A360014 -id:A360014 - cp's OEIS frontend

A360015 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.

1, 2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 100, 102, 104, 106, 110, 112, 114, 116, 118, 120, 122, 124, 128, 130, 132, 134, 136, 138

Offset: 1

Amiram Eldar, Jan 21 2023

Numbers k such that A007814(k) = A051903(k).
The powers of 2 (A000079) are all terms.
The product of any two terms (not necessarily distinct) is also a term.
This sequence is a disjoint union of {1} and the subsequences of numbers m of the form 2^(k-1)*o where o = A000265(m), the odd part of m, is a k-free number, for k >= 2. These subsequences include, for k = 2, numbers of the form 2*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 4*o where o is an odd cubefree number; etc.
The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 0.44541445377638761933... .
The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} (k-1)/(zeta(k)*(2^k-1)) / Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 2.10346728882748723133... . [corrected by Amiram Eldar, Jul 10 2025]
Also numbers whose multiset of prime factors has low (i.e. least) mode 2. Here, 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}. - Gus Wiseman, Jul 14 2023

			From _Gus Wiseman_, Jul 14 2023: (Start)
108 = 2*2*3*3*3 is missing because its mode is not 2.
180 = 2*2*3*3*5 is present because it has low mode 2.
The terms together with their prime factorizations begin:
   1 =
   2 = 2
   4 = 2*2
   6 = 2*3
   8 = 2*2*2
  10 = 2*5
  12 = 2*2*3
  14 = 2*7
  16 = 2*2*2*2
  20 = 2*2*5
  22 = 2*11
  24 = 2*2*2*3
  26 = 2*13
  28 = 2*2*7
  30 = 2*3*5
  32 = 2*2*2*2*2
  34 = 2*17
  36 = 2*2*3*3
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000079, A000265, A007814, A051903, A056911.
Partitions of this type are counted by A241131.
The case of unique mode is A360013, complement here A360014.
For unique minimal prime exponent we have A364061, counted by A364062.
For minimal prime exponent we have A364158, counted by A364159.
A027746 lists prime factors (with multiplicity).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
Cf. A002865, A327473, A327476, A356862, A359178, A362605, A362613, A363486.

q[n_] := IntegerExponent[n, 2] == Max[FactorInteger[n][[;; , 2]]]; q[1] = True; Select[Range[150], q]

is(n) = n == 1 || vecmax(factor(n)[,2]) == valuation(n, 2);

Disjoint union of A360013 and A360014.

a(n) = A360013(n)/2. - Gus Wiseman, Jul 14 2023

A360013 Numbers whose exponent of 2 in their canonical prime factorization is larger than all the other exponents.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 104, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 184, 188, 192, 200, 204, 208, 212, 220, 224, 228, 232, 236, 240, 244, 248, 256

Offset: 1

Amiram Eldar, Jan 21 2023

Numbers k such that A007814(k) > A051903(A000265(k)).
The powers of 2 (A000079), except for 1, are all terms.
The product of any two terms (not necessarily distinct) is also a term.
This sequence is a disjoint union of {2} and the subsequences of numbers m of the form 2^k*o where o = A000265(m), the odd part of m, is a k-free number, for k >= 2. These subsequences include, for k = 2, numbers of the form 4*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 8*o where o is an odd cubefree number; etc.
The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 0.222707226888193809... .
The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} k/(zeta(k)*2*(2^k-1)) / Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 3.10346728882748723133... . [corrected by Amiram Eldar, Jul 10 2025]
This sequence is a subsequence of A360015 and the asymptotic density of this sequence within A360015 is exactly 1/2.
Also even numbers whose multiset of prime factors has unique mode 2. - Gus Wiseman, Jul 10 2023

			From _Gus Wiseman_, Jul 09 2023: (Start)
108 = 2*2*3*3*3 is missing because its mode is not 2.
180 = 2*2*3*3*5 is missing because 2 is not the unique mode.
120 = 2*2*2*3*5 is present because its unique mode is 2.
The terms together with their prime factorizations begin:
   2 = 2
   4 = 2*2
   8 = 2*2*2
  12 = 2*2*3
  16 = 2*2*2*2
  20 = 2*2*5
  24 = 2*2*2*3
  28 = 2*2*7
  32 = 2*2*2*2*2
  40 = 2*2*2*5
  44 = 2*2*11
  48 = 2*2*2*2*3
  52 = 2*2*13
  56 = 2*2*2*7
  60 = 2*2*3*5
  64 = 2*2*2*2*2*2
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000079, A000265, A007814, A051903, A056911.
Equals A360015 \ A360014.
Partitions of this type are counted by A241131.
Allowing any unique mode gives A356862, complement A362605.
Allowing any unique co-mode gives A359178, complement A362606.
Not requiring the mode to be unique gives A360015.
The opposite version is A362616, counted by A362612.
For co-mode instead of mode we have A364061, counted by A364062.
With least prime factor instead of 2, we have A364160, counted by A364193.
With a different factorization, we have the subsequence A335738.
A124010 gives prime signature, ordered A118914.
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. A001222, A002865, A327473, A327476, A362608, A362610, A363723, A363727.

q[n_] := Module[{e = IntegerExponent[n, 2], m}, m = n/2^e; (m == 1 && e > 0) || AllTrue[FactorInteger[m][[;; , 2]], # < e &]]; Select[Range[256], q]

is(n) = {my(e = valuation(n, 2), m = n >> e); (m == 1 && e > 0) || (m > 1 && vecmax(factor(m)[,2]) < e)};

a(n) = 2*A360015(n). - Gus Wiseman, Jul 10 2023

A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

Original entry on oeis.org

2, 4, 8, 16, 18, 32, 50, 54, 64, 98, 108, 128, 162, 242, 250, 256, 324, 338, 450, 486, 500, 512, 578, 648, 686, 722, 882, 972, 1024, 1058, 1250, 1350, 1372, 1458, 1682, 1922, 1944, 2048, 2178, 2250, 2450, 2500, 2646, 2662, 2738, 2916, 3042, 3362, 3698, 3888

