A325858 -id:A325858 - cp's OEIS frontend

A325992 Heinz numbers of integer partitions such that not every ordered pair of distinct parts has a different difference.

30, 60, 90, 105, 110, 120, 150, 180, 210, 220, 238, 240, 270, 273, 300, 315, 330, 360, 385, 390, 420, 440, 450, 462, 476, 480, 506, 510, 525, 540, 546, 550, 570, 600, 627, 630, 660, 690, 714, 720, 735, 750, 770, 780, 806, 810, 819, 840, 858, 870, 880, 900, 910

Offset: 1

Gus Wiseman, Jun 02 2019

nonn

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

			The sequence of terms together with their prime indices begins:
   30: {1,2,3}
   60: {1,1,2,3}
   90: {1,2,2,3}
  105: {2,3,4}
  110: {1,3,5}
  120: {1,1,1,2,3}
  150: {1,2,3,3}
  180: {1,1,2,2,3}
  210: {1,2,3,4}
  220: {1,1,3,5}
  238: {1,4,7}
  240: {1,1,1,1,2,3}
  270: {1,2,2,2,3}
  273: {2,4,6}
  300: {1,1,2,3,3}
  315: {2,2,3,4}
  330: {1,2,3,5}
  360: {1,1,1,2,2,3}
  385: {3,4,5}
  390: {1,2,3,6}

The subset case is A143823.
The maximal case is A325879.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A002033, A056239, A108917, A112798, A143824, A325325, A325868, A325879, A325991, A325993, A325994.

Select[Range[1000],!UnsameQ@@Subtract@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A325855 Number of strict integer partitions of n such that every pair of distinct parts has a different product.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 22, 25, 31, 37, 44, 53, 59, 69, 83, 100, 111, 129, 152, 173, 198, 232, 260, 302, 342, 386, 448, 498, 565, 646, 728, 819, 918, 1039, 1164, 1310, 1462, 1631, 1830, 2053, 2282, 2532, 2825, 3136, 3482, 3869, 4300, 4744

Offset: 0

Gus Wiseman, May 31 2019

nonn

			The a(1) = 1 through a(10) = 10 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)   (43)   (53)   (54)   (64)
                        (41)  (51)   (52)   (62)   (63)   (73)
                              (321)  (61)   (71)   (72)   (82)
                                     (421)  (431)  (81)   (91)
                                            (521)  (432)  (532)
                                                   (531)  (541)
                                                   (621)  (631)
                                                          (721)
                                                          (4321)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..250

The subset case is A196724.
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Heinz numbers of the counterexamples are given by A325993.
Cf. A002033, A108917, A143823, A275972, A325854, A325858, A325876, A325877.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Times@@@Subsets[Union[#],{2}]&]],{n,0,30}]

A325857 Number of integer partitions of n such that every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 74, 97, 125, 165, 209, 269, 335, 428, 527, 664, 804, 1005, 1210, 1496, 1780, 2186, 2586, 3148, 3698, 4473, 5226, 6279, 7290, 8706, 10067, 11950, 13744, 16242, 18605, 21864, 24942, 29184, 33188, 38651, 43782, 50791, 57402, 66300, 74683, 86026, 96658

Offset: 0

Gus Wiseman, May 31 2019

nonn

			The A000041(14) - a(14) = 10 partitions of 14 not satisfying the condition are:
  (6,5,2,1)
  (6,4,3,1)
  (5,4,3,2)
  (5,4,2,2,1)
  (4,4,3,2,1)
  (5,4,2,1,1,1)
  (4,3,3,2,1,1)
  (4,3,2,2,2,1)
  (4,3,2,2,1,1,1)
  (4,3,2,1,1,1,1,1)

Max Alekseyev, Table of n, a(n) for n = 0..80

The subset case is A196723.
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A103300, A108917, A143823, A196724, A325853, A325855, A325858, A325859, A325862.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#],{2}]&]],{n,0,30}]

Terms a(31) onward from Max Alekseyev, Sep 23 2023

A325868 Number of subsets of {1..n} containing n such that every ordered pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 2, 4, 6, 14, 24, 52, 84, 120, 240, 548, 688, 1784, 2600, 4236, 5796, 16200, 17568, 49968, 55648, 101360, 176792, 433736, 430032, 728784, 1360928, 2304840, 2990856, 8682912, 7877376, 25243200, 27946656, 46758912, 81457248, 121546416, 114388320, 442583952

Offset: 1

Gus Wiseman, Jun 02 2019

nonn

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

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..50

Cf. A025582, A108917, A143823, A196723, A196724.

Cf. A325853, A325854, A325858, A325860, A325861, A325864, A325869.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Divide@@@Subsets[#,{2}]&]],{n,10}]

a(21)-a(37) from Fausto A. C. Cariboni, Oct 16 2020

A325880 Number of maximal subsets of {1..n} containing n such that every ordered pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 8, 10, 18, 34, 50, 70, 78, 89, 120, 181, 277, 401, 561, 728, 867, 1031, 1219, 1537, 2013, 2684, 3581, 4973, 6435, 8124, 9974, 12054, 14057, 16890, 19783, 24102, 29539, 37247, 46301, 59825, 74556, 94064, 115057, 141068, 167521, 200790, 232798, 273734

Offset: 1

Gus Wiseman, Jun 02 2019

nonn

Also the number of maximal subsets of {1..n} containing n such that every orderless pair of (not necessarily distinct) elements has a different sum.

