2, 2, 10, 14, 46, 74, 202, 350, 862, 1562, 3610, 6734, 14926, 28394, 61162, 117950, 249022, 484922, 1009210, 1979054, 4076206, 8034314, 16422922, 32491550, 66045982, 131029082, 265246810, 527304974, 1064175886, 2118785834, 4266269482, 8503841150, 17093775742, 34101458042, 68461196410, 136664112494

Offset: 0

Gus Wiseman, Oct 07 2023

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1}    {1,2}    {2}        {1,4}
        {1,2}  {1,2,3}  {1,2}      {2,3}
                        {1,3}      {1,2,3}
                        {2,3}      {1,2,4}
                        {2,4}      {1,3,4}
                        {1,2,3}    {1,4,5}
                        {1,2,4}    {2,3,4}
                        {1,3,4}    {2,3,5}
                        {2,3,4}    {1,2,3,4}
                        {1,2,3,4}  {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

The complement is counted by A117855.
For pairs summing to n + 1 we have A167936.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A008967, A167762, A238628, A365376, A365377, A365381, A365541, A365544.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Tuples[#,2],n]&]],{n,0,10}]

def A366131(n): return (1<>1)<<1) if n else 0 # Chai Wah Wu, Nov 14 2023

From Chai Wah Wu, Nov 14 2023: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) for n > 3.
G.f.: 2*x^2*(1 - x)/((2*x - 1)*(3*x^2 - 1)). (End)

A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Oct 10 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.

Also Heinz numbers of integer partitions containing n/2, where n is the sum of all parts.

			The prime indices of 90 are {1,2,2,3}, with sum 8 and twice maximum 6, so 90 is in the sequence.

Partitions of this type are counted by A086543.
For length instead of maximum we have the complement of A340387.
The complement is A344415, counted by A035363.
A001221 counts distinct prime factors, A001222 with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A334201 adds up all prime indices except the greatest.
A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
Cf. A001414, A100959, A238628, A300061, A301987, A316413, A320924, A344414, A344416, A366318.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[prix[#]]!=Total[prix[#]]/2&]

A366130 Number of subsets of {1..n} with a subset summing to n + 1.

Original entry on oeis.org

0, 0, 1, 2, 7, 15, 38, 79, 184, 378, 823, 1682, 3552, 7208, 14948, 30154, 61698, 124302, 252125, 506521, 1022768, 2051555, 4127633, 8272147, 16607469, 33258510, 66680774, 133467385, 267349211, 535007304, 1071020315, 2142778192, 4288207796

Offset: 0

Gus Wiseman, Oct 07 2023

			The subset S = {1,2,4} has subset {1,4} with sum 4+1 and {2,4} with sum 5+1 and {1,2,4} with sum 6+1, so S is counted under a(4), a(5), and a(6).
The a(0) = 0 through a(5) = 15 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,3}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {2,3,4}    {1,3,5}
                        {1,2,3,4}  {1,4,5}
                                   {2,3,4}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

For pairs summing to n + 1 we have A167762, complement A038754.
For n instead of n + 1 we have A365376, for pairs summing to n A365544.
The complement is counted by A365377 shifted.
The complement for pairs summing to n is counted by A365377.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A004526, A004737, A008967, A046663, A238628, A365381, A365541.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],n+1]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A366130(n):
    a = tuple(set(p.keys()) for p in partitions(n+1,k=n) if max(p.values(),default=0)==1)
    return sum(1 for k in range(2,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any(s<=w for s in a)) # Chai Wah Wu, Nov 24 2023

Diagonal k = n + 1 of A365381.

a(20)-a(32) from Chai Wah Wu, Nov 24 2023

A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 20, 20, 30, 31, 45, 46, 66, 68, 93, 97, 130, 136, 179, 188, 242, 256, 325, 344, 432, 459, 568, 606, 742, 793, 963, 1031, 1240, 1331, 1589, 1707, 2026, 2179, 2567, 2766, 3240, 3493, 4072, 4393, 5094, 5501, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n without two parts (allowing parts to be re-used) summing to n.

