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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A359890 Numbers whose prime indices do not have the same mean as median.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 92, 96, 98, 99, 102, 104, 108, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156, 160, 162, 164, 165
Offset: 1

Views

Author

Gus Wiseman, Jan 22 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:
   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}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
For example, the prime indices of 360 are {1,1,1,2,2,3}, with mean 5/3 and median 3/2, so 360 is in the sequence.

Crossrefs

The LHS (mean of prime indices) is A326567/A326568.
The complement is A359889, counted by A240219.
The odd-length case is A359891, complement A359892.
These partitions are counted by A359894.
The strict case is counted by A359898, odd-length A359900.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359908 lists numbers whose prime indices have integer median.
Cf. A327473, A327476, A348551, A359903, A359911, A359912, A360006-A360009.

Programs

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

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}]

A360071 Regular tetrangle where T(n,k,i) = number of integer partitions of n of length k with i distinct parts.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Jan 28 2023

Keywords

Comments

I call this a tetrangle because it is a sequence of finite triangles. - Gus Wiseman, Jan 30 2023

Examples

			Tetrangle begins:
  1   1     1       1         1           1             1
      1 0   0 1     1 1       0 2         1 2           0 3
            1 0 0   0 1 0     0 2 0       1 1 1         0 3 1
                    1 0 0 0   0 1 0 0     0 2 0 0       0 2 1 0
                              1 0 0 0 0   0 1 0 0 0     0 2 0 0 0
                                          1 0 0 0 0 0   0 1 0 0 0 0
                                                        1 0 0 0 0 0 0
For example, finite triangle n = 5 counts the following partitions:
    (5)
     .    (41)(32)
     .   (311)(221)  .
     .     (2111)    .   .
  (11111)     .      .   .   .

Crossrefs

Row sums are A008284 (partitions by number of parts), reverse A058398.
First columns i = 1 are A051731.
Last columns i = k are A060016.
Column sums are A116608 (partitions by number of distinct parts).
Positive terms are counted by A360072.
A000041 counts partitions, strict A000009.
Other tetrangles: A318393, A318816, A320808, A334433, A345197.
Cf. A055884, A327482, A331195, A360010, A360069.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Length[#]==k&&Length[Union[#]]==i&]],{n,1,9},{k,1,n},{i,1,k}]

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.

A359895 Number of odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 1, 5, 5, 2, 5, 2, 8, 18, 1, 2, 19, 2, 24, 41, 20, 2, 9, 44, 31, 94, 102, 2, 125, 2, 1, 206, 68, 365, 382, 2, 98, 433, 155, 2, 716, 2, 1162, 2332, 196, 2, 17, 1108, 563, 1665, 3287, 2, 3906, 5474, 2005, 3083, 509, 2, 9029
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Comments

The length and median of such a partition are integers with product n.

Examples

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)  (9)
            (111)       (11111)  (222)  (1111111)       (333)
                                 (321)                  (432)
                                                        (531)
                                                        (111111111)
The a(15) = 18 partitions:
  (15)
  (5,5,5)
  (6,5,4)
  (7,5,3)
  (8,5,2)
  (9,5,1)
  (3,3,3,3,3)
  (4,3,3,3,2)
  (4,4,3,2,2)
  (4,4,3,3,1)
  (5,3,3,2,2)
  (5,3,3,3,1)
  (5,4,3,2,1)
  (5,5,3,1,1)
  (6,3,3,2,1)
  (6,4,3,1,1)
  (7,3,3,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

Links

Crossrefs

This is the odd-length case of A240219, complement A359894, strict A359897.
These partitions are ranked by A359891, complement A359892.
The complement is counted by A359896.
The strict case is A359899, complement A359900.
The version for factorizations is A359910.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A008289, A316313, A327472, A327475, A327482, A359889, A359906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]
  • PARI
    \\ P(n, k, m) is g.f. for k parts of max size m.
P(n, k, m)={polcoef(1/prod(i=1, m, 1 - y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)+h); polcoef(P(r, h, m)*P(r, h, r), r))))} \\ Andrew Howroyd, Jan 21 2023

Formula

a(p) = 2 for prime p. - Andrew Howroyd, Jan 21 2023

A359903 Numbers whose prime indices and prime signature have the same mean.

