A356862 -id:A356862 - cp's OEIS frontend

A376250 Numbers with a unique largest prime exponent (A356862) that are not prime powers (A246655).

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 198, 200

Offset: 1

Amiram Eldar, Sep 17 2024

First differs from A059404 at n = 55: A059404(55) = 180 = 2^2 * 3^2 * 5 is not a term of this sequence.
First differs from A360248 at n = 23: a(23) = 90 = 2 * 3^2 * 5 is not a term of A360248.
First differs from A332785 at n = 17: a(17) = 72 = 2^3 * 3^2 is not a term of A332785.
Numbers whose unordered prime signature (i.e., sorted, see A118914) ends with two different integers: {..., k, m} for some 1 <= k < m.
All the factorial numbers above 6 are terms.
The asymptotic density of this sequence is Sum_{k >= 1, p prime} (d(k+1, p) - d(k, p))/((p-1)*p^k) = 0.3660366524547281232052..., where d(k, p) = 0 for k = 1, and (1-1/p)/((1-1/p^k)*zeta(k)) for k > 1, is the density of terms that have in their prime factorization a prime p with the largest exponent that is > k.

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

Equals A356862 \ A246655.

Cf. A000142, A059404, A118914, A332785, A360248.

Select[Range[2, 200], Length[e = FactorInteger[#][[;; , 2]]] > 1 &&  Count[e, Max[e]] == 1 &]

is(k) = if (k == 1, 0, my(e = vecsort(factor(k)[,2])); #e > 1 && e[#e] > e[#e-1]);

A362608 Number of integer partitions of n having a unique mode.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 11, 16, 21, 29, 43, 54, 78, 102, 131, 175, 233, 295, 389, 490, 623, 794, 1009, 1255, 1579, 1967, 2443, 3016, 3737, 4569, 5627, 6861, 8371, 10171, 12350, 14901, 18025, 21682, 26068, 31225, 37415, 44617, 53258, 63313, 75235, 89173, 105645

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

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}.

			The partition (3,3,2,1) has greatest multiplicity 2, and a unique part of multiplicity 2 (namely 3), so is counted under a(9).
The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For parts instead of multiplicities we have A000041(n-1), ranks A102750.
For median instead of mode we have A238478, complement A238479.
These partitions have ranks A356862.
The complement is counted by A362607, ranks A362605.
For co-mode complement we have A362609, ranks A362606.
For co-mode we have A362610, ranks A359178.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A053263, A098859, A237984, A304442, A327472, A360071, A360687, A362612.

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]==1&]],{n,0,30}]

seq(n) = my(A=O(x*x^n)); Vec(sum(m=1, n, sum(j=1, n\m, x^(j*m)*(1-x^j)/(1 - x^(j*m)), A)*prod(j=1, n\m, (1 - x^(j*m))/(1 - x^j) + A)/prod(j=n\m+1, n, 1 - x^j + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023

G.f.: Sum_{m>=1} (Sum_{j>=1} x^(j*m)*(1 - x^j)/(1 - x^(j*m))) * (Product_{j>=1} (1 - x^(j*m))/(1 - x^j)). - Andrew Howroyd, May 04 2023

A362611 Number of modes in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2

Offset: 1

Gus Wiseman, May 05 2023

nonn

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}.

a(n) depends only on the prime signature of n. - Andrew Howroyd, May 08 2023

			The factorization of 450 is 2*3*3*5*5, modes {3,5}, so a(450) = 2.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so a(900) = 3.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so a(1500) = 1.
The factorization of 8820 is 2*2*3*3*5*7*7, modes {2,3,7}, so a(8820) = 3.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Positions of first appearances are A002110.
Positions of 1's are A356862, counted by A362608.
Positions of terms > 1 are A362605, counted by A362607.
For co-mode we have A362613, counted by A362615.
This statistic (mode-count) has triangular form A362614.
A027746 lists prime factors (with multiplicity).
A112798 lists prime indices, length A001222, sum A056239.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362606 ranks partitions with more than one co-mode, counted by A362609.
Cf. A000040, A000720, A001221, A002865, A072774, A215366, A327473, A327476, A359908.

