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 1-8 of 8 results.

A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 6, 4, 2, 2, 3, 2, 2, 2, 8, 2, 3, 2, 3, 2, 2, 2, 4, 4, 2, 6, 3, 2, 2, 2, 10, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 5, 4, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 3, 12, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 5, 8, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Feb 14 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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

Examples

			The unordered prime signature of 2520 is {1,1,2,3}, with median 3/2, so a(2520) = 3.

Crossrefs

The version for divisors is A063655.
For mean instead of two times median we have A088529/A088530.
Prime signature is A124010, unordered A118914.
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for distinct prime factors is A360458.
The version for prime factors is A360459.
Positions of even terms are A360553.
Positions of odd terms are A360554.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices.
A325347 counts partitions w/ integer median, complement A307683.
A329976 counts partitions with median multiplicity 1.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A133464, A359907, A359908, A360454.

Programs

  • Mathematica
    Table[If[n==1,1,2*Median[Last/@FactorInteger[n]]],{n,100}]

A360687 Number of integer partitions of n whose multiplicities have integer median.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676
Offset: 1

Views

Author

Gus Wiseman, Feb 20 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).

Examples

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).

Crossrefs

The case of an odd number of multiplicities is A090794.
For mean instead of median we have A360069, ranks A067340.
These partitions have ranks A360553.
The complement is counted by A360690, ranks A360554.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A241274 Number of partitions p of n such that (number of numbers in p that have multiplicity 1) = (number of numbers in p having multiplicity > 1).

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 3, 7, 7, 10, 12, 16, 14, 23, 25, 33, 41, 59, 72, 101, 126, 171, 216, 280, 344, 436, 535, 666, 788, 970, 1153, 1394, 1649, 1996, 2336, 2796, 3326, 3965, 4689, 5627, 6629, 7926, 9404, 11192, 13273, 15777, 18637, 22057, 26067, 30672, 36122
Offset: 0

Views

Author

Clark Kimberling, Apr 24 2014

Keywords

Examples

			a(6) counts these 3 partitions:  411, 3111, 21111.

Crossrefs

Cf. A241090, A241131, A241132.

Programs

  • Mathematica
    z = 30; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; u[p] == e[p]], {n, 0, z}]

Formula

a(n) + A329976(n) + A330001(n) = A000041(n) for n >= 0.

A330001 Number of partitions p of n such that (number of numbers in p that have multiplicity 1) < (number of numbers in p having multiplicity > 1).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 2, 6, 6, 12, 13, 25, 28, 44, 54, 77, 93, 127, 155, 204, 247, 318, 390, 494, 610, 761, 937, 1172, 1442, 1783, 2194, 2693, 3292, 4028, 4917, 5946, 7221, 8700, 10490, 12584, 15106, 18004, 21523, 25537, 30399, 35945, 42635, 50219, 59382
Offset: 0

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d < r.

Examples

			The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r:  6, 51, 42, 321
These have d = r:  411, 3222, 21111
These have d < r:  33, 222, 2211, 111111
Thus, a(6) = 4.

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] <  r[p]], {n, 0, z}]

Formula

a(n) + A241274(n) + A329976(n) = A000041(n) for n >= 0.

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465
Offset: 1

Views

Author

Gus Wiseman, Feb 22 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).

Examples

			The a(1) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

Crossrefs

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A330145 Number of partitions p of n such that (number of numbers in p that have multiplicity 1) >= (number of numbers in p having multiplicity > 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 7, 13, 16, 24, 30, 43, 52, 73, 91, 122, 154, 204, 258, 335, 423, 545, 684, 865, 1081, 1348, 1675, 2073, 2546, 3123, 3821, 4648, 5656, 6851, 8282, 9966, 12031, 14416, 17315, 20695, 24754, 29477, 35170, 41738, 49638, 58735, 69613, 82119
Offset: 0

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d >= r.

Examples

			The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r:  6, 51, 42, 321
These have d = r:  411, 3222, 21111
These have d < r:  33, 222, 2211, 111111
Thus, a(6) = 7

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] >=  r[p]], {n, 0, z}]

Formula

a(n) + A330001(n) = A000041(n) for all n >= 0.

A330146 Number of partitions p of n such that (number of numbers in p that have multiplicity 1) <= (number of numbers in p having multiplicity > 1).

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 7, 9, 13, 16, 24, 29, 39, 51, 69, 87, 118, 152, 199, 256, 330, 418, 534, 670, 838, 1046, 1296, 1603, 1960, 2412, 2936, 3588, 4342, 5288, 6364, 7713, 9272, 11186, 13389, 16117, 19213, 23032, 27408, 32715, 38810, 46176, 54582, 64692, 76286
Offset: 0

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d <= r.

Examples

			The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r:  6, 51, 42, 321
These have d = r:  411, 3222, 21111
These have d < r:  33, 222, 2211, 111111
Thus, a(6) = 7

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] <=  r[p]], {n, 0, z}]

Formula

a(n) + A329976(n) = A000041(n) for all n >= 0.

A330147 Number of partitions p of n such that (number of numbers in p that have multiplicity 1) != (number of numbers in p having multiplicity > 1).

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 8, 8, 15, 20, 30, 40, 63, 78, 110, 143, 190, 238, 313, 389, 501, 621, 786, 975, 1231, 1522, 1901, 2344, 2930, 3595, 4451, 5448, 6700, 8147, 9974, 12087, 14651, 17672, 21326, 25558, 30709, 36657, 43770, 52069, 61902, 73357, 86921, 102697
Offset: 0

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d != r.

Examples

			The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r:  6, 51, 42, 321
These have d = r:  411, 3222, 21111
These have d < r:  33, 222, 2211, 111111
Thus, a(6) = 8

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[ Count[IntegerPartitions[n], p_ /; d[p] != r[p]], {n, 0, z}]

Formula

a(n) + A241274(n) = A000041(n) for all n >= 0.
Showing 1-8 of 8 results.

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

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