A053263 -id:A053263 - cp's OEIS frontend

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

20, 28, 40, 44, 52, 56, 66, 68, 76, 78, 80, 84, 88, 92, 99, 102, 104, 112, 114, 116, 117, 120, 124, 132, 136, 138, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 190, 198, 200, 204, 207, 208, 212, 220, 222, 224, 228, 230, 232, 234

Offset: 1

Gus Wiseman, Apr 05 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.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 > 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   20: {1,1,3}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   52: {1,1,6}
   56: {1,1,1,4}
   66: {1,2,5}
   68: {1,1,7}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   92: {1,1,9}
   99: {2,2,5}

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal version is A361856, counted by A361849.
These partitions are counted by A361857, reverse A361858.
Including the equal case gives A361868, counted by A361859.
For mean instead of median we have A361907.
A000975 counts subsets with integer median.
A001222 counts prime factors, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Max@@prix[#]>2*Median[prix[#]]&]

A361868 Positive integers > 1 whose prime indices satisfy (maximum) >= 2*(median).

Original entry on oeis.org

12, 20, 24, 28, 40, 42, 44, 48, 52, 56, 60, 63, 66, 68, 72, 76, 78, 80, 84, 88, 92, 96, 99, 102, 104, 112, 114, 116, 117, 120, 124, 126, 130, 132, 136, 138, 140, 144, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 189, 190, 192, 195

Offset: 1

Gus Wiseman, Apr 05 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.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 >= 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal case is A361856, counted by A361849.
These partitions are counted by A361859.
The unequal case is A361867, counted by A361857.
The complement is counted by A361858.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A053263, A111907, A118096, A237753, A237821, A361848, A361855, A361906, A361908.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]>=2*Median[prix[#]]&]

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

Gus Wiseman, May 11 2023

nonn

First differs from A112769 in lacking 300.

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

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

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.

filter:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a,b) -> a[1]Robert Israel, Dec 15 2023

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],FreeQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

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,#]

A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166, 177

Offset: 1

Gus Wiseman, May 10 2023

nonn

Also numbers n whose median prime factor is not a prime factor of n, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime factorization of 60 is 2*2*3*5, with middle parts (2,3), so 60 is in the sequence.

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

Partitions of this type are counted by A238479.
The complement (without 1) is A362618, counted by A238478.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A359893 counts partitions by median.
A359908 ranks partitions with integer median, counted by A325347.
A359912 ranks partitions with non-integer median, counted by A307683.
A362605 ranks partitions with more than one mode, counted by A362607.
A362611 counts modes in prime factorization, triangle version A362614.
A362621 ranks partitions with median equal to maximum, counted by A053263.
A362622 ranks partitions whose maximum is a middle part, counted by A237824.
Contains A006881 and (except for 1) A030229.
Cf. A000040, A171979, A327473, A327476, A356862, A359907, A362616, A362620.

filter:= proc(n) local F,m;
  F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= nops(F);
  m::even and F[m/2] <> F[m/2+1]
end proc:
select(filter, [$2..200]); # Robert Israel, Dec 15 2023

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],FreeQ[prifacs[#],Median[prifacs[#]]]&]

A363124 Number of integer partitions of n with more than one non-mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 9, 19, 28, 46, 65, 98, 132, 190, 251, 348, 451, 603, 768, 1014, 1273, 1648, 2052, 2604, 3233, 4062, 4984, 6203, 7582, 9333, 11349, 13890, 16763, 20388, 24528, 29613, 35502, 42660, 50880, 60883, 72376, 86158, 102120, 121133, 143010

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(7) = 1 through a(10) = 9 partitions:
  (3211)  (3221)   (3321)    (5221)
          (4211)   (4221)    (5311)
          (32111)  (4311)    (6211)
                   (5211)    (32221)
                   (42111)   (43111)
                   (321111)  (52111)
                             (322111)
                             (421111)
                             (3211111)

For middle parts instead of non-modes we have A238479, complement A238478.
For modes instead of non-modes we have A362607, complement A362608.
For co-modes instead of non-modes we have A362609, complement A362610.
The complement is counted by A363125.
For non-co-modes instead of non-modes we have A363128, complement A363129.
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,#]1&]],{n,0,30}]

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}]

