A363124 -id:A363124 - cp's OEIS frontend

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

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

Gus Wiseman, May 16 2023

nonn

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}.

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

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

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.

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

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
nmsi[ms_]:=Select[Union[ms],Count[ms,#]

A363126 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-modes, all 0's removed.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 8, 1, 10, 9, 3, 11, 13, 6, 15, 18, 9, 13, 24, 18, 1, 25, 24, 25, 3, 19, 36, 40, 6, 29, 41, 52, 13, 33, 45, 79, 19, 42, 57, 95, 36, 1, 39, 68, 133, 54, 3, 62, 72, 158, 87, 6, 55, 87, 214, 121, 13, 81, 95, 250, 177, 24

Offset: 0

Gus Wiseman, May 16 2023

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}.

			Triangle begins:
   1
   1
   2
   3
   4   1
   4   3
   8   3
   6   8   1
  10   9   3
  11  13   6
  15  18   9
  13  24  18   1
  25  24  25   3
  19  36  40   6
  29  41  52  13
  33  45  79  19
  42  57  95  36   1
  39  68 133  54   3
Row n = 9 counts the following partitions:
  (9)          (441)       (3321)
  (54)         (522)       (4221)
  (63)         (711)       (4311)
  (72)         (3222)      (5211)
  (81)         (6111)      (42111)
  (333)        (22221)     (321111)
  (432)        (32211)
  (531)        (33111)
  (621)        (51111)
  (222111)     (411111)
  (111111111)  (2211111)
               (3111111)
               (21111111)

Row sums are A000041.
Row lengths are approximately A000196.
Column k = 0 is A047966.
For modes we have A362614, rank statistic A362611.
For co-modes we have A362615, rank statistic A362613.
Columns k > 1 sum to A363124.
Column k = 1 is A363125.
This rank statistic (number of non-modes) is A363127.
For non-co-modes we have A363130, rank statistic A363131.
A008284/A058398 count partitions by length/mean.
A275870 counts collapsible partitions.
A353836 counts partitions by number of distinct run-sums.
A359893 counts partitions by median.
Cf. A002865, A053263, A098859, A237984, A238478, A327472, A353863, A353864, A362612.

nmsi[ms_]:=Select[Union[ms],Count[ms,#]

A363125 Number of integer partitions of n with a unique non-mode.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 18, 24, 24, 36, 41, 45, 57, 68, 72, 87, 95, 105, 131, 136, 149, 164, 199, 203, 232, 246, 276, 298, 335, 347, 409, 399, 467, 488, 567, 569, 636, 662, 757, 767, 878, 887, 1028, 1030, 1168, 1181, 1342, 1388, 1558, 1570, 1789, 1791

Offset: 0

Gus Wiseman, May 16 2023

nonn

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}.

			The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (32211)
                          (211111)  (311111)   (33111)
                                    (2111111)  (51111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

For middle parts instead of non-modes we have A238478, complement A238479.
For modes instead of non-modes we have A362608, complement A362607.
For co-modes instead of non-modes we have A362610, complement A362609.
The complement is counted by A363124.
For non-co-modes instead of non-modes we have A363129, complement A363128.
A000041 counts integer partitions.
A008284/A058398 count partitions by length/mean.
A362611 counts modes in prime factorization, triangle A362614.
A363127 counts non-modes in prime factorization, triangle A363126.
Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

nmsi[ms_]:=Select[Union[ms],Count[ms,#]

A363128 Number of integer partitions of n with more than one non-co-mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 9, 18, 25, 44, 60, 96, 122, 188, 243, 344, 442, 615, 769, 1047, 1308, 1722, 2150, 2791, 3430, 4405, 5401, 6803, 8326, 10408, 12608, 15641, 18906, 23179, 27935, 34061, 40778, 49451, 59038, 71060, 84604, 101386, 120114, 143358

Offset: 0

Gus Wiseman, May 18 2023

nonn

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

			The a(9) = 1 through a(12) = 9 partitions:
  (32211)  (33211)   (33221)    (43311)
           (42211)   (52211)    (44211)
           (322111)  (322211)   (62211)
                     (332111)   (422211)
                     (422111)   (522111)
                     (3221111)  (3222111)
                                (3321111)
                                (4221111)
                                (32211111)

