A340599 -id:A340599 - cp's OEIS frontend

A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 47, 53, 57, 58, 59, 61, 65, 67, 71, 73, 74, 78, 79, 83, 86, 87, 89, 91, 95, 97, 101, 103, 106, 107, 109, 111, 113, 122, 127, 129, 130, 131, 133, 137, 138, 139, 142, 143, 145

Offset: 1

Gus Wiseman, Feb 05 2021

nonn

Also Heinz numbers of strict integer partitions whose greatest part is divisible by their number of parts. These partitions are counted by A340828.

			The sequence of terms together with their prime indices begins:
      2: {1}         31: {11}       71: {20}
      3: {2}         35: {3,4}      73: {21}
      5: {3}         37: {12}       74: {1,12}
      6: {1,2}       38: {1,8}      78: {1,2,6}
      7: {4}         39: {2,6}      79: {22}
     11: {5}         41: {13}       83: {23}
     13: {6}         43: {14}       86: {1,14}
     14: {1,4}       47: {15}       87: {2,10}
     17: {7}         53: {16}       89: {24}
     19: {8}         57: {2,8}      91: {4,6}
     21: {2,4}       58: {1,10}     95: {3,8}
     23: {9}         59: {17}       97: {25}
     26: {1,6}       61: {18}      101: {26}
     29: {10}        65: {3,6}     103: {27}
     30: {1,2,3}     67: {19}      106: {1,16}

Note: Heinz number sequences are given in parentheses below.
The case of equality, and the reciprocal version, are both A002110.
The non-strict reciprocal version is A168659 (A340609).
The non-strict version is A168659 (A340610).
These are the Heinz numbers of partitions counted by A340828.
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up the prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413/A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A257541 gives the rank of the partition with Heinz number n.
A340830 counts strict partitions whose parts are multiples of the length.
Cf. A039900, A047993 (A106529), A064174, A143773 (A316428), A326837, A326849 (A326848), A340599, A340691.

Select[Range[2,100],SquareFreeQ[#]&&Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 11, 26, 77, 220, 677, 2098, 6756, 22101, 74264, 253684, 883795, 3130432, 11275246, 41240180, 153117873, 576634463, 2201600769, 8517634249, 33378499157, 132438117118, 531873247805, 2161293783123, 8883906870289, 36928576428885, 155196725172548, 659272353608609, 2830200765183775

Offset: 0

Gus Wiseman, Feb 05 2021

nonn

We define a multiset partition to be cross-balanced if it uses exactly as many distinct vertices as the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,2}}    {{1,2},{1,2}}
                    {{1},{1},{1}}  {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,2}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The co-balanced version is A319616.
The balanced version is A340600.
The twice-balanced version is A340652.
The version for factorizations is A340654.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A001055, A007717, A064174, A316983, A320663, A324522, A339888, A340655.

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n, G(k,n,k) - G(k-1,n,k) - G(k,n,k-1) + G(k-1,n,k-1)))} \\ Andrew Howroyd, Jan 15 2024

a(11) onwards from Andrew Howroyd, Jan 15 2024

A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.

Original entry on oeis.org

2, 8, 12, 16, 32, 48, 64, 72, 80, 96, 112, 120, 128, 144, 160, 168, 192, 200, 224, 240, 256, 280, 288, 320, 336, 384, 392, 432, 448, 480, 512, 576, 640, 672, 704, 720, 768, 800, 832, 864, 896, 960, 1008, 1024, 1056, 1120, 1152, 1200, 1248, 1280, 1296, 1344

Offset: 1

Gus Wiseman, Jan 28 2021

nonn

			The initial terms and a valid factorization of each:
      2 = 2           168 = 3*7*8        512 = 2*2*2*2*32
      8 = 2*4         192 = 2*2*3*16     576 = 2*2*9*16
     12 = 3*4         200 = 5*5*8        640 = 2*2*10*16
     16 = 4*4         224 = 4*7*8        672 = 2*3*7*16
     32 = 2*2*8       240 = 5*6*8        704 = 2*2*11*16
     48 = 2*3*8       256 = 2*2*4*16     720 = 3*3*5*16
     64 = 2*4*8       280 = 5*7*8        768 = 2*2*2*3*32
     72 = 3*3*8       288 = 2*3*3*16     800 = 2*5*5*16
     80 = 2*5*8       320 = 2*2*5*16     832 = 2*2*13*16
     96 = 2*6*8       336 = 6*7*8        864 = 2*3*9*16
    112 = 2*7*8       384 = 2*2*6*16     896 = 2*2*14*16
    120 = 3*5*8       392 = 7*7*8        960 = 2*2*15*16
    128 = 2*2*2*16    432 = 3*3*3*16    1008 = 3*3*7*16
    144 = 3*6*8       448 = 2*2*7*16    1024 = 2*2*2*4*32
    160 = 4*5*8       480 = 2*3*5*16    1056 = 2*3*11*16

Partitions of the prescribed type are counted by A340611.
The conjugate version is A340689.
A001055 counts factorizations, with strict case A045778.
A047993 counts balanced partitions.
A316439 counts factorizations by product and length.
A340596 counts co-balanced factorizations.
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],2^Length[#]==Max@@#&]!={}&]

cp's OEIS Frontend

A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica