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.

Previous Showing 21-30 of 56 results. Next

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 19, 21, 30, 31, 42, 50, 62, 69, 91, 99, 126, 144, 175, 198, 246, 275, 331, 379, 452, 509, 612, 686, 811, 922, 1076, 1219, 1428, 1604, 1863, 2108, 2434, 2739, 3162, 3551, 4075, 4593, 5240, 5885, 6721, 7527, 8556, 9597, 10870
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 20 2010

Keywords

Comments

A000009(n) <= a(n) <= A000041(n).
Equivalently, the number of partitions of n such that (maximal multiplicity of parts) = (multiplicity of the maximal part), as in the Mathematica program. - Clark Kimberling, Apr 04 2014
Also the number of integer partitions of n whose greatest part is a mode, meaning it appears at least as many times as each of the others. The name "Number of partitions of n such that smaller parts do not occur more frequently than greater parts" seems to describe A100882 = "Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing," which first differs from this at n = 10 due to the partition (3,3,2,1,1). - Gus Wiseman, May 07 2023

Examples

			a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.

Links

Crossrefs

For median instead of mode we have A053263.
The complement is counted by A240302.
The case where the maximum is the only mode is A362612.
A000041 counts integer partitions, strict A000009.
A362608 counts partitions with a unique mode, complement A362607.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes.
Cf. A002865, A008284, A098859, A237984, A275870, A325347, A362610.

Programs

  • Mathematica
    z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)
(* Clark Kimberling, Apr 04 2014 *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
     If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1,
     If[k == -1, j, If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
a[n_] := PartitionsP[n] - b[n, n, -1];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz in A240302 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Commonest[#],Max[#]]&]],{n,0,30}] (* Gus Wiseman, May 07 2023 *)
  • PARI
    { my(N=53, x='x+O('x^N));
my(gf=1+sum(i=1,N,sum(j=1,floor(N/i),x^(i*j)*prod(k=1,i-1,(1-x^(k*(j+1)))/(1-x^k)))));
Vec(gf) } \\ John Tyler Rascoe, Mar 09 2024

Formula

a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k) +p(n,max(i,j),1,k+1) else (if j0 then 0 else 1).
a(n) + A240302(n) = A000041(n). - Clark Kimberling, Apr 04 2014.
G.f.: 1 + Sum_{i, j>0} x^(i*j) * Product_{k=1..i-1} ((1 - x^(k*(j+1)))/(1 - x^k)). - John Tyler Rascoe, Mar 09 2024

A362621 One and numbers whose multiset of prime factors (with multiplicity) has the same median as maximum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169
Offset: 1

Views

Author

Gus Wiseman, May 12 2023

Keywords

Comments

First differs from A334965 in having 750 and lacking 2250.
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 factorization of 108 is 2*2*3*3*3, and the multiset {2,2,3,3,3} has median 3 and maximum 3, so 108 is in the sequence.
The prime factorization of 2250 is 2*3*3*5*5*5, and the multiset {2,3,3,5,5,5} has median 4 and maximum 5, so 2250 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           25: {3,3}           64: {1,1,1,1,1,1}
     2: {1}          27: {2,2,2}         67: {19}
     3: {2}          29: {10}            71: {20}
     4: {1,1}        31: {11}            73: {21}
     5: {3}          32: {1,1,1,1,1}     75: {2,3,3}
     7: {4}          37: {12}            79: {22}
     8: {1,1,1}      41: {13}            81: {2,2,2,2}
     9: {2,2}        43: {14}            83: {23}
    11: {5}          47: {15}            89: {24}
    13: {6}          49: {4,4}           97: {25}
    16: {1,1,1,1}    50: {1,3,3}         98: {1,4,4}
    17: {7}          53: {16}           101: {26}
    18: {1,2,2}      54: {1,2,2,2}      103: {27}
    19: {8}          59: {17}           107: {28}
    23: {9}          61: {18}           108: {1,1,2,2,2}

Crossrefs

Partitions of this type are counted by A053263.
For mode instead of median we have A362619, counted by A171979.
For parts at middle position (instead of median) we have A362622.
The complement is A362980, counted by A237821.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
Cf. A000040, A002865, A237824, A327473, A327476, A359908, A362616, A362620.

