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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A360551 Numbers > 1 whose distinct prime indices have non-integer median.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196
Offset: 1

Views

Author

Gus Wiseman, Feb 16 2023

Keywords

Comments

First differs from A325700 in having 330 and lacking 462.
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. Distinct prime indices are listed by A304038.
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 indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.

Crossrefs

For mean instead of median we have the complement of A326621.
Positions of odd terms in A360457.
The complement (without 1) is A360550, counted by A360686.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices, length A001221, sum A066328.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A348551, A360006, A360009, A360248, A360453.

Programs

  • Mathematica
    Select[Range[2,100],!IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]

A360554 Numbers > 1 whose unordered prime signature has non-integer median.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332, 333
Offset: 1

Views

Author

Gus Wiseman, Feb 16 2023

Keywords

Comments

First differs from A187039 in having 2520 and lacking 1 and 12600.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
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 unordered prime signature of 2520 is {3,2,1,1}, with median 3/2, so 2520 is in the sequence.
The unordered prime signature of 12600 is {3,2,2,1}, with median 2, so 12600 is not in the sequence.

Crossrefs

A subset of A030231.
For mean instead of median we have A070011.
Positions of odd terms in A360460.
The complement is A360553 (without 1), counted by A360687.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A304038, A316413, A326621, A348551, A360006, A360009, A360248, A360453.

Programs

  • Mathematica
    Select[Range[2,100],!IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360557 Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

Original entry on oeis.org

4, 10, 15, 22, 24, 25, 33, 34, 36, 40, 46, 51, 54, 55, 56, 62, 69, 77, 82, 85, 88, 93, 94, 100, 104, 115, 118, 119, 121, 123, 134, 135, 136, 141, 146, 152, 155, 161, 166, 177, 184, 187, 194, 196, 201, 205, 206, 217, 218, 219, 220, 221, 225, 232, 235, 240, 248
Offset: 1

Views

Author

Gus Wiseman, Feb 17 2023

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.
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 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is in the sequence.

Crossrefs

For mean instead of median complement we have A340610, counted by A168659.
For mean instead of median we have A360668, counted by A200727.
Positions of odd terms in A360555.
The complement is A360556 (without 1), counted by A360688.
These partitions are counted by A360691.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551, complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A287352 lists 0-prepended first differences of prime indices.
A325347 counts partitions with integer median, complement A307683.
A355536 lists first differences of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A000975, A026424, A078175, A316413, A360009, A360558, A360669, A360681.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],!IntegerQ[Median[Differences[Prepend[prix[#],0]]]]&]

A361858 Number of integer partitions of n such that the maximum is less than twice the median.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 8, 12, 15, 19, 22, 31, 34, 45, 55, 67, 78, 100, 115, 144, 170, 203, 238, 291, 337, 403, 473, 560, 650, 772, 889, 1046, 1213, 1414, 1635, 1906, 2186, 2533, 2913, 3361, 3847, 4433, 5060, 5808, 6628, 7572, 8615, 9835, 11158, 12698, 14394
Offset: 1

Views

Author

Gus Wiseman, Apr 02 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) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (322)      (71)
                                     (321)     (331)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
The partition y = (3,2,2,1) has maximum 3 and median 2, and 3 < 2*2, so y is counted under a(8).

Crossrefs

For minimum instead of median we have A053263.
For length instead of median we have A237754.
Allowing equality gives A361848, strict A361850.
The equal version is A361849, ranks A361856.
For mean instead of median we have A361852.
Reversing the inequality gives A361857, ranks A361867.
The complement is counted by A361859, ranks A361868.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A027193, A237751, A237755, A237820, A237824, A240219, A361394, A361851, A361860, A361907.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#<2*Median[#]&]],{n,30}]

A361860 Number of integer partitions of n whose median part is the smallest.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 12, 15, 21, 25, 36, 44, 58, 72, 95, 117, 150, 185, 235, 289, 362, 441, 550, 670, 824, 1000, 1223, 1476, 1795, 2159, 2609, 3126, 3758, 4485, 5369, 6388, 7609, 9021, 10709, 12654, 14966, 17632, 20782, 24414, 28684, 33601, 39364, 45996
Offset: 1

Views

Author

Gus Wiseman, Apr 02 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) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

For mean instead of median we have A000005.
For length instead of median we have A006141.
For maximum instead of median we have A053263.
For half-median we have A361861.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A053263, A067659, A111907, A116608, A118096, A237753, A240219, A359907, A361848, A361849.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Min@@#==Median[#]&]],{n,30}]

A360007 Positions of first appearances in the sequence giving the median of the prime indices of n (A360005(n)/2).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 106, 107, 109, 113, 122, 127, 131, 137, 139, 142, 149, 151, 157, 158, 163, 167, 173, 178, 179, 181, 191, 193, 197, 199, 202
Offset: 1

Views

Author

Gus Wiseman, Jan 24 2023

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

Crossrefs

Positions of first appearances in A360005.
The unsorted version is A360006.
For mean instead of median we have A360008.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 counts partitions by median, cf. A359901, A359902.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A359889, A359890, A360009.

Programs

  • Mathematica
    nn=1000;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,2*Median[prix[n]]],{n,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

Formula

Consists of 1, the primes, and all odd-indexed primes times 2.

