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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A363943 Mean of the multiset of prime indices of n, rounded down.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 2, 2, 1, 7, 1, 8, 1, 3, 3, 9, 1, 3, 3, 2, 2, 10, 2, 11, 1, 3, 4, 3, 1, 12, 4, 4, 1, 13, 2, 14, 2, 2, 5, 15, 1, 4, 2, 4, 2, 16, 1, 4, 1, 5, 5, 17, 1, 18, 6, 2, 1, 4, 2, 19, 3, 5, 2, 20, 1, 21, 6, 2, 3, 4, 3, 22, 1, 2, 7
Offset: 1

Views

Author

Gus Wiseman, Jun 29 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.
Extending the terminology introduced at A124943, this is the "low mean" of prime indices.

Examples

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 1.

Crossrefs

Positions of first appearances are 1 and A000040.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A363486, high A363487.
For low median instead of mean we have A363941, triangle A124943.
For high median instead of mean we have A363942, triangle A124944.
The high version is A363944, triangle A363946.
The triangle for this statistic (low mean) is A363945.
Positions of 1's are A363949(n) = 2*A344296(n), counted by A025065.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363947 ranks partitions with rounded mean 1, counted by A363948.
Cf. A102627, A327473, A327476, A327482, A348551, A359889, A363727, A363951.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[meandown[prix[n]],{n,100}]

A359897 Number of strict integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 10, 7, 10, 13, 11, 9, 20, 10, 20, 18, 21, 12, 30, 24, 28, 27, 30, 15, 73, 16, 37, 43, 45, 67, 74, 19, 55, 71, 126, 21, 150, 22, 75, 225, 78, 24, 183, 126, 245, 192, 132, 27, 284, 244, 403, 303, 120, 30, 828
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Examples

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
                          (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
                                 (3,2,1)  (6,1)  (7,1)  (7,2)
                                                        (8,1)
                                                        (4,3,2)
                                                        (5,3,1)

Crossrefs

The non-strict version is A240219, complement A359894, ranked by A359889.
The complement is counted by A359898.
The odd-length case is A359899, complement A359900.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A237984 counts partitions containing their mean, complement A327472.
A240850 counts strict partitions containing their mean, complement A240851.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
Cf. A065795, A066571, A067659, A082550, A102627, A135342, A316313, A327473, A327475, A328966, A359909.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Mean[#]==Median[#]&]],{n,0,30}]

A363944 Mean of the multiset of prime indices of n, rounded up.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 2, 6, 3, 3, 1, 7, 2, 8, 2, 3, 3, 9, 2, 3, 4, 2, 2, 10, 2, 11, 1, 4, 4, 4, 2, 12, 5, 4, 2, 13, 3, 14, 3, 3, 5, 15, 2, 4, 3, 5, 3, 16, 2, 4, 2, 5, 6, 17, 2, 18, 6, 3, 1, 5, 3, 19, 3, 6, 3, 20, 2, 21, 7, 3, 4, 5, 3, 22, 2, 2, 7
Offset: 1

Views

Author

Gus Wiseman, Jun 30 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.
Extending the terminology introduced at A124944, this is the "high mean" of prime indices.

Examples

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 2.

Crossrefs

Positions of first appearances are 1 and A000040.
Positions of 1's are A000079(n>0).
Before rounding up we had A326567/A326568.
For mode instead of mean we have A363487, low A363486.
For median instead of mean we have A363942, triangle A124944.
Rounding down instead of up gives A363943, triangle A363945.
The triangle for this statistic (high mean) is A363946.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363947 ranks partitions with rounded mean 1, counted by A363948.
A363949 ranks partitions with low mean 1, counted by A025065.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124943, A215366, A327473, A327476, A327482, A359889, A362611, A363723, A363724, A363727, A363951.

Programs

  • Mathematica
    prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[meanup[prix[n]],{n,100}]

A360009 Numbers whose prime indices have integer mean and integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110, 111
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).

