cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-34 of 34 results.

A325777 Heinz numbers of integer partitions whose distinct consecutive subsequences do not have different sums.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330
Offset: 1

Views

Author

Gus Wiseman, May 20 2019

Keywords

Comments

First differs from A299729 in lacking 462.
This sequence does not contain all multiples of its elements. For example, it contains 154 (with prime indices {1,4,5}) but not 462 (with prime indices {1,2,4,5}).

Crossrefs

Complement of A325778.
Cf. A002033, A056239, A112798, A143823, A169942, A299702, A301899, A325676, A325683, A325768, A325769, A325770.

Programs

  • Mathematica
    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}]]&]

A305613 Numbers whose multiset of prime factors is not knapsack.

Original entry on oeis.org

30, 60, 70, 72, 84, 90, 120, 140, 144, 150, 168, 180, 210, 216, 240, 252, 270, 280, 286, 288, 300, 308, 330, 336, 350, 360, 378, 390, 420, 432, 440, 450, 480, 490, 495, 504, 510, 525, 528, 540, 560, 570, 572, 576, 588, 594, 600, 616, 630, 646, 648, 660, 672
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2018

Keywords

Comments

A multiset of positive integers is knapsack if every distinct submultiset has a different sum.

Examples

			30 = 2 * 3 * 5 is not knapsack because 2 + 3 = 5.

Crossrefs

Cf. A027746, A108917, A122768, A275972, A276024, A299701, A299729, A301855, A301900, A304793, A305611.

Programs

  • Mathematica
    Select[Range[1000],DivisorSigma[0,#]=!=Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[#],{p_,k_}:>Table[p,{k}]]]]]&]

A326019 Heinz numbers of non-knapsack partitions such that every non-singleton submultiset has a different sum.

Original entry on oeis.org

12, 30, 40, 63, 70, 112, 154, 165, 198, 220, 273, 286, 325, 351, 352, 364, 442, 561, 595, 646, 714, 741, 748, 765, 832, 850, 874, 918, 931, 952, 988, 1045, 1173, 1254, 1334, 1425, 1495, 1539, 1564, 1653, 1672, 1771, 1794, 1798, 1900, 2139, 2176, 2204, 2254
Offset: 1

Views

Author

Gus Wiseman, Jun 03 2019

Keywords

Comments

A subsequence of A299729.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition is knapsack if every distinct submultiset has a different sum.

Examples

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   30: {1,2,3}
   40: {1,1,1,3}
   63: {2,2,4}
   70: {1,3,4}
  112: {1,1,1,1,4}
  154: {1,4,5}
  165: {2,3,5}
  198: {1,2,2,5}
  220: {1,1,3,5}
  273: {2,4,6}
  286: {1,5,6}
  325: {3,3,6}
  351: {2,2,2,6}
  352: {1,1,1,1,1,5}
  364: {1,1,4,6}
  442: {1,6,7}
  561: {2,5,7}
  595: {3,4,7}
  646: {1,7,8}

Crossrefs

Cf. A108917, A275972, A299702, A299729, A304793.
Cf. A325780, A325862, A325863, A326016, A326018.

Programs

  • Mathematica
    hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],!UnsameQ@@hwt/@Divisors[#]&&UnsameQ@@hwt/@Select[Divisors[#],!PrimeQ[#]&]&]

A367108 Triangle read by rows where T(n,k) is the number of integer partitions of n with a unique submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 2, 3, 5, 7, 5, 4, 4, 5, 7, 11, 7, 6, 3, 6, 7, 11, 15, 11, 8, 7, 7, 8, 11, 15, 22, 15, 12, 10, 4, 10, 12, 15, 22, 30, 22, 16, 14, 12, 12, 14, 16, 22, 30, 42, 30, 22, 17, 17, 6, 17, 17, 22, 30, 42, 56, 42, 30, 25, 23, 20, 20, 23, 25, 30, 42, 56
Offset: 1

Views

Author

Gus Wiseman, Nov 18 2023

Keywords

Examples

			Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   3
   5   3   2   3   5
   7   5   4   4   5   7
  11   7   6   3   6   7  11
  15  11   8   7   7   8  11  15
  22  15  12  10   4  10  12  15  22
  30  22  16  14  12  12  14  16  22  30
  42  30  22  17  17   6  17  17  22  30  42
  56  42  30  25  23  20  20  23  25  30  42  56
  77  56  40  31  30  27   7  27  30  31  40  56  77
Row n = 5 counts the following partitions:
  (5)      (41)     (32)     (32)     (41)     (5)
  (41)     (311)    (311)    (311)    (311)    (41)
  (32)     (221)    (221)    (221)    (221)    (32)
  (311)    (2111)   (11111)  (11111)  (2111)   (311)
  (221)    (11111)                    (11111)  (221)
  (2111)                                       (2111)
  (11111)                                      (11111)
Row n = 6 counts the following partitions:
  (6)       (51)      (42)      (33)      (42)      (51)      (6)
  (51)      (411)     (411)     (2211)    (411)     (411)     (51)
  (42)      (321)     (321)     (111111)  (321)     (321)     (42)
  (411)     (3111)    (3111)              (3111)    (3111)    (411)
  (33)      (2211)    (222)               (222)     (2211)    (33)
  (321)     (21111)   (111111)            (111111)  (21111)   (321)
  (3111)    (111111)                                (111111)  (3111)
  (222)                                                       (222)
  (2211)                                                      (2211)
  (21111)                                                     (21111)
  (111111)                                                    (111111)

Crossrefs

Columns k = 0 and k = n are A000041(n).
Column k = 1 and k = n-1 are A000041(n-1).
Column k = 2 appears to be 2*A027336(n).
The version for non-subset-sums is A046663, strict A365663.
Diagonal n = 2k is A108917, complement A366754.
Row sums are A304796, non-unique version A304792.
The non-unique version is A365543.
Cf. A002219, A122768, A275972, A299702, A299729, A301854, A364272, A364911, A365658, A365661, A367094.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Count[Total/@Union[Subsets[#]], k]==1&]], {n,0,10}, {k,0,n}]

Formula

A367094(n,1) = A108917(n).
Previous Showing 31-34 of 34 results.

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