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 1-5 of 5 results.

A360248 Numbers for which the prime indices do not have the same median as the distinct prime indices.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 200
Offset: 1

Views

Author

Gus Wiseman, Feb 07 2023

Keywords

Comments

First differs from A242416 in lacking 180, with prime indices {1,1,2,2,3}.
First differs from A360246 in lacking 126 and having 1950.
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:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
The prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is not in the sequence.
The prime indices of 1950 are {1,2,3,3,6} with median 3 and distinct prime indices {1,2,3,6} with median 5/2, so 1950 is in the sequence.

Crossrefs

These partitions are counted by A360244.
The complement is A360249, counted by A360245.
For multiplicities instead of parts: complement of A360453.
For multiplicities instead of distinct parts: complement of A360454.
For mean instead of median we have A360246, counted by A360242.
The complement for mean instead of median is A360247, counted by A360243.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A325347 = partitions with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median.
A360005 gives median of prime indices (times two).
Cf. A000975, A078174, A316413, A324570, A359890, A360455, A360456.

Programs

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

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[#]]&]

A360249 Numbers for which the prime indices have the same median as the distinct prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 121, 122, 123, 125, 126, 127, 128, 129, 130
Offset: 1

Views

Author

Gus Wiseman, Feb 07 2023

Keywords

Comments

First differs from A072774 in having 90.
First differs from A242414 in having 180.
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 prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is in the sequence.
The prime indices of 180 are {1,1,2,2,3} with median 2 and distinct prime indices {1,2,3} with median 2, so 180 is in the sequence.

Crossrefs

These partitions are counted by A360245.
The complement for mean instead of median is A360246, counted by A360242.
For mean instead of median we have A360247, counted by A360243.
The complement is A360248, counted by A360244.
For multiplicities instead of parts: A360453, counted by A360455.
For multiplicities instead of distinct parts: A360454, counted by A360456.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranks A359889.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A325347 = partitions with integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median.
A359894 = partitions with mean different from median, ranks A359890.
A360005 gives median of prime indices (times two).
Cf. A000975, A078174, A316413, A324570, A359903, A360252, A360253.

Programs

  • Maple
    isA360249 := proc(n)
    local ifs,pidx,pe,medAll,medDist ;
    if n = 1 then
        return true ;
    end if ;
    ifs := ifactors(n)[2] ;
    pidx := [] ;
    for pe in ifs do
        numtheory[pi](op(1,pe)) ;
        pidx := [op(pidx),seq(%,i=1..op(2,pe))] ;
    end do:
    medAll := stats[describe,median](sort(pidx)) ;
    pidx := convert(convert(pidx,set),list) ;
    medDist := stats[describe,median](sort(pidx)) ;
    if medAll = medDist then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 130 do
    if isA360249(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 22 2023
  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Median[prix[#]]==Median[Union[prix[#]]]&]

A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

Original entry on oeis.org

1, 2, 9, 54, 100, 120, 125, 135, 168, 180, 189, 240, 252, 264, 280, 297, 300, 312, 336, 351, 396, 408, 440, 450, 456, 459, 468, 480, 513, 520, 528, 540, 552, 560, 588, 612, 616, 621, 624, 672, 680, 684, 696, 728, 744, 756, 760, 783, 816, 828, 837, 880, 882
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}
   54: {1,2,2,2}
  100: {1,1,3,3}
  120: {1,1,1,2,3}
  125: {3,3,3}
  135: {2,2,2,3}
  168: {1,1,1,2,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  240: {1,1,1,1,2,3}
For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.

Crossrefs

For mean instead of median we have A359903, counted by A360068.
For distinct indices instead of indices we have A360453, counted by A360455.
For distinct indices instead of multiplicities: A360249, counted by A360245.
These partitions are counted by A360456.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranked by A359889.
A325347 counts partitions w/ 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.
A359894 counts partitions with mean different from median, ranks A359890.
A360005 gives median of prime indices (times two).
Cf. A000975, A109297, A109298, A114638, A316413, A324570, A360244, A360248.

Programs

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

A360455 Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 2, 2, 5, 8, 10, 14, 20, 19, 26, 31, 35, 41, 55, 65, 85, 102, 118, 151, 181, 201, 236, 281, 313, 365, 424, 495, 593, 688, 825, 978, 1181, 1374, 1650, 1948, 2323, 2682, 3175, 3680, 4314, 4930, 5718, 6546, 7532, 8557, 9777, 11067, 12622
Offset: 0

Views

Author

Gus Wiseman, Feb 10 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(11) = 8 partitions:
  1   .  .  22    221   3111   .  3311    333     3331     32222
            211                   41111   32211   33211    33221
                                                  42211    44111
                                                  322111   52211
                                                  511111   322211
                                                           332111
                                                           422111
                                                           3221111

Crossrefs

For mean instead of median: A114638, ranks A324570.
For parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360453.
For parts instead of distinct parts: A360456, ranks A360454.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A240219, A326619/A326620, A359894, A359895, A360068, A360243, A360244, A360248.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[Union[#]]&]],{n,0,30}]
Showing 1-5 of 5 results.

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