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 1-4 of 4 results.

A237821 Number of partitions of n such that 2*(least part) <= greatest part.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 16, 25, 35, 48, 68, 92, 123, 164, 216, 282, 367, 471, 604, 769, 975, 1225, 1542, 1924, 2395, 2968, 3669, 4514, 5547, 6781, 8280, 10071, 12229, 14796, 17881, 21537, 25902, 31066, 37206, 44443, 53021, 63098, 74995, 88946, 105350, 124533
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2014

Keywords

Comments

By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023

Examples

			a(6) = 7 counts these partitions:  51, 42, 411, 321, 3111, 2211, 21111.
From _Gus Wiseman_, May 15 2023: (Start)
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
  (21)  (31)   (41)    (42)     (52)
        (211)  (221)   (51)     (61)
               (311)   (321)    (331)
               (2111)  (411)    (421)
                       (2211)   (511)
                       (3111)   (2221)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
  (21)  (31)   (32)    (42)     (43)
        (211)  (41)    (51)     (52)
               (311)   (321)    (61)
               (2111)  (411)    (322)
                       (2211)   (421)
                       (3111)   (511)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
(End)

Crossrefs

The complement is counted by A053263, ranks A081306.
These partitions have ranks A069900.
The case of equality is A118096.
For < instead of <= we have A237820, ranks A362982.
For >= instead of <= we have A237824, ranks A362981.
The conjugate partitions have ranks A362980.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.

Programs

  • Mathematica
    z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)

Formula

G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - Seiichi Manyama, May 27 2023

A362621 One and numbers whose multiset of prime factors (with multiplicity) has the same median as maximum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169
Offset: 1

Views

Author

Gus Wiseman, May 12 2023

Keywords

Comments

First differs from A334965 in having 750 and lacking 2250.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime factorization of 108 is 2*2*3*3*3, and the multiset {2,2,3,3,3} has median 3 and maximum 3, so 108 is in the sequence.
The prime factorization of 2250 is 2*3*3*5*5*5, and the multiset {2,3,3,5,5,5} has median 4 and maximum 5, so 2250 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           25: {3,3}           64: {1,1,1,1,1,1}
     2: {1}          27: {2,2,2}         67: {19}
     3: {2}          29: {10}            71: {20}
     4: {1,1}        31: {11}            73: {21}
     5: {3}          32: {1,1,1,1,1}     75: {2,3,3}
     7: {4}          37: {12}            79: {22}
     8: {1,1,1}      41: {13}            81: {2,2,2,2}
     9: {2,2}        43: {14}            83: {23}
    11: {5}          47: {15}            89: {24}
    13: {6}          49: {4,4}           97: {25}
    16: {1,1,1,1}    50: {1,3,3}         98: {1,4,4}
    17: {7}          53: {16}           101: {26}
    18: {1,2,2}      54: {1,2,2,2}      103: {27}
    19: {8}          59: {17}           107: {28}
    23: {9}          61: {18}           108: {1,1,2,2,2}

Crossrefs

Partitions of this type are counted by A053263.
For mode instead of median we have A362619, counted by A171979.
For parts at middle position (instead of median) we have A362622.
The complement is A362980, counted by A237821.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
Cf. A000040, A002865, A237824, A327473, A327476, A359908, A362616, A362620.

Programs

  • Mathematica
    Select[Range[100],(y=Flatten[Apply[ConstantArray,FactorInteger[#],{1}]];Max@@y==Median[y])&]

A362622 One and numbers whose prime factorization has its greatest part at a middle position.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91
Offset: 1

Views

Author

Gus Wiseman, May 12 2023

Keywords

Examples

			The prime factorization of 150 is 5*5*3*2, with middle parts {3,5}, so 150 is in the sequence.
The prime factorization of 90 is 5*3*3*2, with middle parts {3,3}, so 90 is not in the sequence.

Crossrefs

Partitions of this type are counted by A237824.
For modes instead of middles we have A362619, counted by A171979.
The version for median instead of middles is A362621, counted by A053263.
The complement for median is A362980, counted by A237821.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization.
A362613 counts co-modes in prime factorization.
Cf. A000040, A002865, A237984, A327473, A327476, A359908, A362616, A362620.

Programs

  • Mathematica
    mpm[q_]:=MemberQ[If[OddQ[Length[q]],{Median[q]},{q[[Length[q]/2]],q[[Length[q]/2+1]]}],Max@@q];
Select[Range[100],#==1||mpm[Flatten[Apply[ConstantArray,FactorInteger[#],{1}]]]&]

A363223 Numbers with bigomega equal to median prime index.

Original entry on oeis.org

2, 9, 10, 50, 70, 75, 105, 110, 125, 130, 165, 170, 175, 190, 195, 230, 255, 275, 285, 290, 310, 325, 345, 370, 410, 425, 430, 435, 465, 470, 475, 530, 555, 575, 590, 610, 615, 645, 670, 686, 705, 710, 725, 730, 775, 790, 795, 830, 885, 890, 915, 925, 970
Offset: 1

Views

Author

Gus Wiseman, May 29 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
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.

Examples

			The terms together with their prime indices begin:
    2: {1}
    9: {2,2}
   10: {1,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  130: {1,3,6}
  165: {2,3,5}
  170: {1,3,7}
  175: {3,3,4}

Crossrefs

For maximum instead of median we have A106529, counted by A047993.
For minimum instead of median we have A324522, counted by A006141.
Partitions of this type are counted by A361800.
For twice median we have A362050, counted by A362049.
For maximum instead of length we have A362621, counted by A053263.
A000975 counts subsets with integer median.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A359908 lists numbers whose prime indices have integer median.
A360005 gives twice median of prime indices.
Cf. A000040, A013580, A079309, A240219, A327473, A327476, A361860, A362619, A362622, A362980.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],PrimeOmega[#]==Median[prix[#]]&]

Formula

2*A001222(a(n)) = A360005(a(n)).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.