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 1-7 of 7 results.

A360068 Number of integer partitions of n such that the parts have the same mean as the multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 6, 0, 0, 0, 6, 0, 7, 0, 1, 0, 0, 0, 0, 90, 0, 63, 0, 0, 0, 0, 11, 0, 0, 0, 436, 0, 0, 0, 0, 0, 0, 0, 0, 2157, 0, 0, 240, 1595, 22, 0, 0, 0, 6464, 0, 0, 0, 0, 0, 0, 0, 0, 11628, 4361, 0, 0, 0, 0, 0, 0, 0, 12927, 0, 0, 621, 0
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2023

Keywords

Comments

Note that such a partition cannot be strict for n > 1.
Conjecture: If n is squarefree, then a(n) = 0.

Examples

			The n = 1, 4, 8, 9, 12, 16, 18 partitions (D=13):
  (1)  (22)  (3311)  (333)  (322221)  (4444)      (444222)
             (5111)         (332211)  (43222111)  (444411)
                            (422211)  (43321111)  (552222)
                            (522111)  (53221111)  (555111)
                            (531111)  (54211111)  (771111)
                            (621111)  (63211111)  (822222)
                                                  (D11111)
For example, the partition (4,3,3,3,3,3,2,2,1,1) has mean 5/2, and its multiplicities (1,5,2,2) also have mean 5/2, so it is counted under a(20).

Crossrefs

These partitions are ranked by A359903, for prime factors A359904.
Positions of positive terms are A360070.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326567/A326568 gives mean of prime indices (A112798).
A360069 counts partitions whose multiplicities have integer mean.
Cf. A067340, A067538, A082550, A240219, A316313, A327475, A349156, A359893, A359897, A359905.

Programs

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

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[#]]&]

A359905 Numbers whose prime indices and prime signature both have integer mean.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 91, 94, 97, 100, 101, 103, 105, 107, 109, 110, 111, 113, 115, 118, 121
Offset: 1

Views

Author

Gus Wiseman, Jan 25 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:
     2: {1}          19: {8}            39: {2,6}
     3: {2}          21: {2,4}          41: {13}
     4: {1,1}        22: {1,5}          43: {14}
     5: {3}          23: {9}            46: {1,9}
     7: {4}          25: {3,3}          47: {15}
     8: {1,1,1}      27: {2,2,2}        49: {4,4}
     9: {2,2}        29: {10}           53: {16}
    10: {1,3}        30: {1,2,3}        55: {3,5}
    11: {5}          31: {11}           57: {2,8}
    13: {6}          32: {1,1,1,1,1}    59: {17}
    16: {1,1,1,1}    34: {1,7}          61: {18}
    17: {7}          37: {12}           62: {1,11}

Crossrefs

A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 lists prime signature, mean A088529/A088530.
A316413 lists numbers whose prime indices have integer mean.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A327473, A348551, A359904, A359908, A360005, A360068, 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[100],IntegerQ[Mean[prix[#]]]&&IntegerQ[Mean[prisig[#]]]&]

Formula

Intersection of A316413 and A067340.

A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

Original entry on oeis.org

10, 39, 68, 115, 138, 259, 310, 328, 387, 517, 574, 636, 793, 795, 1034, 1168, 1206, 1241, 1281, 1340, 1534, 1691, 1825, 2212, 2278, 2328, 2343, 2369, 2370, 2727, 2774, 2905, 3081, 3277, 3818, 3924, 4064, 4074, 4247, 4268, 4360, 4539, 4850, 4905, 5243, 5335
Offset: 1

Views

Author

Gus Wiseman, Feb 18 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:
     1: {}
    10: {1,3}
    39: {2,6}
    68: {1,1,7}
   115: {3,9}
   138: {1,2,9}
   259: {4,12}
   310: {1,3,11}
   328: {1,1,1,13}
   387: {2,2,14}
   517: {5,15}
   574: {1,4,13}
   636: {1,1,2,16}
For example, the prime indices of 138 are {1,2,9}, with mean 4, and with first differences (1,7), with mean also 4, so 138 is in the sequence.

Crossrefs

These partitions are counted by A360670.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A301987 lists numbers whose sum of prime indices equals their product.
A316413 lists numbers whose prime indices have integer mean.
A334201 adds up all prime indices except the greatest.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A344415, A348551, A359904, A360008, A360068, A360681.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Mean[prix[#]]==Mean[Differences[prix[#]]]&]

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 42, 49, 60, 66, 70, 78, 84, 90, 102, 105, 114, 120, 126, 132, 138, 140, 150, 154, 156, 168, 174, 186, 198, 204, 210, 222, 228, 234, 246, 258, 264, 270, 276, 280, 282, 286, 294, 306, 308, 312, 315, 318, 330, 342, 348, 350, 354, 366, 372, 378, 385
Offset: 1

Views

Author

Gus Wiseman, Feb 19 2023

Keywords

Comments

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 terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
   30: {1,2,3}
   42: {1,2,4}
   49: {4,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 is not in the sequence.

Crossrefs

For distinct prime indices instead of 0-prepended differences: A360453.
For mean instead of median we have A360680.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
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.
Multisets with integer median:
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.

Programs

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

A360070 Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 272
Offset: 1

Views

Author

Gus Wiseman, Jan 27 2023

Keywords

Comments

Conjecture: No term > 1 is squarefree.

Examples

			A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.

Crossrefs

Positions of positive terms in A360068, ranked by A359903.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326567/A326568 gives mean of prime indices (A112798).
Cf. A005117, A067340, A067538, A240219, A316313, A349156, A359904, A359905, A360069.

Programs

  • Mathematica
    Select[Range[30],Select[IntegerPartitions[#],Mean[#]==Mean[Length/@Split[#]]&]!={}&]

Extensions

a(22)-a(58) from Alois P. Heinz, Jan 29 2023

A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 49, 152, 210, 513, 1444, 1776, 1952, 2310, 2375, 2664, 2760, 2960, 3249, 3864, 3996, 4140, 4144, 5796, 5994, 6072, 6210, 6440, 6512, 6517, 6900, 7176, 7400, 7696, 8694, 9025, 9108, 9384, 10064, 10120, 10350, 10488, 10764, 11248, 11960, 12167
Offset: 1

Views

Author

Gus Wiseman, Feb 19 2023

Keywords

Comments

A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

Examples

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     49: {4,4}
    152: {1,1,1,8}
    210: {1,2,3,4}
    513: {2,2,2,8}
   1444: {1,1,8,8}
   1776: {1,1,1,1,2,12}
   1952: {1,1,1,1,1,18}
   2310: {1,2,3,4,5}
   2375: {3,3,3,8}
   2664: {1,1,1,2,2,12}
   2760: {1,1,1,2,3,9}
   2960: {1,1,1,1,3,12}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with mean 3/2. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with mean also 3/2. So 2760 is in the sequence.

Crossrefs

For indices instead of 0-prepended differences: A359903, counted by A360068.
For median instead of mean we have A360681.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A316413 = numbers whose prime indices have integer mean, complement A348551.
A326619/A326620 gives mean of distinct prime indices.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A359904, A359905, A360008, A360460, A360555, A360556.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#],0]]]&]
Showing 1-7 of 7 results.

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