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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A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2023

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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:
     3: {2}        23: {9}          42: {1,2,4}
     5: {3}        25: {3,3}        43: {14}
     6: {1,2}      26: {1,6}        45: {2,2,3}
     7: {4}        27: {2,2,2}      46: {1,9}
     9: {2,2}      29: {10}         47: {15}
    10: {1,3}      30: {1,2,3}      49: {4,4}
    11: {5}        31: {11}         50: {1,3,3}
    13: {6}        33: {2,5}        51: {2,7}
    14: {1,4}      34: {1,7}        53: {16}
    15: {2,3}      35: {3,4}        54: {1,2,2,2}
    17: {7}        36: {1,1,2,2}    55: {3,5}
    18: {1,2,2}    37: {12}         57: {2,8}
    19: {8}        38: {1,8}        58: {1,10}
    21: {2,4}      39: {2,6}        59: {17}
    22: {1,5}      41: {13}         60: {1,1,2,3}

Crossrefs

For mean instead of median we have A057716, counted by A000065.
These partitions are counted by A238495.
The complement is A364056, counted by A027336, low version A363488.
A000975 counts subsets with integer median, A051293 for mean.
A124943 counts partitions by low median, high version A124944.
A360005 gives twice the median of prime indices, A360459 for prime factors.
A359893 and A359901 count partitions by median.
Cf. A002865, A316413, A325347, A327473, A327476, A363727.

Programs

  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Median[prifacs[#]]>2&]

Formula

A360005(a(n)) > 1.
A360459(a(n)) > 2.

A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1, 0, 2, 4, 0, 0, 1, 0, 2, 5, 3, 0, 0, 1, 0, 4, 7, 0, 3, 0, 0, 1, 0, 4, 8, 5, 4, 0, 0, 0, 1, 0, 4, 14, 7, 4, 0, 0, 0, 0, 1, 0, 7, 21, 8, 0, 5, 0, 0, 0, 0, 1, 0, 7, 22, 11, 10, 0, 5, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jul 07 2023

Keywords

Comments

We use the "rounding half to even" rule, see link.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  2  0  1
  0  2  4  0  0  1
  0  2  5  3  0  0  1
  0  4  7  0  3  0  0  1
  0  4  8  5  4  0  0  0  1
  0  4 14  7  4  0  0  0  0  1
  0  7 21  8  0  5  0  0  0  0  1
  0  7 22 11 10  0  5  0  0  0  0  1
  0  7 36 15 12  0  6  0  0  0  0  0  1
  0 12 32 36 14  0  6  0  0  0  0  0  0  1
  0 12 53 23 23 16  0  7  0  0  0  0  0  0  1
  0 12 80 30 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (31111)    (511)   .  (61)  .  .  (7)
     (22111)    (421)      (52)
     (211111)   (4111)     (43)
     (1111111)  (331)
                (322)
                (3211)
                (2221)

Links

Crossrefs

Row sums are A000041.
The rank statistic for this triangle is A363489.
The version for low mean is A363945, rank statistic A363943.
The version for high mean is A363946, rank statistic A363944.
Column k = 1 is A363947 (A026905 tripled).
A008284 counts partitions by length, A058398 by mean.
A026905 redoubled counts partitions with high mean 2, ranks A363950.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
More triangles: A124943, A124944, A363952, A363953.
Cf. A002865, A025065, A237984, A327472, A327482, A363723, A363724, A363731.

Programs

  • Mathematica
    Table[If[n==k==0,1,Length[Select[IntegerPartitions[n], Round[Mean[#]]==k&]]],{n,0,15},{k,0,n}]

A364156 Ceiling of the mean of the prime factors of n (with multiplicity).

Original entry on oeis.org

0, 2, 3, 2, 5, 3, 7, 2, 3, 4, 11, 3, 13, 5, 4, 2, 17, 3, 19, 3, 5, 7, 23, 3, 5, 8, 3, 4, 29, 4, 31, 2, 7, 10, 6, 3, 37, 11, 8, 3, 41, 4, 43, 5, 4, 13, 47, 3, 7, 4, 10, 6, 53, 3, 8, 4, 11, 16, 59, 3, 61, 17, 5, 2, 9, 6, 67, 7, 13, 5, 71, 3, 73, 20, 5, 8, 9, 6
Offset: 1

Views

Author

Gus Wiseman, Jul 18 2023

Keywords

Examples

			The prime factors of 450 are {2,3,3,5,5}, with mean 18/5, so a(450) = 4.

Crossrefs

For median of prime indices we have triangle A124944, low A124943.
The round version is A067629.
The floor version is A126594.
A027746 lists prime factors, indices A112798.
A078175 lists numbers with integer mean of prime factors.
A123528/A123529 gives mean of prime factors, A326567/A326568 prime indices.
Cf. A026905, A051293, A316413, A327473, A327476, A327482, A363895, A363943, A363948, A363950, A364037.

Programs

  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[If[n==1,0,Ceiling[Mean[prifacs[n]]]],{n,100}]

Formula

Ceiling of A123528(n)/A123529(n).
Previous Showing 21-23 of 23 results.

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