			The a(6) = 1 through a(12) = 7 strict partitions:
  (6)  (7)      (8)      (9)      (10)       (11)       (12)
       (4,2,1)  (5,2,1)  (4,3,2)  (6,3,1)    (5,4,2)    (5,4,3)
                         (5,3,1)  (7,2,1)    (6,3,2)    (7,3,2)
                         (6,2,1)  (4,3,2,1)  (6,4,1)    (7,4,1)
                                             (7,3,1)    (8,3,1)
                                             (8,2,1)    (9,2,1)
                                             (5,3,2,1)  (5,4,2,1)

The second condition alone has bisections A078408 and A365828.
The complement is counted by A365659.
The non-strict version is A365825, complement A238628.
The first condition alone is A365827, complement A140106.
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A004526, A005408, A008967, A035363, A058984, A086543, A100959, A344415, A365376, A365377, A365543.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Tuples[#,2],n]&]], {n,0,30}]

A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 21, 22, 25, 26, 30, 33, 34, 35, 38, 39, 40, 46, 49, 51, 55, 57, 58, 62, 63, 65, 69, 70, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159

Offset: 1

Gus Wiseman, Oct 08 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
     4: {1,1}      38: {1,8}         77: {4,5}
     6: {1,2}      39: {2,6}         82: {1,13}
     9: {2,2}      40: {1,1,1,3}     84: {1,1,2,4}
    10: {1,3}      46: {1,9}         85: {3,7}
    12: {1,1,2}    49: {4,4}         86: {1,14}
    14: {1,4}      51: {2,7}         87: {2,10}
    15: {2,3}      55: {3,5}         91: {4,6}
    21: {2,4}      57: {2,8}         93: {2,11}
    22: {1,5}      58: {1,10}        94: {1,15}
    25: {3,3}      62: {1,11}        95: {3,8}
    26: {1,6}      63: {2,2,4}      106: {1,16}
    30: {1,2,3}    65: {3,6}        111: {2,12}
    33: {2,5}      69: {2,9}        112: {1,1,1,1,4}
    34: {1,7}      70: {1,3,4}      115: {3,9}
    35: {3,4}      74: {1,12}       118: {1,17}

The first condition alone is A001358, counted by A004526.
The complement of the first condition is A100959, counted by A058984.
The partitions with these Heinz numbers are counted by A238628.
The second condition alone is A344415, counted by A035363.
The complement of the second condition is A366319, counted by A086543.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
A334201 adds up all prime indices except the greatest.
A344296 solves for k in A001222(k) >= A056239(k)/2, counted by A025065.
Cf. A001414, A316413, A320924, A325044, A340387, A344414, A344416.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[prix[#]]==2||MemberQ[prix[#],Total[prix[#]]/2]&]

Union of A001358 and A344415.

A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15

Offset: 0

Gus Wiseman, Nov 19 2023

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.

			Triangle begins:
  1
  1  1
  1  2
  1  3  1
  1  3  3
  1  5  3  2
  1  4  7  2  1
  1  6  7  6  2
  1  6 10  6  7
  1  7 12 11  8  3
  1  6 16 11 17  3  2
  1 10 14 20 19 10  2  1
  1  7 22 17 31 14  7  2
  1  9 22 27 37 22 11  6
  1 10 24 27 51 32 16 15
  1 11 27 39 57 43 27 22  4
  1  9 33 34 79 57 36 39  7  2
  1 13 31 51 86 77 45 62 14  4  1
Row n = 9 counts the following partitions:
  (9)  (81)         (711)       (621)      (5211)
       (72)         (6111)      (531)      (4311)
       (63)         (522)       (432)      (4221)
       (54)         (51111)     (33111)    (42111)
       (333)        (441)       (222111)   (3321)
       (111111111)  (411111)    (2211111)  (32211)
                    (3222)                 (321111)
                    (3111111)
                    (22221)
                    (21111111)

Row sums are A000041.
Column k = 1 is A088922.
The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366738 counts semi-sums of partitions, non-binary A304792.
A366741 counts semi-sums of strict partitions, non-binary A365925.
Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]

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A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

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A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

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A366130 Number of subsets of {1..n} with a subset summing to n + 1.

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A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

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A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

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A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

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