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.

Previous Showing 11-15 of 15 results.

A361866 Number of set partitions of {1..n} with block-means summing to an integer.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 75, 267, 1119, 4965, 22694, 117090, 670621, 3866503, 24113829, 161085223, 1120025702, 8121648620, 62083083115, 492273775141, 4074919882483
Offset: 0

Views

Author

Gus Wiseman, Apr 04 2023

Keywords

Examples

			The a(1) = 1 through a(4) = 8 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{13}{24}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
The set partition y = {{1,2},{3,4}} has block-means {3/2,7/2}, with sum 5, so y is counted under a(4).

Crossrefs

For mean instead of sum we have A361865, for median A361864.
For median instead of mean we have A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions with integer mean block-size.
A327475 counts subsets with integer mean, median A000975.
A327481 counts subsets by mean, median A013580.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A067659, A326515, A326516, A326521.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Total[Mean/@#]]&]],{n,6}]

Extensions

a(14)-a(20) from Christian Sievers, May 12 2025

A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.

Original entry on oeis.org

36, 64, 100, 144, 196, 216, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2116, 2304, 2500, 2704, 2744, 2916, 3136, 3364, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832, 6084, 6400, 6724
Offset: 1

Views

Author

Gus Wiseman, Jul 17 2019

Keywords

Examples

			36 has two such factorizations: (3*12) and (4*9).
The sequence of terms together with their prime indices begins:
    36: {1,1,2,2}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   144: {1,1,1,1,2,2}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   324: {1,1,2,2,2,2}
   400: {1,1,1,1,3,3}
   484: {1,1,5,5}
   512: {1,1,1,1,1,1,1,1,1}
   576: {1,1,1,1,1,1,2,2}
   676: {1,1,6,6}
   784: {1,1,1,1,4,4}
   900: {1,1,2,2,3,3}
  1000: {1,1,1,3,3,3}
  1024: {1,1,1,1,1,1,1,1,1,1}
  1156: {1,1,7,7}
  1296: {1,1,1,1,2,2,2,2}
  1444: {1,1,8,8}

Links

Crossrefs

A subsequence of A001597.
Factorizations with integer mean are A326622.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A067538, A067539, A326027, A326516, A326623, A326641, A326643, A326645, A326647.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Length[Select[facs[#],!IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]]>1&]

A327906 Numbers with only one factorization into factors > 1 with integer mean (namely, as a singleton).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 29, 30, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 66, 67, 70, 71, 73, 74, 79, 82, 83, 86, 89, 90, 94, 97, 98, 101, 102, 103, 106, 107, 109, 113, 118, 122, 127, 130, 131, 134, 137, 138, 139, 142
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Examples

			There are 4 factorizations of 24 with integer mean, namely:
  (24)
  (4*6)
  (2*12)
  (2*3*4)
so 24 is not in the sequence.

Links

Crossrefs

Complement of A327907.
Positions of 1's in A326622.
Cf. A001055, A051293, A067538, A078175, A123528/A123529, A316413, A326515, A326516, A326666, A326667, A326671.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Length[Select[facs[#],IntegerQ[Mean[#]]&]]==1&]
  • PARI
    A326622(n, m=n, facsum=0, facnum=0) = if(1==n,facnum > 0 && 1==denominator(facsum/facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326622(n/d, d, facsum+d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024
isA327906(n) = (1==A326622(n)); \\ Antti Karttunen, Nov 10 2024

A327907 Numbers with more than one factorization into at factors > 1 with integer mean.

Original entry on oeis.org

4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 68, 69, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 110, 111, 112, 114, 115, 116
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Examples

			There are 6 factorizations of 60 with integer mean, namely:
  (60)
  (2*30)
  (6*10)
  (3*4*5)
  (2*3*10)
  (2*2*3*5)
so 60 is in the sequence.

Crossrefs

Complement of A327906.
Positions of terms > 1 in A326622.
Cf. A001055, A051293, A067538, A078175, A123528/A123529, A316413, A326515, A326516, A326666, A326667, A326671.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Length[Select[facs[#],IntegerQ[Mean[#]]&]]>1&]

A361863 Number of set partitions of {1..n} such that the median of medians of the blocks is (n+1)/2.

Original entry on oeis.org

1, 2, 3, 9, 26, 69, 335, 1018, 6629, 22805, 182988, 703745
Offset: 1

Views

Author

Gus Wiseman, Apr 04 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).
Since (n+1)/2 is the median of {1..n}, this sequence counts "transitive" set partitions.

Examples

			The a(1) = 1 through a(4) = 9 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{23}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with median 5/2, so is counted under a(4).
The set partition {{1,3},{2,4}} has medians {2,3}, with median 5/2, so is counted under a(4).

Crossrefs

For mean instead of median we have A361910.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A325347 counts partitions w/ integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361864 counts set partitions with integer median of medians, means A361865.
A361866 counts set partitions with integer sum of medians, means A361911.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A326516, A326521, A326537, A359907, A361654, A361801.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],(n+1)/2==Median[Median/@#]&]],{n,6}]
Previous Showing 11-15 of 15 results.

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