			The a(2) = 1 through a(9) = 18 subsets:
  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {2,3,7}    {3,5,8}    {4,6,9}
         {2,3}  {1,3,4}  {1,4,5}  {1,3,6}  {2,4,7}    {4,5,8}    {5,6,9}
                         {2,3,5}  {1,4,6}  {2,6,7}    {1,2,4,8}  {1,2,4,9}
                         {2,4,5}  {1,5,6}  {3,4,7}    {1,2,6,8}  {1,2,6,9}
                                  {2,3,6}  {4,5,7}    {1,3,4,8}  {1,2,7,9}
                                  {2,5,6}  {4,6,7}    {1,3,7,8}  {1,3,4,9}
                                  {3,4,6}  {1,2,5,7}  {1,5,6,8}  {1,3,8,9}
                                  {3,5,6}  {1,3,6,7}  {1,5,7,8}  {1,4,8,9}
                                                      {2,3,6,8}  {1,6,7,9}
                                                      {2,4,7,8}  {1,6,8,9}
                                                                 {2,3,5,9}
                                                                 {2,3,7,9}
                                                                 {2,4,5,9}
                                                                 {2,4,8,9}
                                                                 {2,6,7,9}
                                                                 {2,6,8,9}
                                                                 {3,4,7,9}
                                                                 {3,5,8,9}

Andrew Howroyd, Table of n, a(n) for n = 1..60

Cf. A002033, A108917, A143823, A143824, A196723.

Cf. A325858, A325859, A325861, A325867, A325869, A325879, A325992.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]]],{n,0,10}]

a(n)={
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1<= n, ismaxl(b,w),
         my(s=self()(k+1, b,w));
         b+=1<Andrew Howroyd, Mar 23 2025

a(25) onwards from Andrew Howroyd, Mar 23 2025

A325869 Number of maximal subsets of {1..n} containing n such that every pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 6, 6, 20, 32, 29, 57, 83, 113, 183, 373, 233, 549, 360

Offset: 1

Gus Wiseman, Jun 02 2019

			The a(1) = 1 through a(7) = 6 subsets:
  {1}  {1,2}  {1,2,3}  {1,3,4}  {1,2,3,5}  {1,2,5,6}    {1,2,3,5,7}
                       {2,3,4}  {1,3,4,5}  {2,3,5,6}    {1,2,5,6,7}
                                {2,3,4,5}  {2,4,5,6}    {2,3,4,5,7}
                                           {1,3,4,5,6}  {2,3,5,6,7}
                                                        {2,4,5,6,7}
                                                        {1,3,4,5,6,7}

Cf. A025582, A067992, A108917, A143823, A196723, A196724.

Cf. A325853, A325854, A325858, A325860, A325861, A325864, A325868.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&)/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Divide@@@Subsets[#,{2}]&]]],{n,10}]

A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

210, 420, 462, 630, 840, 858, 910, 924, 1050, 1155, 1260, 1326, 1386, 1470, 1680, 1716, 1820, 1848, 1870, 1890, 1938, 2100, 2145, 2310, 2470, 2520, 2574, 2622, 2652, 2730, 2772, 2926, 2940, 3150, 3234, 3315, 3360, 3432, 3465, 3570, 3640, 3696, 3740, 3780, 3876

Offset: 1

Gus Wiseman, Jun 02 2019

nonn

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

			The sequence of terms together with their prime indices begins:
   210: {1,2,3,4}
   420: {1,1,2,3,4}
   462: {1,2,4,5}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}
   858: {1,2,5,6}
   910: {1,3,4,6}
   924: {1,1,2,4,5}
  1050: {1,2,3,3,4}
  1155: {2,3,4,5}
  1260: {1,1,2,2,3,4}
  1326: {1,2,6,7}
  1386: {1,2,2,4,5}
  1470: {1,2,3,4,4}
  1680: {1,1,1,1,2,3,4}
  1716: {1,1,2,5,6}
  1820: {1,1,3,4,6}
  1848: {1,1,1,2,4,5}
  1870: {1,3,5,7}
  1890: {1,2,2,2,3,4}

The subset case is A196723.
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A056239, A103300, A108917, A112798, A143823, A196724, A325853, A325855, A325858, A325859, A325862, A325992, A325993, A325994.

Select[Range[1000],!UnsameQ@@Plus@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A325993 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different product.