Table[x=Last/@If[n==1,0,FactorInteger[n]];Count[x,Max@@x],{n,100}]

a(n) = if(n==1, 0, my(f=factor(n)[,2], m=vecmax(f)); #select(v->v==m, f)) \\ Andrew Howroyd, May 08 2023

from sympy import factorint
def A362611(n): return list(v:=factorint(n).values()).count(max(v,default=0)) # Chai Wah Wu, May 08 2023

For n > 1, 1 <= a(n) << log n. - Charles R Greathouse IV, May 09 2023

a(n) <= A001221(n), with equality if and only if n is a power of a squarefree number (A072774). - Amiram Eldar, Mar 02 2025

A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 11, 3, 1, 0, 16, 4, 2, 0, 21, 6, 3, 0, 29, 8, 4, 1, 0, 43, 7, 5, 1, 0, 54, 13, 8, 2, 0, 78, 12, 8, 3, 0, 102, 17, 11, 5, 0, 131, 26, 12, 6, 1, 0, 175, 29, 17, 9, 1, 0, 233, 33, 18, 11, 2, 0, 295, 47, 25

Offset: 0

Gus Wiseman, May 04 2023

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}.

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  11   3   1
   0  16   4   2
   0  21   6   3
   0  29   8   4   1
   0  43   7   5   1
   0  54  13   8   2
   0  78  12   8   3
   0 102  17  11   5
   0 131  26  12   6   1
   0 175  29  17   9   1
Row n = 8 counts the following partitions:
  (8)         (53)    (431)
  (44)        (62)    (521)
  (332)       (71)
  (422)       (3311)
  (611)
  (2222)
  (3221)
  (4211)
  (5111)
  (22211)
  (32111)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Alois P. Heinz, Rows n = 0..800, flattened

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.
Column k = 1 is A362608, ranks A356862.
This statistic (mode-count) is ranked by A362611.
For co-modes we have A362615, ranked by A362613.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A002865, A003056, A098859, A240219, A359893, A360071, A362609, A362610, A362612, A372542.

msi[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[msi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - Alois P. Heinz, May 05 2024

A362610 Number of integer partitions of n having a unique part of least multiplicity.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 23, 30, 35, 50, 61, 73, 95, 123, 139, 187, 216, 269, 328, 411, 461, 594, 688, 836, 980, 1211, 1357, 1703, 1936, 2330, 2697, 3253, 3649, 4468, 5057, 6005, 6841, 8182, 9149, 10976, 12341, 14508, 16447, 19380, 21611, 25553, 28628

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

Alternatively, these are partitions with a part appearing fewer times than each of the others.

			The partition (3,3,2,2,2,1,1,1) has least multiplicity 2, and only one part of multiplicity 2 (namely 3), so is counted under a(15).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For parts instead of multiplicities we have A002865, ranks A247180.
For median instead of co-mode we have A238478, complement A238479.
These partitions have ranks A359178.
For mode complement of co-mode we have A362607, ranks A362605.
For mode instead of co-mode we have A362608, ranks A356862.
The complement is counted by A362609, ranks A362606.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A008284, A053263, A098859, A237984, A240219, A304442, A327472, A353863, A353864, A353865, A362612.

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

seq(n) = my(A=O(x*x^n)); Vec(sum(m=2, n+1, sum(j=1, n, x^(j*(m-1))/(1 + if(j*m<=n, x^(j*m)/(1-x^j) )) + A)*prod(j=1, n\m, 1 + x^(j*m)/(1 - x^j) + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023

G.f.: Sum_{m>=2} (Sum_{j>=1} x^(j*(m-1))/(1 + x^(j*m)/(1 - x^j))) * (Product_{j>=1} (1 + x^(j*m)/(1 - x^j))). - Andrew Howroyd, May 04 2023

A362613 Number of co-modes in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A327500 at n = 180.
First differs from A351946 at n = 180.
First differs from A353507 at n = 180.
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 of {a,a,b,b,b,c,c} are {a,c}.
a(n) depends only on the prime signature of n. - Andrew Howroyd, May 08 2023

			The factorization of 180 is 2*2*3*3*5, co-modes {5}, so a(180) = 1.
The factorization of 900 is 2*2*3*3*5*5, co-modes {2,3,5}, so a(900) = 3.
The factorization of 8820 is 2*2*3*3*5*7*7, co-modes {5}, so a(8820) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Positions of first appearances are A002110.
Positions of 1's are A359178, counted by A362610.
Positions of terms > 1 are A362606, counted by A362609.
For mode we have A362611, counted by A362614.
Counting partitions by this statistic (co-mode count) gives A362615.
A027746 lists prime factors (with multiplicity).
A112798 lists prime indices, length A001222, sum A056239.
Cf. A000040, A215366, A327473, A327476, A356862, A359908, A362605.