A207642 Expansion of g.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + x^(n+k)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 3, 3, 2, 4, 4, 4, 4, 5, 6, 6, 6, 6, 8, 9, 8, 10, 10, 10, 12, 14, 14, 14, 15, 16, 19, 20, 20, 22, 24, 24, 26, 28, 30, 34, 34, 35, 38, 40, 42, 46, 50, 50, 54, 58, 60, 63, 66, 70, 76, 80, 84, 88, 92, 96, 102, 108, 112, 120, 126, 131, 140, 146, 151

Offset: 0

Paul D. Hanna, Feb 19 2012

nonn

Conjecture: a(n) is the number of partitions p of n into distinct parts such that max(p) <= 1 + 2*min(p), for n >= 1 (as in the Mathematica program at A241061). - Clark Kimberling, Apr 16 2014

			G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + 4*x^11 + 4*x^12 + 4*x^13 + 5*x^14 + 6*x^15 + 6*x^16 + 6*x^17 + ...
such that, by definition,
A(x) = 1 + x*(1 + x) + x^2*(1 + x^2)*(1 + x^3) + x^3*(1 + x^3)*(1 + x^4)*(1 + x^5) + x^4*(1 + x^4)*(1 + x^5)*(1 + x^6)*(1 + x^7) + x^5*(1 + x^5)*(1 + x^6)*(1 + x^7)*(1 + x^8)*(1 + x^9) + ... + x^n*Product_{k=0..n-1} (1 + x^(n+k)) + ...
Also
A(x) = 1/(1 - x)  +  x^2/((1 - x^2)*(1 - x^3))  +  x^7/((1 - x^3)*(1 - x^4)*(1 - x^5))  +  x^15/((1 - x^4)*(1 - x^5)*(1 - x^6)*(1 - x^7))  +  x^26/((1 - x^5)*(1 - x^6)*(1 - x^7)*(1 - x^8)*(1 - x^9)) + ... + x^(n*(3*n+1)/2)/(Product_{k=0..n} 1 - x^(n+k+1)) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)

Cf. A053263, A087135, A241061.

m:=80; R:=PowerSeriesRing(Integers(), m); [1] cat Coefficients(R!( (&+[x^n*(&*[1+x^(n+j): j in [0..n-1]]) : n in [1..m]]) )); // G. C. Greubel, Jan 12 2019

With[{m = 80}, CoefficientList[Series[Sum[x^n*Product[1+x^(n+j), {j,0, n-1}], {n,0,m}], {x,0,m}], x]] (* G. C. Greubel, Jan 12 2019 *)
nmax = 100; pk = x + x^2; s = 1 + pk; Do[pk = Normal[Series[pk * x*(1 + x^(2*k - 2))*(1 + x^(2*k - 1))/(1 + x^(k - 1)), {x, 0, nmax}]]; s = s + pk, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Jun 18 2019 *)

{a(n)=polcoeff(sum(m=0,n,x^m*prod(k=0,m-1,1+x^(m+k) +x*O(x^n))),n)}
for(n=0,80,print1(a(n),", "))

R = PowerSeriesRing(ZZ, 'x')
m = 80
x = R.gen().O(m)
s = sum(x^n*prod(1+x^(n+j) for j in (0..n-1)) for n in (0..m))
s.coefficients() # G. C. Greubel, Jan 12 2019

G.f.: Sum_{n>=0} x^(n*(3*n+1)/2) / ( Product_{k=0..n} 1 - x^(n+k+1) ). - Paul D. Hanna, Oct 14 2020

a(n) ~ c * exp(r*sqrt(n)) / sqrt(n), where r = 0.926140105877... = 2*sqrt((3/2)*log(z)^2 - polylog(2, 1-z) + polylog(2, 1-z^2)), where z = (-1 + (44 - 3*sqrt(177))^(1/3) + (44 + 3*sqrt(177))^(1/3))/6 = 0.82948354095849703967... is the real root of the equation z^3*(1 - z)/(1 - z^2)^2 = 1 and c = 0.57862299312... - Vaclav Kotesovec, Jun 29 2019, updated Oct 09 2024

cp's OEIS Frontend

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

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Mathematica

A361868 Positive integers > 1 whose prime indices satisfy (maximum) >= 2*(median).

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Mathematica

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

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Maple

Mathematica

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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A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.

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A363124 Number of integer partitions of n with more than one non-mode.

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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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