Original entry on oeis.org

390, 780, 798, 1170, 1365, 1560, 1596, 1914, 1950, 2340, 2394, 2590, 2730, 2886, 3120, 3192, 3510, 3828, 3900, 3990, 4095, 4290, 4386, 4485, 4680, 4788, 5070, 5170, 5180, 5460, 5586, 5742, 5772, 5850, 6042, 6240, 6384, 6630, 6699, 6825, 7020, 7182, 7410, 7656

Offset: 1

Gus Wiseman, Jun 02 2019

nonn

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

			The sequence of terms together with their prime indices begins:
   390: {1,2,3,6}
   780: {1,1,2,3,6}
   798: {1,2,4,8}
  1170: {1,2,2,3,6}
  1365: {2,3,4,6}
  1560: {1,1,1,2,3,6}
  1596: {1,1,2,4,8}
  1914: {1,2,5,10}
  1950: {1,2,3,3,6}
  2340: {1,1,2,2,3,6}
  2394: {1,2,2,4,8}
  2590: {1,3,4,12}
  2730: {1,2,3,4,6}
  2886: {1,2,6,12}
  3120: {1,1,1,1,2,3,6}
  3192: {1,1,1,2,4,8}
  3510: {1,2,2,2,3,6}
  3828: {1,1,2,5,10}
  3900: {1,1,2,3,3,6}
  3990: {1,2,3,4,8}

The subset case is A196724.
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Heinz numbers of the counterexamples are given by A325993.
Cf. A002033, A056239, A108917, A112798, A143823, A292886, A325858, A325877, A325991, A325992, A325994.

Select[Range[1000],!UnsameQ@@Times@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A326035 Number of uniform knapsack partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 6, 9, 10, 12, 12, 17, 16, 20, 25, 27, 29, 35, 39, 44, 57, 53, 66, 75, 84, 84, 114, 112, 131, 133, 162, 167, 209, 192, 242, 250, 289, 279, 363, 348, 417, 404, 502, 487, 608, 557, 706, 682, 835, 773, 1004, 922, 1149, 1059, 1344, 1257, 1595

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition is uniform if all parts appear with the same multiplicity, and knapsack if every distinct submultiset has a different sum.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..650

Cf. A002033, A047966, A072774, A108917, A275972, A276024, A299702.

Cf. A325592, A325858, A326015, A326016, A326017, A326036, A326037.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ks[n_]:=Select[IntegerPartitions[n],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[ks[n],SameQ@@Length/@Split[#]&]],{n,30}]

A363994 a(n) is the number of partitions of n whose difference multiset has no duplicates; see Comments.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 7, 6, 10, 11, 11, 15, 18, 18, 25, 29, 28, 38, 44, 47, 54, 67, 68, 84, 88, 102, 114, 137, 132, 167, 180, 204, 214, 261, 264, 315, 328, 377, 414, 476, 473, 564, 603, 677, 708, 820, 846, 969, 1028, 1131, 1214, 1364, 1414, 1596, 1701, 1858

Offset: 0

Clark Kimberling, Sep 08 2023

nonn

If M is a multiset of real numbers, then the difference multiset of M consists of the differences of pairs of numbers in M. For example, the difference multiset of {1,2,2,5} is {0,1,1,3,3,4}.

			The partitions of 8 are [8], [7,1], [6,2], [6,1,1], [5,3], [5,2,1], [5,1,1,1], [4,4], [4,3,1], [4,2,2], [4,2,1,1], [4,1,1,1,1], [3,3,2], [3,3,1,1], [3,2,2,1], [3,2,1,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
The 7 partitions whose difference multiset is duplicate-free are [8], [7,1], [6,2], [5,3], [5,2,1], [4,4], [4,3,1].

Cf. A000041, A325858, A325876, A364612.

s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}]
g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Differences, s[n, k]]]
t[n_] := t[n] = Table[g[n, k], {k, 1, PartitionsP[n]}];
a[n_] := Count[t[n], True];
Table[a[n], {n, 1, 20}]

from collections import Counter
from itertools import combinations
from sympy.utilities.iterables import partitions
def A363994(n): return sum(1 for p in partitions(n) if max(list(Counter(abs(d[0]-d[1]) for d in combinations(list(Counter(p).elements()),2)).values()),default=1)==1) # Chai Wah Wu, Sep 17 2023

a(n) = A000041(n) - A364612(n).

a(n) = A325876(n) - (1 - n mod 2) for n > 0. - Andrew Howroyd, Sep 17 2023

More terms from Alois P. Heinz, Sep 12 2023

cp's OEIS Frontend

A325992 Heinz numbers of integer partitions such that not every ordered pair of distinct parts has a different difference.

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A325855 Number of strict integer partitions of n such that every pair of distinct parts has a different product.

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A325857 Number of integer partitions of n such that every orderless pair of distinct parts has a different sum.

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A325868 Number of subsets of {1..n} containing n such that every ordered pair of distinct elements has a different quotient.

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A325880 Number of maximal subsets of {1..n} containing n such that every ordered pair of distinct elements has a different difference.

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A325869 Number of maximal subsets of {1..n} containing n such that every pair of distinct elements has a different quotient.

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A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.

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A325993 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different product.

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A326035 Number of uniform knapsack partitions of n.

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