Table[x=Last/@If[n==1,0,FactorInteger[n]];Count[x,Min@@x],{n,100}]

a(n) = if(n==1, 0, my(f=factor(n)[,2], m=vecmin(f)); #select(v->v==m, f)) \\ Andrew Howroyd, May 08 2023

from sympy import factorint
def A362613(n):
    v = factorint(n).values()
    w = min(v,default=0)
    return sum(1 for e in v if e<=w) # Chai Wah Wu, May 08 2023

A359178 Numbers with a unique smallest prime exponent.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117

Offset: 1

Jens Ahlström, Jan 08 2023

nonn

180 is the smallest number with a unique smallest prime exponent that is not a member of A130091.

			2 = 2^1 is a term since it has 1 as a unique smallest exponent.
6 = 2^1 * 3^1 is not a term since it has two primes with the same smallest exponent.
180 = 2^2 * 3^2 * 5^1 is a term since it has 1 as a unique smallest exponent.

Jens Ahlström, Table of n, a(n) for n = 1..9999
Wikipedia, Mode (statistics).

For parts instead of multiplicities we have A247180, counted by A002865.
For greatest instead of smallest we have A356862, counted by A362608.
The complement is A362606, counted by A362609.
Partitions of this type are counted by A362610.
These are the positions of 1's in A362613, for modes A362611.
A001221 counts prime exponents and A001222 adds them up.
A027746 lists prime factors, A112798 indices, A124010 exponents.
Cf. A102750, A130091, A327473, A327476, A359908, A362605, A362615.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Count[e, Min[e]] == 1]; Select[Range[2, 200], q] (* Amiram Eldar, Jan 08 2023 *)

isok(n) = if (n>1, my(f=factor(n), e = vecmin(f[,2])); #select(x->(x==e), f[,2], 1) == 1); \\ Michel Marcus, Jan 27 2023

from sympy import factorint
def ok(k):
  c = sorted(factorint(k).values())
  return len(c) == 1 or c[0] != c[1]
print([k for k in range(2, 118) if ok(k)])

from itertools import count, islice
from sympy import factorint
def A359178_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(f:=list(factorint(n).values())).count(min(f))==1,count(max(startvalue,2)))
A359178_list = list(islice(A359178_gen(),20)) # Chai Wah Wu, Feb 08 2023

A362607 Number of integer partitions of n with more than one mode.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 13, 13, 23, 23, 33, 45, 56, 64, 90, 101, 137, 169, 208, 246, 320, 379, 469, 567, 702, 828, 1035, 1215, 1488, 1772, 2139, 2533, 3076, 3612, 4333, 5117, 6113, 7168, 8557, 10003, 11862, 13899, 16385, 19109, 22525, 26198, 30729, 35736

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

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}.

			The partition (3,2,2,1,1) has greatest multiplicity 2, and two parts of multiplicity 2 (namely 1 and 2), so is counted under a(9).
The a(3) = 1 through a(9) = 9 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)
              (41)  (51)    (52)   (62)    (63)
                    (321)   (61)   (71)    (72)
                    (2211)  (421)  (431)   (81)
                                   (521)   (432)
                                   (3311)  (531)
                                           (621)
                                           (32211)
                                           (222111)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 301 terms from John Tyler Rascoe)

For parts instead of multiplicities we have A002865.
For median instead of mode we have A238479, complement A238478.
These partitions have ranks A362605.
The complement is counted by A362608, ranks A356862.
For co-mode we have A362609, ranks A362606.
For co-mode complement we have A362610, ranks A359178.
A000041 counts integer partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A008284, A098859, A237984, A275870, A304442, A327472, A353864, A353865, A360071, A362612.

b:= proc(n, i, m, t) option remember; `if`(n=0, `if`(t=2, 1, 0), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, 2, t))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..51);  # Alois P. Heinz, May 05 2024

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]>1&]],{n,0,30}]

