A360015 -id:A360015 - cp's OEIS frontend

A363953 Number of integer partitions of n with high mode k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 2, 2, 1, 1, 1, 0, 7, 2, 1, 2, 1, 1, 1, 0, 9, 4, 2, 2, 2, 1, 1, 1, 0, 13, 6, 2, 2, 2, 2, 1, 1, 1, 0, 18, 7, 4, 3, 3, 2, 2, 1, 1, 1, 0, 26, 10, 5, 2, 3, 3, 2, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Jul 07 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}.

Extending the terminology of A124944, the "high mode" in a multiset is the greatest mode.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  1  1  1
  0  3  1  1  1  1
  0  4  2  2  1  1  1
  0  7  2  1  2  1  1  1
  0  9  4  2  2  2  1  1  1
  0 13  6  2  2  2  2  1  1  1
  0 18  7  4  3  3  2  2  1  1  1
  0 26 10  5  2  3  3  2  2  1  1  1
  0 32 15  8  4  4  4  3  2  2  1  1  1
  0 47 19  9  5  3  4  4  3  2  2  1  1  1
  0 60 26 13  7  5  5  5  4  3  2  2  1  1  1
  0 79 34 18 10  6  5  5  5  4  3  2  2  1  1  1
Row n = 9 counts the following partitions:
  .  (711)        (522)     (333)   (441)  (54)   (63)   (72)  (81)  (9)
     (6111)       (4221)    (3321)  (432)  (531)  (621)
     (5211)       (3222)
     (51111)      (32211)
     (4311)       (22221)
     (42111)      (222111)
     (411111)
     (33111)
     (321111)
     (3111111)
     (2211111)
     (21111111)
     (111111111)

Row sums are A000041.
For median: A124944 (low A124943), rank statistic A363942 (low A363941).
Column k = 1 is A241131 (partitions w/ high mode 1), ranks A360013, A360015.
The rank statistic for this triangle is A363487, low A363486.
For mean: A363946 (low A363945), rank statistic A363944 (low A363943).
The low version is A363952.
A008284 counts partitions by length, A058398 by mean.
A362612 counts partitions (max part) = (unique mode), ranks A362616.
A362614 counts partitions by number of modes, rank statistic A362611.
A362615 counts partitions by number of co-modes, rank statistic A362613.
Cf. A362610, A362608, A362607, A362609; A359178, A356862, A362605, A362606.
Cf. A002865, A025065, A026905, A237984, A363723, A363724, A363731.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], If[Length[#]==0,0,Last[modes[#]]]==k&]],{n,0,15},{k,0,n}]

A364062 Number of integer partitions of n with unique co-mode 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 6, 2, 8, 6, 9, 6, 16, 7, 21, 12, 23, 18, 39, 17, 47, 32, 59, 40, 86, 44, 110, 72, 131, 95, 188, 103, 233, 166, 288, 201, 389, 244, 490, 347, 587, 440, 794, 524, 974, 727, 1187, 903, 1547, 1106, 1908, 1459, 2303, 1826, 2979, 2198

Offset: 0

Gus Wiseman, Jul 12 2023

nonn

These are partitions with at least one 1 but with fewer 1's than each of the other parts.

We define a co-mode in a multiset to be an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			The a(n) partitions for n = 5, 7, 11, 13, 15:
  (221)    (331)      (551)          (661)            (771)
  (11111)  (2221)     (33221)        (4441)           (44331)
           (1111111)  (33311)        (33331)          (55221)
                      (222221)       (44221)          (442221)
                      (2222111)      (332221)         (3322221)
                      (11111111111)  (2222221)        (3333111)
                                     (22222111)       (22222221)
                                     (1111111111111)  (222222111)
                                                      (111111111111111)

For high (or unique) mode we have A241131, ranks A360013.
For low mode we have A241131, ranks A360015.
Allowing any unique co-mode gives A362610, ranks A359178.
These partitions have ranks A364061.
Adding all 1-free partitions gives A364159, ranks A364158.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A237984 counts partitions containing their mean, ranks A327473.
A327472 counts partitions not containing their mean, ranks A327476.
A362608 counts partitions w/ unique mode, ranks A356862, complement A362605.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Cf. A002865, A027336, A325347, A098859, A360014, A362562, A362606, A363720, A363723, A363725, A363726.

comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],comodes[#]=={1}&]],{n,0,30}]

A364158 Numbers whose multiset of prime factors has low (i.e. least) co-mode 2.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 16, 18, 22, 26, 30, 32, 34, 36, 38, 42, 46, 50, 54, 58, 62, 64, 66, 70, 74, 78, 82, 86, 90, 94, 98, 100, 102, 106, 108, 110, 114, 118, 122, 126, 128, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194

Offset: 1

Gus Wiseman, Jul 14 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Except for 1, this is the lists of all even numbers whose prime factorization contains at most as many 2's as non-2 parts.
Extending the terminology of A124943, the "low co-mode" of a multiset is the least co-mode.