Offset: 1

Gus Wiseman, Jul 12 2023

nonn

Also numbers whose multiset of prime factors has unique co-mode 2. Here, a co-mode in a multiset is 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 terms together with their prime factors begin:
    2 = 2
    4 = 2*2
    8 = 2*2*2
   16 = 2*2*2*2
   18 = 2*3*3
   32 = 2*2*2*2*2
   50 = 2*5*5
   54 = 2*3*3*3
   64 = 2*2*2*2*2*2
   98 = 2*7*7
  108 = 2*2*3*3*3
  128 = 2*2*2*2*2*2*2

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

For any unique co-mode: A359178, counted by A362610, complement A362606.
For high mode: A360013, positions of 1's in A363487, counted by A241131.
For low mode: A360015, positions of 1's in A363486, counted by A241131.
Partitions of this type are counted by A364062.
For low co-mode: A364158, positions of 1's in A364192, counted by A364159.
Positions of 1's in A364191, high A364192.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions w/ unique mode, count A362608, complement A362605.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Cf. A000265, A007814, A327473, A327476, A362616, A360014, A363722, A363723, A363725, A363727, A363730.

filter:= proc(n) local F,F2,Fo;
  F:= ifactors(n)[2];
  F2,Fo:= selectremove(t -> t[1]=2, F);
  Fo = [] or F2[1,2] < min(Fo[..,2])
end proc:
select(filter, 2*[$1..5000]); # Robert Israel, Apr 22 2024

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

from sympy import factorint
from itertools import count, islice
def A364061_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(l:=(~n&n-1).bit_length()) < min(factorint(m:=n>>l).values(),default=0) or m==1, count(max(startvalue+startvalue&1,2),2))
A364061_list = list(islice(A364061_gen(),30)) # Chai Wah Wu, Jul 14 2023

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} (1-1/2^(k-1))*(s(k)-s(k+1)) = 1.16896822653093929144..., where s(k) = Product_{primes p >= 3} (1 + 1/(p^(k-1)*(p-1))) is the sum of reciprocals of the odd k-full numbers (numbers whose prime factorization has no exponent that is smaller than k). - Amiram Eldar, Aug 30 2024

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}]

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).

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}]

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[#]]&]

A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 8, 6, 15, 13, 19, 25, 33, 36, 54, 58, 80, 96, 122, 141, 188, 217, 274, 326, 408, 474, 600, 695, 859, 1012, 1233, 1440, 1763, 2050, 2475, 2899, 3476, 4045, 4850, 5630, 6695, 7797, 9216, 10689, 12628, 14611, 17162, 19875, 23253

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(3) = 1 through a(10) = 8 partitions:
  (21)  (31)  (41)  (51)    (61)   (71)    (81)      (91)
                    (321)   (421)  (431)   (531)     (541)
                    (2211)         (521)   (621)     (631)
                                   (3311)  (32211)   (721)
                                           (222111)  (4321)
                                                     (4411)
                                                     (33211)
                                                     (42211)

First differences of A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A360014.
Equal case of A381544 (ranks A381439).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max = 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 = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A051904, A091605, A116861, A237984, A239455, A362608, A363724, A381079, A381437, A381438.

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

A382526 Number of integer partitions of n with fewer ones than greatest multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 41, 56, 72, 94, 124, 158, 205, 262, 331, 419, 531, 663, 829, 1033, 1281, 1581, 1954, 2393, 2936, 3584, 4366, 5300, 6433, 7764, 9374, 11277, 13548, 16225, 19425, 23166, 27623, 32842, 39004, 46212, 54719, 64610, 76251

Offset: 0

Gus Wiseman, Apr 05 2025

nonn

			The a(2) = 1 through a(9) = 12 partitions:
  (2)  (3)  (4)   (5)    (6)    (7)     (8)      (9)
            (22)  (32)   (33)   (43)    (44)     (54)
                  (221)  (42)   (52)    (53)     (63)
                         (222)  (322)   (62)     (72)
                                (331)   (332)    (333)
                                (2221)  (422)    (432)
                                        (2222)   (441)
                                        (3221)   (522)
                                        (22211)  (3222)
                                                 (3321)
                                                 (4221)
                                                 (22221)

The complement (greater than or equal to) is A241131 except first, ranks A360015.
The opposite version (greater than) is A241131 shifted except first, ranks A360013.
These partitions have ranks A382856, complement A360015.
The weak version (less than or equal to) is A381544, ranks A381439.
For equality we have 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]

A382856 Numbers whose prime indices do not have a mode of 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 108, 109, 111, 113, 115

Offset: 1

Gus Wiseman, Apr 07 2025

nonn

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}

The case of non-unique mode is A024556.
The complement is A360015 except first.
Partitions of this type are are counted by A382526 except first, complement A241131.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A112798 lists prime indices, length A001222, sum A056239.
A116598 counts ones in partitions, rank statistic A007814.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A362611 counts modes in prime indices, triangle A362614.
For co-mode see A359178, A362613, A364061 (A364062), A364158 (A364159).
Cf. A000265, A002865, A106529, A327473, A327476, A362605, A363486, A356862, A360013, A360014, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],FreeQ[Commonest[prix[#]],1]&]

cp's OEIS Frontend

A360015 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A360013 Numbers whose exponent of 2 in their canonical prime factorization is larger than all the other exponents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

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

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

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A364160 Numbers whose least prime factor has the greatest exponent.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

Views

Author

Keywords