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 1-5 of 5 results.

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752
Offset: 0

Views

Author

Gus Wiseman, Nov 14 2021

Keywords

Comments

Equivalently, partitions whose length does not divide their sum.
By conjugation, also the number of integer partitions of n with greatest part not dividing n.

Examples

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Crossrefs

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Programs

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

Formula

a(n > 0) = A000041(n) - A067538(n).

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Nov 14 2021

Keywords

Comments

Equivalently, partitions whose length does not divide their sum.
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.

Examples

			The terms and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

Crossrefs

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

Programs

  • Maple
    q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021
  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]

A360552 Numbers > 1 whose distinct prime factors have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103
Offset: 1

Views

Author

Gus Wiseman, Feb 16 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).

Examples

			The prime factors of 900 are {2,2,3,3,5,5}, with distinct parts {2,3,5}, with median 3, so 900 is in the sequence.

Crossrefs

For mean instead of median we have A078174, complement of A176587.
The complement is A100367 (without 1).
Positions of even terms in A360458.
- 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.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A323171/A323172 = mean of distinct prime factors, indices A326619/A326620.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A056239, A078175, A175352, A316413, A326621, A360009.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[First/@FactorInteger[#]]]&]

A175397 Numbers such that both arithmetic means of distinct and all prime factors are not integers.

Original entry on oeis.org

1, 6, 10, 12, 14, 18, 22, 24, 26, 28, 30, 34, 36, 38, 40, 46, 48, 52, 54, 56, 58, 62, 66, 70, 72, 74, 76, 80, 82, 86, 88, 90, 94, 96, 98, 100, 102, 104, 106, 108, 118, 120, 122, 124, 130, 132, 134, 136, 138, 142, 144, 146, 148, 150, 152, 154, 158, 160, 162, 165, 166, 172, 174, 176, 178, 182, 184
Offset: 1

Views

Author

Jaroslav Krizek, May 01 2010

Keywords

Comments

Contains all even semiprimes. - Robert Israel, Nov 10 2024

Examples

			For a(13) = 36: 36 = 2^2*3^3; both (2+2+3+3)/4 and (2+3)/2 are not integers.

Links

Crossrefs

Subsequence of A176552, A175352 and A176587. Complement of A175418. Cf. A174894.

Programs

  • Maple
    filter:= proc(n) local F,t,m;
  F:= ifactors(n)[2]; m:= nops(F);
  not (add(t[1],t=F)/m)::integer and not (add(t[1]*t[2],t=F)/add(t[2],t=F))::integer
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Nov 10 2024

Extensions

a(27) corrected, and more terms from Robert Israel, Nov 10 2024

A175418 Complement of A175397, where A175397 = numbers such that both arithmetic means of distinct and all prime factors are not integers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 44, 45, 47, 49, 50, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Comments

For these numbers hold that both arithmetic means of distinct and all prime factors are integers or only one of these means is an integer.
Includes all prime powers and odd semiprimes. - Robert Israel, Nov 10 2024

Links

Crossrefs

Cf. A174894, A176552, A175352, A175397, A176587.

Programs

  • Maple
    filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  (add(t[1],t=F)/nops(F))::integer or (add(t[1]*t[2],t=F)/add(t[2],t=F))::integer
end proc:
select(filter, [$2..100]); # Robert Israel, Nov 10 2024

Extensions

a(49) corrected by Robert Israel, Nov 10 2024
Showing 1-5 of 5 results.

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