Examples

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}

Crossrefs

For just integer mean we have A316413 (counted by A067538).
The mean of prime indices is given by A326567/A326568.
The complement is A348551 \/ A359912 (counted by A349156 and A307683).
These partitions are counted by A359906.
For just integer median we have A359908 (counted by A325347).
The median of prime indices is given by A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A327473, A359889, A359890, A359903, A359905, A360006.

Programs

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

Formula

Intersection of A316413 and A359908.

A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 46, 61, 82, 102, 136, 174, 224, 283, 360, 449, 569, 708, 883, 1089, 1352, 1659, 2042, 2492, 3039, 3695, 4492, 5426, 6555, 7889, 9482, 11360, 13602, 16231, 19348, 23005, 27313, 32364, 38303, 45227, 53341, 62800, 73829
Offset: 1

Views

Author

Gus Wiseman, Jun 26 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 mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (3221)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

For mean instead of mode we have A240219, see A359894, A359889, A359895, A359897, A359899.
Including mean also gives A363719, ranks A363727.
For mean instead of median we have A363723, see A363724, A363731.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
A362608 counts partitions with a unique mode, ranks A356862.
Cf. A027193, A237984, A325347, A362562, A363720, A363725, A363726.

Programs

  • Mathematica
    modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Median[#]}==modes[#]&]],{n,30}]

A363942 High median in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 1, 6, 4, 3, 1, 7, 2, 8, 1, 4, 5, 9, 1, 3, 6, 2, 1, 10, 2, 11, 1, 5, 7, 4, 2, 12, 8, 6, 1, 13, 2, 14, 1, 2, 9, 15, 1, 4, 3, 7, 1, 16, 2, 5, 1, 8, 10, 17, 2, 18, 11, 2, 1, 6, 2, 19, 1, 9, 3, 20, 1, 21, 12, 3, 1, 5, 2, 22, 1, 2
Offset: 1

Views

Author

Gus Wiseman, Jul 01 2023

Keywords

Comments

The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest 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 prime indices of 90 are {1,2,2,3}, with high median 2, so a(90) = 2.
The prime indices of 150 are {1,2,3,3}, with high median 3, so a(150) = 3.

Crossrefs

Positions of first appearances are 1 and A000040.
The triangle for this statistic (high median) is A124944, low A124943.
Regular median of prime indices is A360005(n)/2.
For mode instead of median we have A363487, low A363486.
The low version is A363941.
For mean instead of median we have A363944, triangle A363946, low A363943.
A061395 give maximum prime index, A055396 minimum.
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
Cf. A025065, A215366, A316413, A326567/A326568, A359889, A359908, A359912, A363727, A363949, A363950.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
merr[y_]:=If[Length[y]==0,0, If[OddQ[Length[y]],y[[(Length[y]+1)/2]],y[[1+Length[y]/2]]]];
Table[merr[prix[n]],{n,100}]

A360244 Number of integer partitions of n where the parts do not have the same median as the distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 11, 17, 23, 37, 42, 68, 87, 110, 153, 209, 261, 352, 444, 573, 750, 949, 1187, 1508, 1909, 2367, 2938, 3662, 4507, 5576, 6826, 8359, 10203, 12372, 15011, 18230, 21996, 26518, 31779, 38219, 45682, 54660, 65112, 77500, 92089, 109285
Offset: 0

Views

Author

Gus Wiseman, Feb 05 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) = 17 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (4211)     (3222)
                          (3211)    (5111)     (3321)
                          (4111)    (22211)    (4311)
                          (22111)   (32111)    (5211)
                          (31111)   (41111)    (6111)
                          (211111)  (221111)   (22221)
                                    (311111)   (33111)
                                    (2111111)  (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (33111) has median 1, and the distinct parts {1,3} have median 2, so y is counted under a(9).

