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.

Showing 1-6 of 6 results.

A381542 Numbers > 1 whose greatest prime index equals their greatest prime multiplicity.

Original entry on oeis.org

2, 9, 12, 18, 36, 40, 112, 120, 125, 135, 200, 250, 270, 336, 352, 360, 375, 500, 540, 560, 567, 600, 675, 750, 784, 832, 1000, 1008, 1056, 1080, 1125, 1134, 1350, 1500, 1680, 1760, 1800, 2176, 2250, 2268, 2352, 2401, 2464, 2496, 2673, 2700, 2800, 2835, 3000
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}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   125: {3,3,3}
   135: {2,2,2,3}
   200: {1,1,1,3,3}
   250: {1,3,3,3}
   270: {1,2,2,2,3}
   336: {1,1,1,1,2,4}
   352: {1,1,1,1,1,5}
   360: {1,1,1,2,2,3}

Crossrefs

Counting partitions by the LHS gives A008284, rank statistic A061395.
Counting partitions by the RHS gives A091602, rank statistic A051903.
For length instead of maximum we have A106529, counted by A047993 (balanced partitions).
For number of distinct factors instead of max index we have A212166, counted by A239964.
Partitions of this type are counted by A240312.
Including number of distinct parts gives A381543, counted by A382302.
A000005 counts divisors.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
A381544 counts partitions without more ones than any other part, ranks A381439.
Cf. A000720, A047966, A048767, A130091, A317090, A381632.

Programs

  • Mathematica
    Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==Max@@FactorInteger[#][[All,2]]&]

Formula

A061395(a(n)) = A051903(a(n)).

A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 2, 2, 4, 3, 3, 4, 4, 3, 6, 5, 8, 8, 13, 13, 16, 17, 21, 22, 25, 26, 32, 34, 37, 44, 47, 55, 62, 72, 78, 94, 103, 118, 132, 151, 163, 189, 205, 230, 251, 284, 307, 346, 377, 420, 462, 515, 562, 629, 690, 763
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2025

Keywords

Examples

			The a(n) partitions for n = 1, 2, 10, 13, 14, 19, 20, 21:
  1  .  32221   332221   333221   4333321     43333211    43333221
        322111  333211   3322211  43322221    44322221    433332111
                3322111  3332111  433321111   433222211   443222211
                4321111           443221111   443321111   444321111
                                  543211111   4332221111  4332222111
                                  4322221111              4333221111
                                                          4432221111
                                                          5432211111

Links

Crossrefs

Without the middle statistic we have A000009, ranked by A055932.
Counting partitions by the LHS gives A008284 (strict A008289), rank statistic A061395.
Counting partitions by the middle statistic gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116608/A365676, rank statistic A001221.
Without the LHS we have A239964, ranked by A212166.
Without the RHS we have A240312, ranked by A381542.
The Heinz numbers of these partitions are listed by A381543.
A000041 counts integer partitions.
A047993 counts partitions with max part = length, ranks A106529.
A116598 counts ones in partitions, rank statistic A007814.
A381438 counts partitions by last part part of section-sum partition.
Cf. A047966, A130091, A237984, A239455, A241131, A351293, A362608, A363719, A381079, A381544, A382303.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#==Max@@Length/@Split[#]==Length[Union[#]]&]],{n,0,30}]
  • PARI
    A_x(N) = {if(N<1,[0],my(x='x+O('x^(N+1))); concat([0],Vec(sum(i=1,N, prod(j=1,i, (x^j-x^((i+1)*j))/(1-x^j)) - prod(j=1,i, (x^j-x^(i*j))/(1-x^j))))))}
A_x(60) \\ John Tyler Rascoe, Mar 25 2025

Formula

G.f.: Sum_{i>0} (B(i+1,i,x) - B(i,i,x)) where B(a,c,x) = Product_{j=1..c} (x^j - x^(a*j))/(1 - x^j). - John Tyler Rascoe, Mar 25 2025

A381544 Number of integer partitions of n not containing more ones than any other part.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2025

Keywords

Examples

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)

Crossrefs

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A381439.
The case of equality is 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]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

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]
Showing 1-6 of 6 results.

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