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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A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

Original entry on oeis.org

2, 9, 24, 54, 72, 80, 108, 125, 216, 224, 400, 704, 960, 1215, 1250, 1568, 1664, 2000, 2401, 2500, 2688, 2880, 4352, 4800, 5000, 5103, 6075, 7290, 7744, 8064, 8448, 8640, 8960, 9375, 9728, 10000, 10976, 14400, 14580, 18816, 19968, 21632, 23552, 24000, 24057
Offset: 1

Views

Author

Gus Wiseman, Mar 24 2025

Keywords

Comments

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.

Examples

			The terms together with their prime indices begin:
      2: {1}
      9: {2,2}
     24: {1,1,1,2}
     54: {1,2,2,2}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    108: {1,1,2,2,2}
    125: {3,3,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    400: {1,1,1,1,3,3}
    704: {1,1,1,1,1,1,5}
    960: {1,1,1,1,1,1,2,3}

Crossrefs

For (length) instead of (sum of distinct) we have A000961.
Including number of parts gives A062457 (degenerate).
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
Partitions of this type are counted by A381079.
A001222 counts prime factors, distinct A001221.
A047993 counts partitions with max part = length, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, complement A351293.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A000720, A048767, A130091, A246655, A317090, A380955, A381437, A381439.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Last/@FactorInteger[#]==Total[Union[prix[#]]]&]

Formula

A051903(a(n)) = A066328(a(n)).

A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 8, 6, 15, 13, 19, 25, 33, 36, 54, 58, 80, 96, 122, 141, 188, 217, 274, 326, 408, 474, 600, 695, 859, 1012, 1233, 1440, 1763, 2050, 2475, 2899, 3476, 4045, 4850, 5630, 6695, 7797, 9216, 10689, 12628, 14611, 17162, 19875, 23253
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2025

Keywords

Examples

			The a(3) = 1 through a(10) = 8 partitions:
  (21)  (31)  (41)  (51)    (61)   (71)    (81)      (91)
                    (321)   (421)  (431)   (531)     (541)
                    (2211)         (521)   (621)     (631)
                                   (3311)  (32211)   (721)
                                           (222111)  (4321)
                                                     (4411)
                                                     (33211)
                                                     (42211)

Crossrefs

First differences of A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A360014.
Equal case of A381544 (ranks A381439).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A051904, A091605, A116861, A237984, A239455, A362608, A363724, A381079, A381437, A381438.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,1]==Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A382526 Number of integer partitions of n with fewer ones than greatest multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 41, 56, 72, 94, 124, 158, 205, 262, 331, 419, 531, 663, 829, 1033, 1281, 1581, 1954, 2393, 2936, 3584, 4366, 5300, 6433, 7764, 9374, 11277, 13548, 16225, 19425, 23166, 27623, 32842, 39004, 46212, 54719, 64610, 76251
Offset: 0

Views

Author

Gus Wiseman, Apr 05 2025

Keywords

Examples

			The a(2) = 1 through a(9) = 12 partitions:
  (2)  (3)  (4)   (5)    (6)    (7)     (8)      (9)
            (22)  (32)   (33)   (43)    (44)     (54)
                  (221)  (42)   (52)    (53)     (63)
                         (222)  (322)   (62)     (72)
                                (331)   (332)    (333)
                                (2221)  (422)    (432)
                                        (2222)   (441)
                                        (3221)   (522)
                                        (22211)  (3222)
                                                 (3321)
                                                 (4221)
                                                 (22221)

Crossrefs

The complement (greater than or equal to) is A241131 except first, ranks A360015.
The opposite version (greater than) is A241131 shifted except first, ranks A360013.
These partitions have ranks A382856, complement A360015.
The weak version (less than or equal to) is A381544, ranks A381439.
For equality we have A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,1]

A382856 Numbers whose prime indices do not have a mode of 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 108, 109, 111, 113, 115
Offset: 1

Views

Author

Gus Wiseman, Apr 07 2025

Keywords

Examples

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}

Crossrefs

The case of non-unique mode is A024556.
The complement is A360015 except first.
Partitions of this type are are counted by A382526 except first, complement A241131.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A112798 lists prime indices, length A001222, sum A056239.
A116598 counts ones in partitions, rank statistic A007814.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A362611 counts modes in prime indices, triangle A362614.
For co-mode see A359178, A362613, A364061 (A364062), A364158 (A364159).
Cf. A000265, A002865, A106529, A327473, A327476, A362605, A363486, A356862, A360013, A360014, A381437.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],FreeQ[Commonest[prix[#]],1]&]
Previous Showing 11-14 of 14 results.

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