			The terms together with their prime factorizations begin:
   1 =
   2 = 2
   4 = 2*2
   6 = 2*3
   8 = 2*2*2
  10 = 2*5
  14 = 2*7
  16 = 2*2*2*2
  18 = 2*3*3
  22 = 2*11
  26 = 2*13
  30 = 2*3*5
  32 = 2*2*2*2*2
  34 = 2*17
  36 = 2*2*3*3

Partitions of this type are counted by A364159.
Positions of 1's in A364191, high A364192, modes A363486, high A363487.
For median we have A363488, positions of 1 in A363941, triangle A124943.
For mode instead of co-mode we have A360015, counted by A241131.
A027746 lists prime factors (with multiplicity), length A001222.
A362611 counts modes in prime factorization, triangle A362614
A362613 counts co-modes in prime factorization, triangle A362615
Ranking 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. A215366, A327473, A327476, A360005.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Select[Range[100],#==1||Min[comodes[prifacs[#]]]==2&]

A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 15, 20, 23, 32, 40, 50, 61, 82, 95, 126, 149, 188, 228, 292, 337, 430, 510, 633, 748, 933, 1083, 1348, 1579, 1925, 2262, 2761, 3197, 3893, 4544, 5458, 6354, 7634, 8835, 10577, 12261, 14546, 16864, 19990, 23043, 27226, 31428

Offset: 0

Gus Wiseman, Jul 16 2023

nonn

Also integer partitions of n with least co-mode 1. Here, we define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (221)    (321)     (331)      (431)
                            (11111)  (2211)    (421)      (521)
                                     (111111)  (2221)     (3221)
                                               (1111111)  (3311)
                                                          (22211)
                                                          (11111111)

For mode instead of co-mode we have A241131, ranks A360015.
The case with only one 1 is A364062, ranks A364061.
Counts partitions ranked by A364158.
Counts positions of 1's in A364191, high A364192.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
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. A027336, A124943, A237984, A327472, A363486, A363487.

Table[Length[Select[IntegerPartitions[n-1],Count[#,1]

A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, Mar 02 2025

nonn

First differs from A335740 in lacking 72, which has prime indices {1,1,1,2,2} and section-sum partition (3,3,1).
Also numbers whose section-sum partition of prime indices (A381436) ends with a number > 1.
Includes all squarefree numbers (A005117) except 2.

			The terms together with their prime indices begin:
     3: {2}        25: {3,3}        45: {2,2,3}
     5: {3}        26: {1,6}        46: {1,9}
     6: {1,2}      27: {2,2,2}      47: {15}
     7: {4}        29: {10}         49: {4,4}
     9: {2,2}      30: {1,2,3}      50: {1,3,3}
    10: {1,3}      31: {11}         51: {2,7}
    11: {5}        33: {2,5}        53: {16}
    13: {6}        34: {1,7}        54: {1,2,2,2}
    14: {1,4}      35: {3,4}        55: {3,5}
    15: {2,3}      36: {1,1,2,2}    57: {2,8}
    17: {7}        37: {12}         58: {1,10}
    18: {1,2,2}    38: {1,8}        59: {17}
    19: {8}        39: {2,6}        61: {18}
    21: {2,4}      41: {13}         62: {1,11}
    22: {1,5}      42: {1,2,4}      63: {2,2,4}
    23: {9}        43: {14}         65: {3,6}

The LHS (exponent of 2) is A007814.
The complement is A360013 = 2*A360015 (if we prepend 1), counted by A241131 (shifted right and starting with 1 instead of 0).
The case of equality is A360014, inclusive A360015.
The RHS (greatest exponent of an odd prime factor) is A375669.
These are positions of terms > 1 in A381437.
Partitions of this type are counted by A381544.
A000040 lists the primes, differences A001223.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381436 gives section-sum partition of prime indices, Heinz number A381431.
A381438 counts partitions by last part part of section-sum partition.
Cf. A000720, A001221, A001222, A001694, A003557, A005117, A066328, A130091, A181819, A212166, A380955.

Select[Range[100],FactorInteger[2*#][[1,2]]-1<=Max@@Last/@Rest[FactorInteger[2*#]]&]

Positive integers n such that A007814(n) <= A375669(n).

A364191 Low co-mode in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 1, 8, 3, 2, 1, 9, 2, 3, 1, 2, 4, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 4, 1, 2, 6, 16, 1, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 2, 8, 4, 1, 22, 3, 2, 1

Offset: 1

Gus Wiseman, Jul 16 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "low co-mode" in a multiset is the least co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 2.

