A355743 -id:A355743 - cp's OEIS frontend

A382076 Number of integer partitions of n whose run-sums are not all equal.

0, 0, 0, 1, 1, 5, 6, 13, 15, 27, 37, 54, 64, 99, 130, 172, 220, 295, 372, 488, 615, 788, 997, 1253, 1547, 1955, 2431, 3005, 3706, 4563, 5586, 6840, 8332, 10139, 12305, 14879, 17933, 21635, 26010, 31181, 37314, 44581, 53156, 63259, 75163, 89124, 105553, 124752, 147210

Offset: 0

Gus Wiseman, Apr 02 2025

nonn

Also the number of integer partitions of n that cannot be partitioned into distinct constant multisets with a common sum. Multiset partitions of this type are ranked by A005117 /\ A326534 /\ A355743, while twice-partitions are counted by A382524, strict case of A279789.

			The partition (3,2,1,1,1) has runs ((3),(2),(1,1,1)) with sums (3,2,3) so is counted under a(8).
The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)     (43)      (53)
              (41)    (51)     (52)      (62)
              (221)   (321)    (61)      (71)
              (311)   (411)    (322)     (332)
              (2111)  (2211)   (331)     (431)
                      (21111)  (421)     (521)
                               (511)     (611)
                               (2221)    (3221)
                               (3211)    (3311)
                               (4111)    (4211)
                               (22111)   (5111)
                               (31111)   (22211)
                               (211111)  (32111)
                                         (311111)
                                         (2111111)

The complement is counted by A304442, ranks A353833.
For distinct instead of equal block-sums we have A381717.
This is the strict case of A381993, see A381995, zeros A381871.
A050361 counts factorizations into distinct prime powers, see A381715.
A304405 counts partitions with weakly decreasing run-sums, ranks A357875.
A304406 counts partitions with weakly increasing run-sums, ranks A357861.
A304428 counts partitions with strictly decreasing run-sums, ranks A357862.
A304430 counts partitions with strictly increasing run-sums, ranks A357864.
A317141 counts coarsenings of prime indices, refinements A300383.
A326534 ranks multiset partitions with a common sum.
A353837 counts partitions with distinct run-sums.
A354584 lists run-sums of weakly increasing prime indices.
A355743 ranks multiset partitions into constant blocks.
Cf. A000688, A005117, A006171, A047966, A279784, A381453, A381455, A381635, A381636, A381994, A382204.

Table[Length[Select[IntegerPartitions[n],!SameQ@@Total/@Split[#]&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 13, 25, 33, 54, 54, 99, 124, 166, 207, 295, 352, 488, 591, 780, 987, 1253, 1488, 1951, 2419, 2993, 3665, 4563, 5508, 6840, 8270, 10127, 12289, 14869, 17781, 21635, 25992, 31167, 37184, 44581, 53008, 63259, 75076, 89080, 105531, 124752, 146842, 173516, 204141, 239921, 281461, 329929, 385852

Offset: 0

Gus Wiseman, Mar 17 2025

nonn

			The multiset partition {{2},{2},{1,1},{1,1}} has both properties (constant blocks and common sum), so (2,2,1,1,1,1) is not counted under a(8). We can also use {{2,2},{1,1,1,1}}.
The a(3) = 1 through a(8) = 13 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (32111)
                             (211111)  (311111)

Twice-partitions of this type (constant with equal) are counted by A279789.
Multiset partitions of this type are ranked by A326534 /\ A355743.
For distinct instead of equal block-sums we have A381717.
These partitions are ranked by A381871, zeros of A381995.
For strict instead of constant blocks we have A381994, see A381719, A382080.
The strict case is A382076.
Normal multiset partitions of this type are counted by A382204.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000688, A006171, A047966, A265947, A279784, A381453, A381455, A381635, A381636, A381992.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]==0&]],{n,0,30}]

a(31)-a(54) from Robert Price, Mar 31 2025

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

Intersection of A005117 and A355743.

A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42

Offset: 1

Gus Wiseman, Jul 31 2022

nonn

			The a(30) = 14 numbers: 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30.

The complement is counted by A025528, with 1's A065515.
For primes instead of prime-powers we have A062298, with 1's A065855.
The version treating 1 as a prime-power is A085970.
One more than the partial sums of A143731.
A000688 counts factorizations into prime-powers.
A001222 counts prime-power divisors.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A000720, A023894, A036234, A050361, A054685, A069513, A355743, A356064, A356065, A356066.

Table[Length[Select[Range[n],!PrimePowerQ[#]&]],{n,100}]

a(n) = A085970(n) + 1.

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Mar 22 2025

nonn

Also numbers with a unique factorization into prime powers with distinct sums of prime indices.

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, sum A056239.

			The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

For distinct blocks instead of block-sums we have A004709, counted by A000726.
Twice-partitions of this type are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
These are the positions of 1 in A381635.
For no choices we have A381636 (zeros of A381635), counted by A381717.
For strict instead of constant blocks we have A381870, counted by A382079.
Partitions of this type (unique into constant with distinct) are counted by A382301.
Normal multiset partitions of this type are counted by A382203.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
Cf. A006171, A279784, A295935.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]

A354911 Number of factorizations of n into relatively prime prime-powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 1, 1, 0, 5, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

Original entry on oeis.org

13, 26, 29, 37, 39, 43, 47, 52, 58, 61, 65, 71, 73, 74, 78, 79, 86, 87, 89, 91, 94, 101, 104, 107, 111, 113, 116, 117, 122, 129, 130, 137, 139, 141, 142, 143, 145, 146, 148, 149, 151, 156, 158, 163, 167, 169, 172, 173, 174, 178, 181, 182, 183, 185, 188, 193

Offset: 1

Gus Wiseman, Jul 25 2022

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.

