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 1-3 of 3 results.

A357710 Number of integer compositions of n with integer geometric mean.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 8, 4, 15, 17, 22, 48, 40, 130, 88, 287, 323, 543, 1084, 1145, 2938, 3141, 6928, 9770, 15585, 29249, 37540, 78464, 103289, 194265, 299752, 475086, 846933, 1216749, 2261920, 3320935, 5795349, 9292376, 14825858, 25570823, 39030115, 68265801, 106030947, 178696496
Offset: 0

Views

Author

Gus Wiseman, Oct 15 2022

Keywords

Examples

			The a(6) = 4 through a(9) = 15 compositions:
  (6)       (7)        (8)         (9)
  (33)      (124)      (44)        (333)
  (222)     (142)      (2222)      (1224)
  (111111)  (214)      (11111111)  (1242)
            (241)                  (1422)
            (412)                  (2124)
            (421)                  (2142)
            (1111111)              (2214)
                                   (2241)
                                   (2412)
                                   (2421)
                                   (4122)
                                   (4212)
                                   (4221)
                                   (111111111)

Crossrefs

The unordered version (partitions) is A067539, ranked by A326623.
Compositions with integer average are A271654, partitions A067538.
Subsets whose geometric mean is an integer are A326027.
The version for factorizations is A326028.
The strict case is A339452, partitions A326625.
These compositions are ranked by A357490.
A011782 counts compositions.
Cf. A025047, A051293, A078174, A078175, A102627, A320322, A326622, A326624, A326641, A357182, A357183.

Programs

  • Mathematica
    Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,0,15}]
  • Python
    from math import prod, factorial
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A357710(n): return sum(factorial(s)//prod(factorial(d) for d in p.values()) for s,p in partitions(n,size=True) if integer_nthroot(prod(a**b for a, b in p.items()),s)[1]) if n else 0 # Chai Wah Wu, Sep 24 2023

Extensions

More terms from David A. Corneth, Oct 17 2022

A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 15, 16, 17, 24, 31, 32, 36, 42, 63, 64, 69, 70, 81, 88, 98, 104, 127, 128, 136, 170, 255, 256, 277, 278, 282, 292, 325, 326, 337, 344, 354, 360, 394, 418, 424, 511, 512, 513, 514, 515, 528, 547, 561, 568, 640, 682, 768, 769, 785, 792, 896
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2022

Keywords

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The terms together with their corresponding compositions begin:
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  24: (1,4)
  31: (1,1,1,1,1)
  32: (6)
  36: (3,3)
  42: (2,2,2)
  63: (1,1,1,1,1,1)
  64: (7)
  69: (4,2,1)

Crossrefs

For regular mean we have A096199, counted by A271654 (partitions A067538).
Subsets whose geometric mean is an integer are counted by A326027.
The unordered version (partitions) is A326623, counted by A067539.
The strict case is counted by A339452, partitions A326625.
These compositions are counted by A357710.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A102627, A301987, A326622, A326624, A326028, A326567/A326568, A326641, A326645, A335405, A357184.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],IntegerQ[GeometricMean[stc[#]]]&]

A339175 Number of compositions (ordered partitions) of n into distinct parts such that the average of the parts is an integer.

Original entry on oeis.org

1, 1, 1, 3, 1, 11, 1, 7, 19, 9, 1, 101, 1, 13, 193, 231, 1, 131, 1, 1411, 163, 21, 1, 3533, 3601, 25, 289, 7083, 1, 29195, 1, 3295, 451, 33, 98641, 84725, 1, 37, 649, 254007, 1, 768323, 1, 9907, 444643, 45, 1, 3420605, 2197441, 133849, 1153, 17331, 1, 12132635, 209641
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 05 2020

Keywords

Examples

			a(8) = 7 because we have [8], [7, 1], [1, 7], [6, 2], [2, 6], [5, 3] and [3, 5].

Links

Crossrefs

Cf. A032020, A067538, A102627, A339452.
Showing 1-3 of 3 results.

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