Original entry on oeis.org

1, 2, 9, 88, 100, 125, 624, 756, 792, 810, 880, 900, 1312, 2401, 4617, 4624, 6240, 7392, 7560, 7920, 8400, 9261, 9604, 9801, 10648, 12416, 23424, 33984, 37760, 45792, 47488, 60912, 66176, 71552, 73920, 75200, 78720, 83592, 89216, 89984, 91264, 91648, 99456
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.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Examples

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      9: {2,2}
     88: {1,1,1,5}
    100: {1,1,3,3}
    125: {3,3,3}
    624: {1,1,1,1,2,6}
    756: {1,1,2,2,2,4}
    792: {1,1,1,2,2,5}
    810: {1,2,2,2,2,3}
    880: {1,1,1,1,3,5}
    900: {1,1,2,2,3,3}
   1312: {1,1,1,1,1,13}
   2401: {4,4,4,4}
   4617: {2,2,2,2,2,8}
   4624: {1,1,1,1,7,7}
   6240: {1,1,1,1,1,2,3,6}
   7392: {1,1,1,1,1,2,4,5}
   7560: {1,1,1,2,2,2,3,4}
   7920: {1,1,1,1,2,2,3,5}
Example: 810 has prime indices {1,2,2,2,2,3} and prime exponents (1,4,1), both of which have mean 2, so 810 is in the sequence.
Example: 78720 has prime indices {1,1,1,1,1,1,1,2,3,13} and prime exponents (7,1,1,1), both of which have mean 5/2, so 78720 is in the sequence.

Crossrefs

Prime indices are A112798, sum A056239, mean A326567/A326568.
Prime signature is A124010, sum A001222, mean A088529/A088530.
For prime factors instead of indices we have A359904.
Partitions with these Heinz numbers are counted by A360068.
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A316413 lists numbers whose prime indices have integer mean.
A360005 gives median of prime indices (times two).
Cf. A240219, A316313, A326622, A327473, A327476, A348551, A359905, A359908, A360008, A360069.

Programs

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

A363946 Triangle read by rows where T(n,k) is the number of integer partitions of n with high mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 3, 2, 0, 1, 0, 1, 6, 3, 0, 0, 1, 0, 1, 6, 4, 3, 0, 0, 1, 0, 1, 11, 5, 4, 0, 0, 0, 1, 0, 1, 11, 13, 0, 4, 0, 0, 0, 1, 0, 1, 18, 9, 8, 5, 0, 0, 0, 0, 1, 0, 1, 18, 21, 10, 0, 5, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jun 30 2023

Keywords

Comments

Extending the terminology of A124944, the "high mean" of a multiset is obtained by taking the mean and rounding up.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  3  2  0  1
  0  1  6  3  0  0  1
  0  1  6  4  3  0  0  1
  0  1 11  5  4  0  0  0  1
  0  1 11 13  0  4  0  0  0  1
  0  1 18  9  8  5  0  0  0  0  1
  0  1 18 21 10  0  5  0  0  0  0  1
  0  1 29 28 12  0  6  0  0  0  0  0  1
  0  1 29 32 18 14  0  6  0  0  0  0  0  1
  0  1 44 43 23 16  0  7  0  0  0  0  0  0  1
  0  1 44 77 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (1111111)  (4111)    (511)  (61)  .  .  (7)
                (3211)    (421)  (52)
                (31111)   (331)  (43)
                (2221)    (322)
                (22111)
                (211111)

Crossrefs

Row sums are A000041.
Column k = 2 is A026905 redoubled, ranks A363950.
For median instead of mean we have triangle A124944, low A124943.
For mode instead of mean we have rank stat A363486, high A363487.
For median instead of mean we have rank statistic A363942, low A363941.
The rank statistic for this triangle is A363944.
The version for low mean is A363945, rank statistic A363943.
For mode instead of mean we have triangle A363953, low A363952.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A002865, A025065, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363948.

Programs

  • Mathematica
    meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meanup[#]==k&]],{n,0,15},{k,0,n}]

A360241 Number of integer partitions of n whose distinct parts have integer mean.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 6, 13, 13, 22, 19, 43, 34, 56, 66, 97, 92, 156, 143, 233, 256, 322, 341, 555, 542, 710, 831, 1098, 1131, 1644, 1660, 2275, 2484, 3035, 3492, 4731, 4848, 6063, 6893, 8943, 9378, 12222, 13025, 16520, 18748, 22048, 24405, 31446, 33698, 41558
Offset: 0

Views

Author

Gus Wiseman, Feb 02 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (511)      (53)
                    (1111)           (51)      (3211)     (62)
                                     (222)     (31111)    (71)
                                     (321)     (1111111)  (422)
                                     (3111)               (2222)
                                     (111111)             (3221)
                                                          (3311)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).

Crossrefs

For parts instead of distinct parts we have A067538, ranked by A316413.
The strict case is A102627.
These partitions are ranked by A326621.
For multiplicities instead of distinct parts: A360069, ranked by A067340.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, also A327482.
A116608 counts partitions by number of distinct parts.
A326619/A326620 gives mean of distinct prime indices.
A326622 counts factorizations with integer mean, strict A328966.
A360071 counts partitions by number of parts and number of distinct parts.
The following count partitions:
- A360242 mean(parts) != mean(distinct parts), ranked by A360246.
- A360243 mean(parts) = mean(distinct parts), ranked by A360247.
- A360250 mean(parts) > mean(distinct parts), ranked by A360252.
- A360251 mean(parts) < mean(distinct parts), ranked by A360253.
Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.

Programs

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