These are numbers divisible by a prime number not of the form prime(q^k) where q is a prime number and k >= 1.

			The terms together with their prime indices begin:
   13: {6}
   26: {1,6}
   29: {10}
   37: {12}
   39: {2,6}
   43: {14}
   47: {15}
   52: {1,1,6}
   58: {1,10}
   61: {18}
   65: {3,6}
   71: {20}
   73: {21}
   74: {1,12}
   78: {1,2,6}
   79: {22}
   86: {1,14}
   87: {2,10}

Heinz numbers of the partitions counted by A023893.
Allowing prime index 1 gives A356066.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A023894 counts partitions into prime-powers, strict A054685.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
A355743 = numbers whose prime indices are prime-powers, squarefree A356065.
Cf. A076610, A085970, A106244, A302492, A302493, A302601, A330946, A354911.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@PrimePowerQ/@DeleteCases[primeMS[#],1]&]

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 9, 5, 9, 2, 23, 2, 11, 10, 24, 2, 33, 2, 36, 12, 15, 2, 87, 7, 17, 17, 53, 2, 96, 2, 79, 16, 21, 14, 196, 2, 23, 18, 154, 2, 166, 2, 99, 54, 27, 2, 431, 9, 85, 22, 128, 2, 303, 18, 261, 24, 33, 2, 771, 2, 35, 73, 331, 20, 422, 2, 198, 28, 216, 2, 1369

Offset: 0

Gus Wiseman, Apr 22 2025

nonn

			The partition (4,4,2,2,2,2,1,1,1,1,1,1,1,1) has two partitions into constant blocks with a common sum: {{4,4},{2,2,2,2},{1,1,1,1,1,1,1,1}} and {{4},{4},{2,2},{2,2},{1,1,1,1},{1,1,1,1}}, so is counted under a(24).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (2211)               (2222)
                                     (3111)               (22211)
                                     (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)

Twice-partitions of this type (constant with common) are counted by A279789.
Multiset partitions of this type are ranked by A383309.
The complement is counted by A381993, ranks A381871.
For sets we have the complement of A381994, see A381719, A382080.
Normal multiset partitions of this type are counted by A382203, sets A381718.
For distinct instead of equal block-sums we have A382427.
These partitions are ranked by A383014, nonzeros of A381995.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers, see A381715.
A323774 counts partitions into constant blocks with a common sum
Constant blocks with distinct sums: A381635, A381636, A381717.
Permutation with equal run-sums: A383096, A383098, A383100, A383110
Cf. A006171, A047966, A279784, A295935, A323774, A326534, A353864, A355743, A381716, A382076, A382204.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]>0&]],{n,0,30}]

Multiset systems of this type have MM-numbers A383309 = A326534 /\ A355743.

Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.

More terms from Jakub Buczak, May 03 2025

A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 9, 14, 16, 25, 30, 41, 52, 69, 83, 105, 129, 164, 208, 263, 315, 388, 449, 573, 694

Offset: 0

Gus Wiseman, Mar 26 2025

			The a(4) = 3 through a(8) = 14 partitions and their unique multiset partition into constant blocks with distinct sums:
  {4}     {5}       {6}         {7}        {8}
  {22}    {1}{4}    {33}        {1}{6}     {44}
  {1}{3}  {2}{3}    {1}{5}      {2}{5}     {1}{7}
          {11}{3}   {2}{4}      {3}{4}     {2}{6}
          {1}{22}   {11}{4}     {11}{5}    {3}{5}
          {2}{111}  {11}{22}    {1}{33}    {11}{6}
                    {1}{2}{3}   {3}{22}    {2}{33}
                    {1}{11}{3}  {1}{2}{4}  {11}{33}
                                {3}{1111}  {11}{222}
                                           {1}{2}{5}
                                           {1}{3}{4}
                                           {1}{3}{22}
                                           {1}{4}{111}
                                           {1}{111}{22}

For distinct blocks instead of block-sums we have A000726, ranks A004709.
Twice-partitions of this type (constant with distinct) are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
For no choices we have A381717, ranks A381636, zeros of A381635.
The Heinz numbers of these partitions are A381991, positions of 1 in A381635.
Normal multiset partitions of this type are counted by A382203.
For at least one choice we have A382427.
For strict instead of constant blocks we have A382460, ranks A381870.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A293511, A295935, A353864, A381633, A381716, A381990, A381992, A381993, A382079.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,10}]

A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 14, 19, 28, 39, 50, 70, 91, 120, 161, 203, 260, 338, 426, 556, 695, 863, 1082, 1360, 1685

Offset: 0

Gus Wiseman, Mar 26 2025

Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.

			The partition (3,2,2,2,1) can be partitioned as {{1},{2},{3},{2,2}} or {{1},{3},{2,2,2}}, so is counted under a(10).
The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (1111)  (221)    (51)      (61)
                            (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
The complement is counted by A381717, ranks A381636, zeros of A381635.
For strict instead of constant blocks we have A381992, ranks A382075.
For a unique choice we have A382301, ranks A381991.
Normal multiset partitions of this type are counted by A382203, sets A381718.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A295935, A300385, A353864, A381633, A381716, A381990, A381993, A382079, A382876.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]

cp's OEIS Frontend

A382076 Number of integer partitions of n whose run-sums are not all equal.

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A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

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Mathematica

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A356065 Squarefree numbers whose prime indices are all prime-powers.

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Mathematica

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A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

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Mathematica

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A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

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Mathematica

A354911 Number of factorizations of n into relatively prime prime-powers.

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A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

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Mathematica

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

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Mathematica

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A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

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