A143823 -id:A143823 - cp's OEIS frontend

A325684 Number of minimal complete rulers of length n.

1, 1, 1, 2, 3, 4, 5, 12, 12, 24, 40, 46, 92, 133, 192, 308, 546, 710, 1108, 1754, 2726, 3878, 5928, 9260, 14238, 20502, 30812, 48378, 72232, 105744, 160308, 241592, 362348, 540362, 797750, 1183984, 1786714

Offset: 0

Gus Wiseman, May 13 2019

A complete ruler of length n is a subset of {0..n} containing 0 and n and such that the differences of distinct terms (up to sign) cover an initial interval of positive integers.

Also the number of maximal (most coarse) compositions of n whose consecutive subsequence-sums cover an initial interval of positive integers.

			The a(1) = 1 through a(7) = 12 rulers:
  {0,1}  {0,1,2}  {0,1,3}  {0,1,2,4}  {0,1,2,5}  {0,1,4,6}    {0,1,2,3,7}
                  {0,2,3}  {0,1,3,4}  {0,1,3,5}  {0,2,5,6}    {0,1,2,4,7}
                           {0,2,3,4}  {0,2,4,5}  {0,1,2,3,6}  {0,1,2,5,7}
                                      {0,3,4,5}  {0,1,3,5,6}  {0,1,3,5,7}
                                                 {0,3,4,5,6}  {0,1,3,6,7}
                                                              {0,1,4,5,7}
                                                              {0,1,4,6,7}
                                                              {0,2,3,6,7}
                                                              {0,2,4,6,7}
                                                              {0,2,5,6,7}
                                                              {0,3,5,6,7}
                                                              {0,4,5,6,7}
The a(1) = 1 through a(9) = 24 compositions:
  (1)  (11)  (12)  (112)  (113)  (132)   (1114)  (1133)   (1143)
             (21)  (121)  (122)  (231)   (1123)  (1241)   (1332)
                   (211)  (221)  (1113)  (1132)  (1322)   (2331)
                          (311)  (1221)  (1222)  (1412)   (3411)
                                 (3111)  (1231)  (1421)   (11115)
                                         (1312)  (2141)   (11124)
                                         (1321)  (2231)   (11142)
                                         (2131)  (3311)   (11241)
                                         (2221)  (11114)  (11322)
                                         (2311)  (11132)  (12141)
                                         (3211)  (23111)  (12222)
                                         (4111)  (41111)  (12231)
                                                          (12312)
                                                          (13221)
                                                          (14112)
                                                          (14121)
                                                          (14211)
                                                          (21141)
                                                          (21321)
                                                          (22221)
                                                          (22311)
                                                          (24111)
                                                          (42111)
                                                          (51111)

Cf. A000079, A103295, A126796, A143823, A169942, A325677, A325683, A325685.

fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Accumulate/@Select[Join@@Permutations/@IntegerPartitions[n],SubsetQ[ReplaceList[#,{_,s__,_}:>Plus[s]],Range[n]]&]]],{n,0,15}]

a(16)-a(36) from Fausto A. C. Cariboni, Feb 27 2022

A325778 Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, May 20 2019

nonn

First differs from A299702 in having 462.

The enumeration of these partitions by sum is given by A325769.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Complement of A325777.

Cf. A002033, A056239, A112798, A143823, A169942, A299702, A301899, A325676, A325768, A325769, A325770, A325779.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{_,s__,_}:>{s}]]&]

A325854 Number of strict integer partitions of n such that every pair of distinct parts has a different quotient.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 6, 8, 9, 12, 13, 16, 20, 23, 30, 33, 41, 47, 52, 61, 75, 90, 98, 116, 132, 151, 173, 206, 226, 263, 297, 337, 387, 427, 488, 555, 623, 697, 782, 886, 984, 1108, 1240, 1374, 1545, 1726, 1910, 2120, 2358, 2614, 2903, 3218, 3567, 3933

Offset: 0

Gus Wiseman, May 31 2019

nonn

Also the number of strict integer partitions of n such that every pair of (not necessarily distinct) parts has a different product.

			The a(1) = 1 through a(10) = 9 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)
                                           (431)  (81)   (91)
                                           (521)  (432)  (532)
                                                  (531)  (541)
                                                  (621)  (631)
                                                         (721)
The two strict partitions of 13 such that not every pair of distinct parts has a different quotient are (9,3,1) and (6,4,2,1).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

The subset case is A325860.
The maximal case is A325861.
The integer partition case is A325853.
The strict integer partition case is A325854.
Heinz numbers of the counterexamples are given by A325994.
Cf. A108917, A143823, A196724, A275972, A325768, A325855, A325858, A325868, A325869, A325876, A325877.

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

A325856 Number of integer partitions of n such that every pair of distinct parts has a different product.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 171, 225, 287, 369, 467, 592, 740, 931, 1155, 1435, 1767, 2178, 2661, 3254, 3953, 4798, 5793, 6991, 8390, 10069, 12022, 14346, 17054, 20255, 23960, 28334, 33390, 39308, 46148, 54116, 63295, 73967, 86224

Offset: 0

Gus Wiseman, May 31 2019

nonn

			The five partitions of 15 not satisfying the condition are:
  (8,4,2,1)
  (6,4,3,2)
  (6,3,3,2,1)
  (6,3,2,2,1,1)
  (6,3,2,1,1,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..150

The subset case is A196724.
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Cf. A002033, A108917, A143823, A325853, A325854, A325857, A325858, A325877.

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

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

Original entry on oeis.org

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

A325865 Number of maximal subsets of {1..n} of which every subset has a different sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 14, 23, 27, 40, 64, 104, 180, 275, 399, 554, 679, 872, 1117, 1431, 1920, 2520, 3530, 4751, 6644, 8855, 12021, 15461, 19939, 25109, 31656, 38750, 46204, 55650, 65942, 78045, 91304, 106592, 124761, 145701, 172343, 201217, 238739, 280601, 339746, 400394

Offset: 0

Gus Wiseman, Jun 01 2019

nonn

			The a(1) = 1 through a(6) = 14 subsets:
  {1}  {1,2}  {1,2}  {1,3}    {1,2,4}  {1,2,4}
              {1,3}  {1,2,4}  {1,2,5}  {1,2,5}
              {2,3}  {2,3,4}  {1,3,5}  {1,2,6}
                              {2,3,4}  {1,3,5}
                              {2,4,5}  {1,3,6}
                              {3,4,5}  {1,4,6}
                                       {2,3,4}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,5}
                                       {3,4,6}
                                       {3,5,6}
                                       {4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..60

Cf. A002033, A108917, A143823, A196723, A275972.

Cf. A325860, A325864, A325866, A325867, A325877, A325878, A325879, A325880.

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

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

a(18) onwards from Andrew Howroyd, Mar 23 2025

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

cp's OEIS Frontend

A325684 Number of minimal complete rulers of length n.

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A325778 Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.

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A325854 Number of strict integer partitions of n such that every pair of distinct parts has a different quotient.

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A325856 Number of integer partitions of n such that every pair of distinct parts has a different product.

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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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A325865 Number of maximal subsets of {1..n} of which every subset 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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