Crossrefs

For mean instead of median: A360242, ranks A360246, complement A360243.
These partitions are ranked by A360248.
The complement is A360245, ranked by A360249.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A240219 counts partitions with mean equal to median, ranks A359889.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median.
A359894 counts partitions with mean different from median, ranks A359890.
A360071 counts partitions by number of parts and number of distinct parts.
Cf. A000975, A027193, A067659, A326619/A326620, A326621, A359902, A360068, A360241, A360250, A360251.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Median[#]!=Median[Union[#]]&]],{n,0,30}]

A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020
Offset: 1

Views

Author

Gus Wiseman, Jun 19 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 mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
Without loss of generality, we may assume there is a unique middle-part (A238478).
Includes all constant partitions.

Examples

			The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
  1  2   4     6       8         C             E               G
     11  22    33      44        66            77              88
         1111  222     2222      444           2222222         4444
               111111  3221      3333          3222221         5443
                       11111111  4332          3322211         6442
                                 5331          4222211         7441
                                 222222        11111111111111  22222222
                                 322221                        32222221
                                 422211                        33222211
                                 111111111111                  42222211
                                                               52222111
                                                               1^16

Crossrefs

For unequal instead of equal: A363720, ranks A363730, unique mode A363725.
The odd-length case is A363721.
These partitions have ranks A363727, nonprime A363722.
The case of non-constant partitions is A363728, ranks A363729.
The version for factorizations is A363741, see A359909, A359910.
Just two statistics:
- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
- (mean) = (mode) gives A363723, see A363724, A363731.
- (median) = (mode) gives A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363742.

Programs

  • Mathematica
    modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], {Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

A363941 Low median in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 2, 8, 1, 2, 1, 9, 1, 3, 1, 2, 1, 10, 2, 11, 1, 2, 1, 3, 1, 12, 1, 2, 1, 13, 2, 14, 1, 2, 1, 15, 1, 4, 3, 2, 1, 16, 2, 3, 1, 2, 1, 17, 1, 18, 1, 2, 1, 3, 2, 19, 1, 2, 3, 20, 1, 21, 1, 3, 1, 4, 2, 22, 1, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Jul 01 2023

Keywords

Comments

The low median (see A124943) in a multiset is either the middle part (for odd length), or the least 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 prime indices of 90 are {1,2,2,3}, with low median 2, so a(90) = 2.
The prime indices of 150 are {1,2,3,3}, with low median 2, so a(150) = 2.

Crossrefs

Positions of first appearances are 1 and A000040.
The triangle for this statistic (low median) is A124943, high A124944.
Median of prime indices is A360005(n)/2.
For mode instead of median we have A363486, high A363487.
Positions of 1's are A363488.
The high version is A363942.
A067538 counts partitions with integer mean, ranked by A316413.
A112798 lists prime indices, length A001222, sum A056239.
A363943 gives low mean of prime indices, triangle A363945.
A363944 gives high mean of prime indices, triangle A363946.
Cf. A025065, A215366, A326567/A326568, A344296, A359889, A359908, A363727, A363740.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mell[y_]:=If[Length[y]==0,0, If[OddQ[Length[y]],y[[(Length[y]+1)/2]],y[[Length[y]/2]]]];
Table[mell[prix[n]],{n,30}]

A363949 Numbers whose prime indices have mean 1 when rounded down.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 560, 576, 600, 640
Offset: 1

Views

Author

Gus Wiseman, Jul 02 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.

Examples

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Crossrefs

These partitions are counted by A025065.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A360015, counted by A241131.
For median instead of mean we have A363488, counted by A027336.
Positions of 1's in A363943, triangle A363945.
For the usual rounding (not low or high) we have A363948, counted by A363947.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
For mean 2 instead of 1 we have A363950, counted by A026905 redoubled.
Cf. A051293, A327473, A327476, A344296, A359889, A360013, A363723, A363727, A363946, A363951.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Floor[Mean[prix[#]]]==1&]

Formula

a(n) = 2*A344296(n).
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