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.

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A363486 Low 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, 1, 6, 1, 2, 1, 7, 2, 8, 1, 2, 1, 9, 1, 3, 1, 2, 1, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 1, 13, 1, 14, 1, 2, 1, 15, 1, 4, 3, 2, 1, 16, 2, 3, 1, 2, 1, 17, 1, 18, 1, 2, 1, 3, 1, 19, 1, 2, 1, 20, 1, 21, 1, 3, 1, 4, 1, 22, 1, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Jun 23 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}.
Extending the terminology of A124943, the "low mode" in a multiset is its least mode.

Crossrefs

Positions of first appearances are 1 and A000040.
Positions of 1's are A360013, counted by A241131.
For greatest instead of least we have A363487.
The version for median is A363941, triangle A124943.
The high version for median is A363942, triangle A124944.
The version for mean instead of mode is A363943, high A363944.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A356862 ranks partitions with a unique mode, counted by A362608.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A362616 ranks partitions (max part) = (unique mode), counted by A362612.
Cf. A000961, A215366, A327473, A327476, A360015, A363723.

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]&];
Table[If[n==1,0,First[modes[prix[n]]]],{n,30}]

A363487 High 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, 1, 6, 4, 3, 1, 7, 2, 8, 1, 4, 5, 9, 1, 3, 6, 2, 1, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 1, 13, 4, 14, 1, 2, 9, 15, 1, 4, 3, 7, 1, 16, 2, 5, 1, 8, 10, 17, 1, 18, 11, 2, 1, 6, 5, 19, 1, 9, 4, 20, 1, 21, 12, 3, 1, 5, 6, 22, 1, 2
Offset: 1

Views

Author

Gus Wiseman, Jul 04 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}.
Extending the terminology of A124944, the "high mode" in a multiset is its greatest mode.

Crossrefs

Positions of first appearances are 1 and A000040.
Positions of 1's are A360015, counted by A241131.
For low instead of high mode we have A363486.
The version for low median is A363941, triangle A124943.
The version for high median is A363942, triangle A124944.
The version for mean instead of mode is A363944, low A363943.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A356862 ranks partitions with a unique mode, counted by A362608.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A362616 ranks partitions (max part) = (unique mode), counted by A362612.
Cf. A000961, A215366, A327473, A327476, A359908, A360013, A363723, 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]&];
Table[If[n==1,0,Last[modes[prix[n]]]],{n,30}]

A362616 Numbers in whose prime factorization the greatest factor is the unique mode.

Original entry on oeis.org

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, 150, 151, 157, 162, 163, 167
Offset: 1

Views

Author

Gus Wiseman, May 05 2023

Keywords

Comments

First differs from A329131 in lacking 450 and having 1500.
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}.

Examples

			The factorization of 90 is 2*3*3*5, modes {3}, so 90 is missing.
The factorization of 450 is 2*3*3*5*5, modes {3,5}, so 450 is missing.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so 900 is missing.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so 1500 is present.
The terms together with their prime indices begin:
     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}
    25: {3,3}        64: {1,1,1,1,1,1}    109: {29}

Crossrefs

First term with given bigomega is A000079.
For median instead of mode we have A053263.
Partitions of this type are counted by A362612.
A112798 lists prime 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 more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362614 counts partitions by number of modes, ranked by A362611.
A362615 counts partitions by number of co-modes, ranked by A362613.
Cf. A000040, A002865, A327473, A327476, A358137, A360687.

Programs

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

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

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[#]]]&]

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}]

A362620 Numbers whose greatest prime factor is not a mode, meaning it appears fewer times than some other.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212
Offset: 1

Views

Author

Gus Wiseman, May 11 2023

Keywords

Comments

First differs from A112769 in lacking 300.

Examples

			The prime factorization of 90 is 2*3*3*5, with modes {3} and maximum 5, so 90 is in the sequence.

Links

Crossrefs

Partitions of this type are counted by A240302.
The complement is A362619, counted by A171979.
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, A362616.

Programs

  • Maple
    filter:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a,b) -> a[1]Robert Israel, Dec 15 2023
  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],FreeQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

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

Views

Author

Gus Wiseman, Jul 14 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.
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.

Examples

			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

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Jul 16 2023

Keywords

Comments

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}.

Examples

			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)

Crossrefs

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.

Programs

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