For parts instead of multiplicities we have
For middles instead of non-co-modes we have A238479, complement A238478.
For modes instead of non-co-modes we have A362607, complement A362608.
For co-modes instead of non-co-modes we have A362609, complement A362610.
For non-modes instead of non-co-modes we have A363124, complement A363125.
The complement is counted by A363129.
A000041 counts integer partitions.
A008284/A058398 count partitions by length/mean.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
A363127 counts non-modes in prime factorization, triangle A363126.
A363131 counts non-co-modes in prime factorization, triangle A363130.
Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

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

A363129 Number of integer partitions of n with a unique non-co-mode.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 18, 24, 37, 43, 64, 81, 99, 129, 162, 201, 247, 303, 364, 457, 535, 653, 765, 943, 1085, 1315, 1517, 1830, 2096, 2516, 2877, 3432, 3881, 4622, 5235, 6189, 7003, 8203, 9261, 10859, 12199, 14216, 15985, 18544, 20777, 24064, 26897

Offset: 0

Gus Wiseman, May 18 2023

nonn

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

			The a(4) = 1 through a(9) = 18 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (3221)     (3222)
                          (3211)    (4211)     (3321)
                          (4111)    (5111)     (4221)
                          (22111)   (22211)    (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (22221)
                                    (311111)   (33111)
                                    (2111111)  (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

For parts instead of multiplicities we have A002133.
For middles instead of non-co-modes we have A238478, complement A238479.
For modes instead of non-co-modes we have A362608, complement A362607.
For co-modes instead of non-co-modes we have A362610, complement A362609.
For non-modes instead of non-co-modes we have A363125, complement A363124.
The complement is counted by A363128.
A000041 counts integer partitions.
A008284/A058398 count partitions by length/mean.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
A363127 counts non-modes in prime factorization, triangle A363126.
A363131 counts non-co-modes in prime factorization, triangle A363130.
Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

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

A363130 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-co-modes, all 0's removed.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 9, 10, 12, 11, 18, 1, 15, 24, 3, 13, 37, 6, 25, 43, 9, 19, 64, 18, 29, 81, 25, 33, 99, 44, 42, 129, 59, 1, 39, 162, 93, 3, 62, 201, 116, 6, 55, 247, 175, 13, 81, 303, 224, 19, 84, 364, 309, 35, 103, 457, 389, 53, 105, 535, 529, 86

Offset: 0

Gus Wiseman, May 18 2023

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

			Triangle begins:
   1
   1
   2
   3
   4   1
   4   3
   8   3
   6   9
  10  12
  11  18   1
  15  24   3
  13  37   6
  25  43   9
  19  64  18
  29  81  25
  33  99  44
Row n = 9 counts the following partitions:
  (9)          (441)       (32211)
  (54)         (522)
  (63)         (711)
  (72)         (3222)
  (81)         (3321)
  (333)        (4221)
  (432)        (4311)
  (531)        (5211)
  (621)        (6111)
  (222111)     (22221)
  (111111111)  (33111)
               (42111)
               (51111)
               (321111)
               (411111)
               (2211111)
               (3111111)
               (21111111)

Row sums are A000041.
Row lengths are approximately A000196.
Column k = 0 is A047966.
For modes instead of non-co-modes we have A362614, rank stat A362611.
For co-modes instead of non-co-modes we have A362615, rank stat A362613.
For non-modes instead of non-co-modes we have A363126, rank stat A363127.
Columns k > 1 sum to A363128.
Column k = 1 is A363129.
This rank statistic (number of non-co-modes) is A363131.
A008284/A058398 count partitions by length/mean.
A275870 counts collapsible partitions.
A353836 counts partitions by number of distinct run-sums.
A359893 counts partitions by median.
Cf. A002865, A053263, A098859, A237984, A327472, A353863, A362612, A363124, A363125.

ncomsi[ms_]:=Select[Union[ms],Count[ms,#]>Min@@Length/@Split[ms]&];
DeleteCases[Table[Length[Select[IntegerPartitions[n] , Length[ncomsi[#]]==k&]],{n,0,15},{k,0,Sqrt[n]}],0,{2}]

cp's OEIS Frontend

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

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A363126 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-modes, all 0's removed.

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A363125 Number of integer partitions of n with a unique non-mode.

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A363128 Number of integer partitions of n with more than one non-co-mode.

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A363129 Number of integer partitions of n with a unique non-co-mode.

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A363130 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-co-modes, all 0's removed.

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Mathematica