A360069 Number of integer partitions of n whose multiset of multiplicities has integer mean.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 9, 13, 16, 25, 26, 39, 42, 62, 67, 95, 107, 147, 168, 225, 245, 327, 381, 471, 565, 703, 823, 1038, 1208, 1443, 1743, 2088, 2439, 2937, 3476, 4163, 4921, 5799, 6825, 8109, 9527, 11143, 13122, 15402, 17887, 20995, 24506, 28546, 33234, 38661
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 13 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)    (4111)     (521)
                                     (3111)    (211111)   (2222)
                                     (111111)  (1111111)  (3311)
                                                          (5111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example,  the partition (3,2,1,1,1,1) has multiplicities (1,1,4) with mean 2, so is counted under a(9). On the other hand, the partition (3,2,2,1,1) has multiplicities (1,2,2) with mean 5/3, so is not counted under a(9).

Crossrefs

These partitions are ranked by A067340 (prime signature has integer mean).
Parts instead of multiplicities: A067538, strict A102627, ranked by A316413.
The case where the parts have integer mean also is ranked by A359905.
A000041 counts integer partitions, strict A000009.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326622 counts factorizations with integer mean, strict A328966.
Cf. A082550, A240219, A316313, A325347, A326669, A327475, A349156, A360068.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[Length/@Split[#]]]&]],{n,0,30}]

A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 100, 112, 125, 180, 250, 252, 300, 352, 360, 392, 396, 405, 450, 468, 504, 540, 588, 600, 612, 675, 684, 720, 756, 792, 828, 832, 882, 900, 936, 1008, 1044, 1116, 1125, 1176, 1188, 1200, 1224, 1332, 1350, 1368, 1372, 1404, 1440, 1452, 1476
Offset: 1

Views

Author

Gus Wiseman, Feb 10 2023

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.
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 terms together with their prime indices begin:
    1: {}
    2: {1}
    9: {2,2}
   12: {1,1,2}
   18: {1,2,2}
   40: {1,1,1,3}
  100: {1,1,3,3}
  112: {1,1,1,1,4}
  125: {3,3,3}
  180: {1,1,2,2,3}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  300: {1,1,2,3,3}
  352: {1,1,1,1,1,5}
  360: {1,1,1,2,2,3}
For example, the prime indices of 756 are {1,1,2,2,2,4} with distinct parts {1,2,4} with median 2 and multiplicities {1,2,3} with median 2, so 756 is in the sequence.

Crossrefs

Without taking median we have A109298, unordered A109297.
For mean instead of median we have A324570, counted by A114638.
For indices instead of multiplicities we have A360249, counted by A360245.
For indices instead of distinct indices we have A360454, counted by A360456.
These partitions are counted by A360455.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranks A359889.
A316413 = numbers whose prime indices have integer mean, distinct A326621.
A325347 = partitions with integer median, strict A359907, ranks A359908.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median.
A360005 gives median of prime indices (times two).
Cf. A000975, A359890, A359903, A360068, A360244, A360247, A360248.

Programs

  • Mathematica
    Select[Range[100],#==1||Median[Last/@FactorInteger[#]]== Median[PrimePi/@First/@FactorInteger[#]]&]

A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 10, 15, 23, 34, 46, 67, 90, 121, 164, 219, 285, 375, 483, 622, 799, 1017, 1284, 1621, 2033, 2537, 3158, 3915, 4832, 5953, 7303, 8930, 10896, 13248, 16071, 19451, 23482, 28272, 33977, 40736, 48741, 58201, 69367, 82506, 97986, 116139
Offset: 1

Views

Author

Gus Wiseman, Apr 02 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(4) = 1 through a(9) = 15 partitions:
  (211)  (311)   (411)    (421)     (422)      (522)
         (2111)  (3111)   (511)     (521)      (621)
                 (21111)  (3211)    (611)      (711)
                          (4111)    (4211)     (4221)
                          (22111)   (5111)     (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (33111)
                                    (311111)   (42111)
                                    (2111111)  (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 >= 2*2, so y is counted under a(10).

Crossrefs

For length instead of median we have A237752.
For minimum instead of median we have A237821.
Reversing the inequality gives A361848.
The equal case is A361849, ranks A361856.
The unequal case is A361857, ranks A361867.
The complement is counted by A361858.
These partitions have ranks A361868.
For mean instead of median we have A361906.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A027193, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#>=2*Median[#]&]],{n,30}]

A359913 Numbers whose multiset of prime factors has integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81
Offset: 1

Views

Author

Gus Wiseman, Jan 25 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 terms together with their prime factors begin:
   2: {2}
   3: {3}
   4: {2,2}
   5: {5}
   7: {7}
   8: {2,2,2}
   9: {3,3}
  11: {11}
  12: {2,2,3}
  13: {13}
  15: {3,5}
  16: {2,2,2,2}
  17: {17}
  18: {2,3,3}
  19: {19}
  20: {2,2,5}
  21: {3,7}
  23: {23}
  24: {2,2,2,3}

Crossrefs

Prime factors are listed by A027746.
The complement is A072978, for prime indices A359912.
For mean instead of median we have A078175, for prime indices A316413.
For prime indices instead of factors we have A359908, counted by A325347.
Positions of even terms in A360005.
A067340 lists numbers whose prime signature has integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, strict A359907.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A067538, A175352, A348551, A359890, A359905, A360007, A360006, A360069.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[Flatten[ConstantArray@@@FactorInteger[#]]]]&]
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