Programs

  • Mathematica
    Select[Range[100],(y=Flatten[Apply[ConstantArray,FactorInteger[#],{1}]];Max@@y==Median[y])&]

A363730 Numbers whose prime indices have different mean, median, and mode.

Original entry on oeis.org

42, 60, 66, 70, 78, 84, 102, 114, 130, 132, 138, 140, 150, 154, 156, 165, 170, 174, 180, 182, 186, 190, 195, 204, 220, 222, 228, 230, 231, 246, 255, 258, 260, 266, 276, 282, 285, 286, 290, 294, 308, 310, 315, 318, 322, 330, 340, 345, 348, 354, 357, 360, 364
Offset: 1

Views

Author

Gus Wiseman, Jun 24 2023

Keywords

Comments

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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.
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 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 indices of 180 are {1,1,2,2,3}, with mean 9/5, median 2, modes {1,2}, so 180 is in the sequence.
The prime indices of 108 are {1,1,2,2,2}, with mean 8/5, median 2, modes {2}, so 108 is not in the sequence.
The terms together with their prime indices begin:
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}
  140: {1,1,3,4}
  150: {1,2,3,3}

Crossrefs

These partitions are counted by A363720
For equal instead of unequal we have A363727, counted by A363719.
The version for factorizations is A363742, equal A363741.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
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.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A000961, A327473, A327476, A359908, A363722, A363725, A363729.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],{Mean[prix[#]]}!={Median[prix[#]]}!=modes[prix[#]]&]

Formula

All three of A326567(a(n))/A326568(a(n)), A360005(a(n))/2, and A363486(a(n)) = A363487(a(n)) are different.

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

Views

Author

Gus Wiseman, Jul 12 2023

Keywords

Comments

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}.

Examples

			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

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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}&]
  • Python
    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

Formula

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

A362622 One and numbers whose prime factorization has its greatest part at a middle position.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91
Offset: 1

Views

Author

Gus Wiseman, May 12 2023

Keywords

Examples

			The prime factorization of 150 is 5*5*3*2, with middle parts {3,5}, so 150 is in the sequence.
The prime factorization of 90 is 5*3*3*2, with middle parts {3,3}, so 90 is not in the sequence.

Crossrefs

Partitions of this type are counted by A237824.
For modes instead of middles we have A362619, counted by A171979.
The version for median instead of middles is A362621, counted by A053263.
The complement for median is A362980, counted by A237821.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization.
A362613 counts co-modes in prime factorization.
Cf. A000040, A002865, A237984, A327473, A327476, A359908, A362616, A362620.

Programs

  • Mathematica
    mpm[q_]:=MemberQ[If[OddQ[Length[q]],{Median[q]},{q[[Length[q]/2]],q[[Length[q]/2+1]]}],Max@@q];
Select[Range[100],#==1||mpm[Flatten[Apply[ConstantArray,FactorInteger[#],{1}]]]&]

A363127 Number of non-modes in the multiset of prime factors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, May 16 2023

Keywords

Comments

A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.

Examples

			The prime factorization of 13860 is 2*2*3*3*5*7*11, with non-modes {5,7,11}, so a(13860) = 3.

Links

Crossrefs

Positions of first appearances converge to A088860.
For modes instead of non-modes we have A362611, triangle A362614.
For co-modes instead of non-modes we have A362613, triangle A362615.
The triangle for this rank statistic (number of non-modes) is A363126.
For non-co-modes instead of non-modes we have A363131, triangle A363130.
A027746 lists prime factors, A112798 indices, A124010 exponents.
A047966 counts uniform partitions, ranks A072774.
A363124 counts partitions with more than one non-mode, complement A363125.
Cf. A001221, A001222, A002865, A051903, A056239, A353863, A362616.

Programs

  • Maple
    f:= proc(n) local F,m;
  F:= ifactors(n)[2][..,2];
  m:= max(F);
  nops(select(`<`,F,m))
end proc;
map(f, [$1..100]); # Robert Israel, Aug 01 2025
  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
nmsi[ms_]:=Select[Union[ms],Count[ms,#]

A363952 Number of integer partitions of n with low mode k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 4, 2, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 9, 3, 2, 0, 0, 0, 1, 0, 13, 5, 2, 1, 0, 0, 0, 1, 0, 18, 6, 3, 2, 0, 0, 0, 0, 1, 0, 26, 9, 3, 2, 1, 0, 0, 0, 0, 1, 0, 32, 13, 5, 3, 2, 0, 0, 0, 0, 0, 1, 0, 47, 16, 7, 3, 2, 1, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jul 07 2023

Keywords

Comments

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 A124943, the "low mode" of a multiset is the least mode.

