A050320 -id:A050320 - cp's OEIS frontend

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A000688 at a(144) = 9, A000688(144) = 10.
First differs from A295879 at a(128) = 15, A295879(128) = 13.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into prime powers > 1.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The prime indices of 36 are {1,1,2,2}, with the following 4 partitions into a multiset of constant multisets:
  {{1,1},{2,2}}
  {{1},{1},{2,2}}
  {{2},{2},{1,1}}
  {{1},{1},{2},{2}}
with block-sums: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, which are all different, so a(36) = 4.
The prime indices of 144 are {1,1,1,1,2,2}, with the following 10 partitions into a multiset of constant multisets:
  {{2,2},{1,1,1,1}}
  {{1},{2,2},{1,1,1}}
  {{2},{2},{1,1,1,1}}
  {{1,1},{1,1},{2,2}}
  {{1},{1},{1,1},{2,2}}
  {{1},{2},{2},{1,1,1}}
  {{2},{2},{1,1},{1,1}}
  {{1},{1},{1},{1},{2,2}}
  {{1},{1},{2},{2},{1,1}}
  {{1},{1},{1},{1},{2},{2}}
with block-sums: {4,4}, {1,3,4}, {2,2,4}, {2,2,4}, {1,1,2,4}, {1,2,2,3}, {2,2,2,2}, {1,1,1,1,4}, {1,1,2,2,2}, {1,1,1,1,2,2}, of which 9 are distinct, so a(144) = 9.
The a(n) partitions for n = 4, 8, 16, 32, 36, 64, 72, 128:
  (2)   (3)    (4)     (5)      (42)    (6)       (43)     (7)
  (11)  (21)   (22)    (32)     (222)   (33)      (322)    (43)
        (111)  (31)    (41)     (411)   (42)      (421)    (52)
               (211)   (221)    (2211)  (51)      (2221)   (61)
               (1111)  (311)            (222)     (4111)   (322)
                       (2111)           (321)     (22111)  (331)
                       (11111)          (411)              (421)
                                        (2211)             (511)
                                        (3111)             (2221)
                                        (21111)            (3211)
                                        (111111)           (4111)
                                                           (22111)
                                                           (31111)
                                                           (211111)
                                                           (1111111)

Robert Price, Table of n, a(n) for n = 1..1000

Before taking sums we had A000688.
Positions of 1 are A005117.
There is a chain from the prime indices of n to a singleton iff n belongs to A300273.
The lower version is A381453.
For distinct blocks we have A381715, before sum A050361.
For distinct block-sums we have A381716, before sums A381635 (zeros A381636).
Other multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326) see A381441 (upper).
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633) see A381634, A293243.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002577, A002846, A018818, A213242, A213427, A275870, A289078, A299200, A299202, A300385.

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

a(s) = 1 for any squarefree number s.

a(p^k) = A000041(k) for any prime p.

A174726 a(n) = (A002033(n-1) - A008683(n))/2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 2, 4, 1, 7, 1, 8, 1, 1, 1, 13, 1, 1, 1, 10, 1, 7, 1, 4, 4, 1, 1, 24, 1, 4, 1, 4, 1, 10, 1, 10, 1, 1, 1, 22, 1, 1, 4, 16, 1, 7, 1, 4, 1, 7, 1, 38, 1, 1, 4, 4, 1

Offset: 1

Mats Granvik, Mar 28 2010

nonn

a(n) is the number of permutation matrices with a negative contribution to the determinant that is the Möbius function. See A174725 for how the determinant is defined. - Mats Granvik, May 26 2017
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an odd number of factors > 1. The unordered case is A339890. For example, the a(n) factorizations for n = 8, 12, 24, 30, 32, 36 are:
  (8)      (12)     (24)     (30)     (32)         (36)
  (2*2*2)  (2*2*3)  (2*2*6)  (2*3*5)  (2*2*8)      (2*2*9)
           (2*3*2)  (2*3*4)  (2*5*3)  (2*4*4)      (2*3*6)
           (3*2*2)  (2*4*3)  (3*2*5)  (2*8*2)      (2*6*3)
                    (2*6*2)  (3*5*2)  (4*2*4)      (2*9*2)
                    (3*2*4)  (5*2*3)  (4*4*2)      (3*2*6)
                    (3*4*2)  (5*3*2)  (8*2*2)      (3*3*4)
                    (4*2*3)           (2*2*2*2*2)  (3*4*3)
                    (4*3*2)                        (3*6*2)
                    (6*2*2)                        (4*3*3)
                                                   (6*2*3)
                                                   (6*3*2)
                                                   (9*2*2)
(End)