For prime factors instead of indices we have A067695, high A359612.
For mode instead of co-mode we have A363486, high A363487, triangle A363952.
For median instead of co-mode we have A363941, high A363942.
Positions of 1's are A364158, counted by A364159.
The high version is A364192 = positions of 1's in A364061.
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
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. A124943, A241131, A327473, A327476, A360005, A360015, A363488.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Min[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A067695(n)).

A067695(n) = A000040(a(n)).

A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 3, 1, 7, 1, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 3, 13, 4, 14, 5, 3, 9, 15, 2, 4, 1, 7, 6, 16, 1, 5, 4, 8, 10, 17, 3, 18, 11, 4, 1, 6, 5, 19, 7, 9, 4, 20, 2, 21, 12, 2, 8, 5, 6, 22, 3, 2

Offset: 1

Gus Wiseman, Jul 16 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "high co-mode" in a multiset is the greatest co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 4.

For prime factors instead of indices we have A359612, low A067695.
For mode instead of co-mode we have A363487 (triangle A363953), low A363486 (triangle A363952).
The version for median instead of co-mode is A363942, low A363941.
Positions of 1's are A364061, counted by A364062.
The low version is A364191, 1's at A364158 (counted by A364159).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
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. A241131, A327473, A327476, A360005, A360015, A362612, A363740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Max[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A359612(n)).

A359612(n) = A000040(a(n)).

A381544 Number of integer partitions of n not containing more ones than any other part.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A381439.
The case of equality is A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 104, 112, 116, 120, 124, 128, 136, 144, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 200, 208, 212, 224, 232, 236, 240, 244, 248, 256, 264, 268, 272, 280, 284, 288, 292

Offset: 1

Gus Wiseman, Jul 07 2023

nonn

The multiset of prime factors of n is row n of A027746.

The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest of the two middle parts (for even length).

			The terms together with their prime indices begin:
     2: {1}             64: {1,1,1,1,1,1}      136: {1,1,1,7}
     4: {1,1}           68: {1,1,7}            144: {1,1,1,1,2,2}
     8: {1,1,1}         72: {1,1,1,2,2}        148: {1,1,12}
    12: {1,1,2}         76: {1,1,8}            152: {1,1,1,8}
    16: {1,1,1,1}       80: {1,1,1,1,3}        160: {1,1,1,1,1,3}
    20: {1,1,3}         88: {1,1,1,5}          164: {1,1,13}
    24: {1,1,1,2}       92: {1,1,9}            168: {1,1,1,2,4}
    28: {1,1,4}         96: {1,1,1,1,1,2}      172: {1,1,14}
    32: {1,1,1,1,1}    104: {1,1,1,6}          176: {1,1,1,1,5}
    40: {1,1,1,3}      112: {1,1,1,1,4}        184: {1,1,1,9}
    44: {1,1,5}        116: {1,1,10}           188: {1,1,15}
    48: {1,1,1,1,2}    120: {1,1,1,2,3}        192: {1,1,1,1,1,1,2}
    52: {1,1,6}        124: {1,1,11}           200: {1,1,1,3,3}
    56: {1,1,1,4}      128: {1,1,1,1,1,1,1}    208: {1,1,1,1,6}

Partitions of this type are counted by A027336.
Median of prime indices is A360005(n)/2.
For mode instead of median we have A360013, low A360015.
The low version is A363488, positions of 1's in A363941.
Positions of 1's in A363942.
A112798 lists prime indices, length A001222, sum A056239.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A124943 counts partitions by low median, high A124944.
Cf. A072978, A215366, A316413, A359908, A363727, A363740, A363949.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
merr[y_]:=If[Length[y]==0,0,If[OddQ[Length[y]],y[[(Length[y]+1)/2]], y[[1+Length[y]/2]]]];
Select[Range[100],merr[prifacs[#]]==2&]

A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 6, 0, 0, 0, 96, 0

Offset: 0

Gus Wiseman, Apr 09 2025

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 24 are {1,1,1,2}, with permutations (1,1,1,2) and (2,1,1,1), so a(4) = 2.

For anti-run permutations we have A335407, see also A335125, A382858.
This is the restriction of A382771 to the factorials A000142, equal A382857.
A022559 counts prime indices of n!, sum A081401.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A181821, A351013, A351202, A360015, A382773, A382879.

Table[Length[Select[Permutations[prix[n!]],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

a(n) = A382771(n!).

cp's OEIS Frontend

A363953 Number of integer partitions of n with high mode k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A364062 Number of integer partitions of n with unique co-mode 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A364158 Numbers whose multiset of prime factors has low (i.e. least) co-mode 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A364191 Low co-mode in the multiset of prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A381544 Number of integer partitions of n not containing more ones than any other part.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

Views

Author

Keywords

Comments