A362612 -id:A362612 - cp's OEIS frontend

A363485 Number of integer partitions of n covering an initial interval of positive integers with more than one mode.

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 3, 1, 2, 6, 5, 3, 8, 4, 8, 11, 13, 9, 17, 17, 19, 25, 24, 23, 44, 35, 39, 54, 55, 63, 83, 79, 86, 104, 119, 125, 157, 164, 178, 220, 237, 251, 297, 324, 357, 413, 439, 486, 562, 607, 673, 765, 828, 901, 1040, 1117, 1220

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(n) partitions for n = {3, 6, 12, 15, 16, 18}:
  (21)  (321)   (332211)    (54321)       (443221)    (4433211)
        (2211)  (3222111)   (433221)      (3332221)   (5432211)
                (22221111)  (443211)      (4332211)   (43332111)
                            (33222111)    (33322111)  (333222111)
                            (322221111)   (43222111)  (333321111)
                            (2222211111)              (3322221111)
                                                      (32222211111)
                                                      (222222111111)

For parts instead of multiplicities we have A025147, complement A096765.
For co-mode we have A363264, complement A363263.
The complement is counted by A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A071178 counts maxima in prime factorization, modes A362611.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A105039, A117989, A243737, A362612.

Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]>1&]]],{n,0,30}]

A364160 Numbers whose least prime factor has the greatest exponent.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116

Offset: 1

Gus Wiseman, Jul 14 2023

nonn

First differs from A334298 in having 600 and lacking 180.
Also numbers whose minimum part in prime factorization is a unique mode.
If k is a term, then so are all powers of k. - Robert Israel, Sep 17 2024

			The prime factorization of 600 is 2*2*2*3*5*5, and 3 > max(1,2), so 600 is in the sequence.
The prime factorization of 180 is 2*2*3*3*5, but 2 <= max(2,1), so 180 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           29: {10}              67: {19}
     2: {1}          31: {11}              68: {1,1,7}
     3: {2}          32: {1,1,1,1,1}       71: {20}
     4: {1,1}        37: {12}              72: {1,1,1,2,2}
     5: {3}          40: {1,1,1,3}         73: {21}
     7: {4}          41: {13}              76: {1,1,8}
     8: {1,1,1}      43: {14}              79: {22}
     9: {2,2}        44: {1,1,5}           80: {1,1,1,1,3}
    11: {5}          45: {2,2,3}           81: {2,2,2,2}
    12: {1,1,2}      47: {15}              83: {23}
    13: {6}          48: {1,1,1,1,2}       84: {1,1,2,4}
    16: {1,1,1,1}    49: {4,4}             88: {1,1,1,5}
    17: {7}          52: {1,1,6}           89: {24}
    19: {8}          53: {16}              92: {1,1,9}
    20: {1,1,3}      56: {1,1,1,4}         96: {1,1,1,1,1,2}
    23: {9}          59: {17}              97: {25}
    24: {1,1,1,2}    60: {1,1,2,3}         99: {2,2,5}
    25: {3,3}        61: {18}             101: {26}
    27: {2,2,2}      63: {2,2,4}          103: {27}
    28: {1,1,4}      64: {1,1,1,1,1,1}    104: {1,1,1,6}

Robert Israel, Table of n, a(n) for n = 1..10000

Allowing any unique mode gives A356862, complement A362605.
Allowing any unique co-mode gives A359178, complement A362606.
The even case is A360013, counted by A241131.
For greatest instead of least we have A362616, counted by A362612.
These partitions are counted by A364193.
A027746 lists prime factors (with multiplicity).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Cf. A098859, A327473, A327476, A360014, A360015, A362610, A364061, A364062.

filter:= proc(n) local F,i;
  F:= ifactors(n)[2];
  if nops(F) = 1 then return true fi;
  i:= min[index](F[..,1]);
  andmap(t -> F[t,2] < F[i,2], {$1..nops(F)} minus {i})
end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Sep 17 2024

Select[Range[100],First[Last/@FactorInteger[#]] > Max@@Rest[Last/@FactorInteger[#]]&]

A364193 Number of integer partitions of n where the least part is the unique mode.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 7, 9, 13, 17, 24, 32, 43, 58, 75, 97, 130, 167, 212, 274, 346, 438, 556, 695, 865, 1082, 1342, 1655, 2041, 2511, 3067, 3756, 4568, 5548, 6728, 8130, 9799, 11810, 14170, 16980, 20305, 24251, 28876, 34366, 40781, 48342, 57206, 67597, 79703

Offset: 0

Gus Wiseman, Jul 16 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (3211)     (611)
                                     (3111)    (4111)     (2222)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For greatest part and multiple modes we have A171979.
Allowing multiple modes gives A240303.
For greatest instead of least part we have A362612, ranks A362616.
For mean instead of least part we have A363723.
These partitions have ranks A364160.
A000041 counts integer partitions.
A362611 counts modes in prime factorization, A362613 co-modes.
A362614 counts partitions by number of modes, co-modes A362615.
A363486 gives least mode in prime indices, A363487 greatest.
A363952 counts partitions by low mode, A363953 high.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A327472, A360015.

Table[If[n==0,0,Length[Select[IntegerPartitions[n], Last[Length/@Split[#]]>Max@@Most[Length/@Split[#]]&]]],{n,0,30}]

A363265 Number of integer factorizations of n with a unique mode.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 6, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Jun 27 2023

nonn

An integer factorization of n is a multiset of positive integers > 1 with product n.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
Conjecture: 9 is missing from this sequence.

			The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:
  (2)  (4)    (16)       (24)       (48)         (72)
       (2*2)  (4*4)      (2*2*6)    (3*4*4)      (2*6*6)
              (2*2*4)    (2*2*2*3)  (2*2*12)     (3*3*8)
              (2*2*2*2)             (2*2*2*6)    (2*2*18)
                                    (2*2*3*4)    (2*2*2*9)
                                    (2*2*2*2*3)  (2*2*3*6)
                                                 (2*3*3*4)
                                                 (2*2*2*3*3)

The complement for partitions is A362607, ranks A362605.
The version for partitions is A362608, ranks A356862.
A001055 counts factorizations, strict A045778, ordered A074206.
A089723 counts constant factorizations.
A316439 counts factorizations by length, A008284 partitions.
A339846 counts even-length factorizations, A339890 odd-length.
Cf. A240219, A326622, A333487, A335434, A347438, A362610, A362611, A362612, A362614, A363723.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n],Length[modes[#]]==1&]],{n,100}]

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A363485 Number of integer partitions of n covering an initial interval of positive integers with more than one mode.

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A364160 Numbers whose least prime factor has the greatest exponent.

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A364193 Number of integer partitions of n where the least part is the unique mode.

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A363265 Number of integer factorizations of n with a unique mode.

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