Mats Granvik, Table of n, a(n) for n = 1..10000

The even version is A174725.
The unordered case is A339890, with even version A339846.
A001055 counts factorizations, with strict case A045778.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A332304 counts strict compositions of odd length.
Cf. A002033, A024430, A027187, A050320, A052841, A058695, A160786, A316439.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],OddQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

a(n) = (A002033(n-1) - A008683(n))/2. - Mats Granvik, May 26 2017

For n > 0, a(n) + A174725(n) = A074206(n). - Gus Wiseman, Jan 04 2021

A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

Original entry on oeis.org

4, 4, -3, 5, -6, 4, 3, -15, 25, -10, -84, 369, -1067, 2610, -5824, 12246, -24622, 47577, -88233, 155962, -259086, 393455, -512281, 456609, 191219, -2396571, 8213890, -21761143, 50923029, -110269263, 225991429, -444168664, 844390152, -1561482492, 2817844569

Offset: 0

Gus Wiseman, Oct 19 2024

sign

Harvey P. Dale, Table of n, a(n) for n = 0..400

The version for primes is A007442, noncomposites A030016, composites A377036.
For squarefree instead of nonsquarefree numbers we have A377041.
For antidiagonal-sums we have A377047, absolute A377048.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A084758, A112925, A120992, A251092, A376311, A376591.

nn=20;
Table[First[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k]],{k,0,nn}]
With[{nsf=Select[Range[1000],!SquareFreeQ[#]&]},Table[Differences[nsf,n],{n,0,40}]][[;;,1]] (* Harvey P. Dale, Nov 28 2024 *)

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 127, 168, 208, 267, 343, 431, 536, 676, 836, 1045, 1283, 1582, 1949, 2395, 2895, 3549, 4298, 5216, 6281, 7569, 9104, 10953, 13078, 15652, 18627, 22207, 26325, 31278, 37002, 43708, 51597, 60807, 71533, 84031

Offset: 0

Gus Wiseman, Mar 15 2025

nonn

			The partition y = (3,3,3,2,2,1,1,1,1) has only one multiset partition into a set of sets, namely {{1},{3},{1,2},{1,3},{1,2,3}}, but this does not have distinct sums, so y is counted under a(17).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279785.
For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
Normal multiset partitions of this type are counted by A381718, see A116539.
These partitions are ranked by A381806, zeros of A381634 and A381633.
The complement is counted by A381992, ranked by A382075.
For distinct blocks we have A382078, complement A382077, unique A382079.
MM-numbers of these multiset partitions (strict blocks with distinct sum) are A382201.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
Cf. A293243, A293511, A358914, A381078, A381441, A381454.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,10}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94, 104, 106, 116, 118, 122, 124, 128, 134, 136, 142, 146, 148, 152, 158, 164, 166, 172, 178, 184, 188, 194, 202, 206, 212, 214, 218, 226, 232, 236, 244

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Consists of 1 and all numbers that are even and have no odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
      1: {}              44: {1,1,5}          106: {1,16}
      2: {1}             46: {1,9}            116: {1,1,10}
      4: {1,1}           52: {1,1,6}          118: {1,17}
      6: {1,2}           58: {1,10}           122: {1,18}
      8: {1,1,1}         62: {1,11}           124: {1,1,11}
     10: {1,3}           64: {1,1,1,1,1,1}    128: {1,1,1,1,1,1,1}
     14: {1,4}           68: {1,1,7}          134: {1,19}
     16: {1,1,1,1}       74: {1,12}           136: {1,1,1,7}
     20: {1,1,3}         76: {1,1,8}          142: {1,20}
     22: {1,5}           82: {1,13}           146: {1,21}
     26: {1,6}           86: {1,14}           148: {1,1,12}
     28: {1,1,4}         88: {1,1,1,5}        152: {1,1,1,8}
     32: {1,1,1,1,1}     92: {1,1,9}          158: {1,22}
     34: {1,7}           94: {1,15}           164: {1,1,13}
     38: {1,8}          104: {1,1,1,6}        166: {1,23}
For example, the factorizations of 88 are (2*2*2*11), (2*2*22), (2*4*11), (2*44), (4*22), (8*11), (88), none of which has odd minimum, so 88 is in the sequence.

The version looking at greatest factor is A000079.
The version for twice-balanced is A340656, with complement A340657.
These factorization are counted by A340832.
The complement is A340855.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Select[Range[100],Function[n,n==1||EvenQ[n]&&Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]=={}]]

A340855 Numbers that can be factored into factors > 1, the least of which is odd.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

These are numbers that are odd or have an odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
     3: {2}          27: {2,2,2}      48: {1,1,1,1,2}
     5: {3}          29: {10}         49: {4,4}
     7: {4}          30: {1,2,3}      50: {1,3,3}
     9: {2,2}        31: {11}         51: {2,7}
    11: {5}          33: {2,5}        53: {16}
    12: {1,1,2}      35: {3,4}        54: {1,2,2,2}
    13: {6}          36: {1,1,2,2}    55: {3,5}
    15: {2,3}        37: {12}         56: {1,1,1,4}
    17: {7}          39: {2,6}        57: {2,8}
    18: {1,2,2}      40: {1,1,1,3}    59: {17}
    19: {8}          41: {13}         60: {1,1,2,3}
    21: {2,4}        42: {1,2,4}      61: {18}
    23: {9}          43: {14}         63: {2,2,4}
    24: {1,1,1,2}    45: {2,2,3}      65: {3,6}
    25: {3,3}        47: {15}         66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).

The version looking at greatest factor is A057716.
The version for twice-balanced is A340657, with complement A340656.
These factorization are counted by A340832.
The complement is A340854.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors, listed by A161906.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Select[Range[100],Function[n,n>1&&(OddQ[n]||Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]!={})]]

A377048 Antidiagonal-sums of the absolute value of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

Original entry on oeis.org

4, 12, 13, 22, 28, 31, 39, 64, 85, 132, 395, 1103, 2650, 5868, 12297, 24694, 47740, 88731, 157744, 265744, 418463, 605929, 805692, 1104513, 2396645, 8213998, 21761334, 50923517, 110270883, 225997492, 444193562, 844498084, 1561942458, 2819780451, 4973173841

Offset: 1

Gus Wiseman, Oct 19 2024

nonn

These are the row-sums of the absolute value triangle version of A377046.

			The third antidiagonal of A377046 is (9, 1, -3), so a(3) = 13.

The version for primes is A376681, noncomposites A376684, composites A377035.
For squarefree instead of nonsquarefree numbers we have A377040.
The non-absolute version is A377047.
For leading column we have A377049.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A376591, A376592.

nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,nn},{j,i}]

A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Gus Wiseman, Mar 12 2025

nonn

First differs from A212166 in lacking 360.
First differs from A293511 in having 600.
Also numbers with a unique factorization into squarefree numbers with distinct sums of prime indices (A056239).
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.

			For n = 600 the unique multiset partition is {{1},{1,3},{1,2,3}}. The unique factorization is 2*10*30.

Without distinct block-sums we have A000961, ones in A050320.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
For distinct blocks instead of sums we have A293511, ones in A050326.
These are the positions of ones in A381633, see A381634, A381806, A381990.
Normal multiset partitions of this type are counted by A381718, see A279785.
For constant instead of strict blocks we have A381991, ones in A381635.
A001055 counts multiset partitions of prime indices, strict A045778.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A317141 counts coarsenings of prime indices, refinements A300383.
A321469 counts factorizations with distinct sums of prime indices, ones A166684.
Cf. A000720, A001222, A005117, A066328, A293243, A299202, A300385.

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

A382077 Number of integer partitions of n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 171, 217, 283, 361, 449, 574, 721, 900, 1126, 1397, 1731, 2143, 2632, 3223, 3961, 4825, 5874, 7131, 8646, 10452, 12604, 15155, 18216, 21826, 26108, 31169, 37156, 44202, 52492, 62233, 73676, 87089, 102756, 121074

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240306 at a(14) = 76, A240306(14) = 77.

First differs from A381992 at a(17) = 171, A381992(17) = 170.

			For y = (3,2,2,2,1,1,1), we have the multiset partition {{1},{2},{1,2},{1,2,3}}, so y is counted under a(12).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)

Factorizations of this type are counted by A050345.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions of this type are counted by A116539.
The MM-numbers of these multiset partitions are A302494.
Twice-partitions of this type are counted by A358914.
For distinct block-sums instead of blocks we have A381992, ranked by A382075.
The complement is counted by A382078, unique A382079.
These partitions are ranked by A382200, complement A293243.
For normal multisets instead of integer partitions we have A382214, complement A292432.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A292444, A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870, A381990.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]>0&]],{n,0,9}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A377047 Antidiagonal-sums of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

Original entry on oeis.org

4, 12, 7, 22, 14, 17, 39, 0, 37, 112, -337, 1103, -2570, 5868, -12201, 24670, -47528, 88283, -155910, 259140, -393399, 512341, -456546, -191155, 2396639, -8213818, 21761218, -50922953, 110269343, -225991348, 444168748, -844390064, 1561482582, -2817844477

Offset: 1

Gus Wiseman, Oct 19 2024

sign

These are the row-sums of the triangle-version of A377046.

			The third antidiagonal of A377046 is (9, 1, -3), so a(3) = 7.

The version for primes is A140119, noncomposites A376683, composites A377034.
For squarefree instead of nonsquarefree numbers we have A377039.
The absolute value version is A377048.
For leading column we have A377049.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A376311, A376591, A376592, A377051.

nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

cp's OEIS Frontend

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

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A174726 a(n) = (A002033(n-1) - A008683(n))/2.

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A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

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A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

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A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

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A340855 Numbers that can be factored into factors > 1, the least of which is odd.

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A377048 Antidiagonal-sums of the absolute value of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

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A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.

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A382077 Number of integer partitions of n that can be partitioned into a set of sets.

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