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.

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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}]]]]]&]

A316362 Heinz numbers of strict integer partitions such that not every distinct subset has a different average.

Original entry on oeis.org

30, 105, 110, 210, 238, 273, 330, 385, 390, 462, 506, 510, 546, 570, 627, 690, 714, 770, 806, 858, 870, 910, 930, 935, 966, 1001, 1110, 1131, 1155, 1190, 1230, 1254, 1290, 1326, 1330, 1365, 1394, 1410, 1430, 1482, 1495, 1518, 1590, 1729, 1770, 1785, 1786, 1794
Offset: 1

Views

Author

Gus Wiseman, Jun 30 2018

Keywords

Comments

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

Examples

			462 is the Heinz number of (5,4,2,1), and the subsets {1,5}, and {2,4} have the same average, so 462 belongs to the sequence.

Crossrefs

Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299702, A301899, A316313, A316314, A316361.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[3000],SquareFreeQ[#]&&!UnsameQ@@Mean/@Union[Subsets[primeMS[#]]]&]

A316465 Heinz numbers of integer partitions such that every nonempty submultiset has an integer average.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 29, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 79, 81, 82, 83, 85, 87, 89, 91, 94, 97, 101, 103, 107, 109, 110, 111, 113, 115, 118, 121, 125, 127, 128
Offset: 1

Views

Author

Gus Wiseman, Jul 06 2018

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Supersequence of A000961. - David A. Corneth, Jul 06 2018

Examples

			Sequence of partitions begins (), (1), (2), (1,1), (3), (4), (1,1,1), (2,2), (3,1), (5), (6), (1,1,1,1), (7), (8), (4,2), (5,1), (9), (3,3), (2,2,2).

Crossrefs

Cf. A056239, A067538, A122768, A237984, A296150, A316313, A316314, A316440, A316557.

Programs

  • Mathematica
    Select[Range[100],And@@IntegerQ/@Mean/@Union[Rest[Subsets[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]&]

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).

A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2023

Keywords

Comments

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

Examples

			Triangle begins:
  1
  1  1
  1  2
  1  3  1
  1  3  3
  1  5  3  2
  1  4  7  2  1
  1  6  7  6  2
  1  6 10  6  7
  1  7 12 11  8  3
  1  6 16 11 17  3  2
  1 10 14 20 19 10  2  1
  1  7 22 17 31 14  7  2
  1  9 22 27 37 22 11  6
  1 10 24 27 51 32 16 15
  1 11 27 39 57 43 27 22  4
  1  9 33 34 79 57 36 39  7  2
  1 13 31 51 86 77 45 62 14  4  1
Row n = 9 counts the following partitions:
  (9)  (81)         (711)       (621)      (5211)
       (72)         (6111)      (531)      (4311)
       (63)         (522)       (432)      (4221)
       (54)         (51111)     (33111)    (42111)
       (333)        (441)       (222111)   (3321)
       (111111111)  (411111)    (2211111)  (32211)
                    (3222)                 (321111)
                    (3111111)
                    (22221)
                    (21111111)

Crossrefs

Row sums are A000041.
Column k = 1 is A088922.
The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366738 counts semi-sums of partitions, non-binary A304792.
A366741 counts semi-sums of strict partitions, non-binary A365925.
Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

Programs

  • Mathematica
    DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]
Previous Showing 51-55 of 55 results.

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