Examples

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   0   1
   0   3   1   0   1
   0   4   2   0   0   1
   0   7   2   1   0   0   1
   0   9   3   2   0   0   0   1
   0  13   5   2   1   0   0   0   1
   0  18   6   3   2   0   0   0   0   1
   0  26   9   3   2   1   0   0   0   0   1
   0  32  13   5   3   2   0   0   0   0   0   1
   0  47  16   7   3   2   1   0   0   0   0   0   1
   0  60  21  10   4   3   2   0   0   0   0   0   0   1
   0  79  30  13   6   3   2   1   0   0   0   0   0   0   1
   0 104  38  17   7   4   3   2   0   0   0   0   0   0   0   1
Row n = 8 counts the following partitions:
  .  (71)        (62)     (53)   (44)  .  .  .  (8)
     (611)       (422)    (332)
     (521)       (3221)
     (5111)      (2222)
     (431)       (22211)
     (4211)
     (41111)
     (3311)
     (32111)
     (311111)
     (221111)
     (2111111)
     (11111111)

Crossrefs

Row sums are A000041.
For median: A124943 (high A124944), rank statistic A363941 (high A363942).
Column k = 1 is A241131 (partitions w/ low mode 1), ranks A360015, A360013.
The rank statistic for this triangle is A363486.
For mean: A363945 (high A363946), rank statistic A363943 (high A363944).
The high version is A363953.
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, A067538, A237984, A363723, A363724, A363731.

Programs

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

A362619 One and all numbers whose greatest prime factor is a mode, meaning it appears at least as many times as each of the others.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83
Offset: 1

Views

Author

Gus Wiseman, May 09 2023

Keywords

Comments

First differs from A304678 in having 300.

Examples

			The prime factorization of 300 is 2*2*3*5*5, with modes {2,5} and maximum 5, so 300 is in the sequence.

Links

Crossrefs

Partitions of this type are counted by A171979.
The case of a unique mode is A362616, counted by A362612.
The complement is A362620, counted by A240302.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with a more than one mode, counted by A362607.
A362606 ranks partitions with a more than one co-mode, counted by A362609.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
A362621 ranks partitions with median equal to maximum, counted by A053263.
Cf. A000040, A002865, A237824, A237984, A327473, A327476, A359908.

Programs

  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],MemberQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

90, 270, 525, 550, 756, 810, 1666, 1911, 1950, 2268, 2430, 2625, 2695, 2700, 2750, 5566, 6762, 6804, 6897, 7128, 7290, 8100, 8500, 9310, 9750, 10285, 10478, 11011, 11550, 11662, 12250, 12375, 12495, 13125, 13377, 13750, 14014, 14703, 18865, 19435, 20412, 21384
Offset: 1

Views

Author

Gus Wiseman, Jun 24 2023

Keywords

Comments

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.
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 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 indices of 6897 are {2,5,5,8}, with mean 5, median 5, and modes {5}, so 6897 is in the sequence.
The terms together with their prime indices begin:
     90: {1,2,2,3}
    270: {1,2,2,2,3}
    525: {2,3,3,4}
    550: {1,3,3,5}
    756: {1,1,2,2,2,4}
    810: {1,2,2,2,2,3}
   1666: {1,4,4,7}
   1911: {2,4,4,6}
   1950: {1,2,3,3,6}
   2268: {1,1,2,2,2,2,4}
   2430: {1,2,2,2,2,2,3}

Crossrefs

For just primes instead of prime powers we have A363722.
Including prime-powers gives A363727, counted by A363719.
These partitions are counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A000961 lists the prime powers, complement A024619.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
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.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725, A363741.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[1000],!PrimePowerQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 07 2023

Keywords

Comments

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.

Examples

			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)

Crossrefs

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.

Programs

  • Mathematica
    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}]
Previous Showing 21-30 of 56 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.