G_x(N)={my(x='x+O('x^(N-1)), Ib(k,j) = if(k>j,1,0), A_x(u)=sum(i=1,N-u, sum(j=u+1, N-u, (x^(i*(u+j))*(1-x^u)*(1-x^j))/((1-x^(u*i))*(1-x^(j*i))) * prod(k=1,N-i*(u+j), (1-x^(k*(i+Ib(k,j))))/(1-x^k)))));
concat([0,0,0],Vec(sum(u=1,N, A_x(u))))}
G_x(51) \\ John Tyler Rascoe, Apr 05 2024

G.f.: Sum_{u>0} A(u,x) where A(u,x) = Sum_{i>0} Sum_{j>u} ( x^(i*(u+j))*(1-x^u)*(1-x^j) )/( (1-x^(u*i))*(1-x^(j*i)) ) * Product_{k>0} ( (1-x^(k*(i+[k>j])))/(1-x^k) ) is the g.f. for partitions of this kind with least mode u and [] is the Iverson bracket. - John Tyler Rascoe, Apr 05 2024

A362612 Number of integer partitions of n such that the greatest part is the unique mode.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 9, 10, 12, 15, 16, 19, 23, 26, 32, 37, 41, 48, 58, 65, 75, 88, 101, 115, 135, 151, 176, 200, 228, 261, 300, 336, 385, 439, 498, 561, 641, 717, 818, 921, 1036, 1166, 1321, 1477, 1667, 1867, 2099, 2346, 2640, 2944, 3303, 3684

Offset: 0

Gus Wiseman, May 03 2023

nonn

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}.

			The a(1) = 1 through a(10) = 7 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    221    33      331      44        333        55
              1111  11111  222     2221     332       441        442
                           111111  1111111  2222      3321       3331
                                            22211     22221      22222
                                            11111111  111111111  222211
                                                                 1111111111

John Tyler Rascoe, Table of n, a(n) for n = 0..500

For median instead of mode we have A053263, complement A237821.
These partitions have ranks A362616.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362607 counts partitions with more than one mode, ranks A362605.
A362608 counts partitions with a unique mode, ranks A356862.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A238478, A238479, A327472, A362609, A362610.

Table[Length[Select[IntegerPartitions[n],Commonest[#]=={Max[#]}&]],{n,0,30}]

A_x(N)={my(x='x+O('x^N), g=sum(i=1, N, sum(j=1, N/i, x^(i*j)*prod(k=1,i-1,(1-x^(j*k))/(1-x^k))))); concat([0],Vec(g))}
A_x(60) \\ John Tyler Rascoe, Apr 03 2024

G.f.: Sum_{i, j>0} x^(i*j) * Product_{k=1,i-1} ((1-x^(j*k))/(1-x^k)). - John Tyler Rascoe, Apr 03 2024

A362605 Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154

Offset: 1

Gus Wiseman, May 05 2023

nonn

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}.

			The prime indices of 180 are {1,1,2,2,3}, with modes {1,2}, so 180 is in the sequence, and the sequence differs from A182853.
The terms together with their prime indices begin:
     6: {1,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    21: {2,4}
    22: {1,5}
    26: {1,6}
    30: {1,2,3}
    33: {2,5}
    34: {1,7}
    35: {3,4}
    36: {1,1,2,2}
    38: {1,8}
    39: {2,6}
    42: {1,2,4}
    46: {1,9}
    51: {2,7}
    55: {3,5}

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

The first term with bigomega n appears to be A166023(n).
The complement is A356862, counted by A362608.
For co-mode complement we have A359178, counted by A362610.
For co-mode we have A362606, counted by A362609.
Partitions of this type are counted by A362607.
These are the positions of terms > 1 in A362611.
A112798 lists prime indices, length A001222, sum A056239.
A362614 counts partitions by number of modes, ranks A362611.
A362615 counts partitions by number of co-modes, ranks A362613.
Cf. A002865, A215366, A327473, A327476, A353864, A359908, A362612.

q:= n-> (l-> nops(l)>1 and l[-1]=l[-2])(sort(map(i-> i[2], ifactors(n)[2]))):
select(q, [$1..250])[];  # Alois P. Heinz, May 10 2023

Select[Range[100],Count[Last/@FactorInteger[#], Max@@Last/@FactorInteger[#]]>1&]

is(n) = {my(e = factor(n)[, 2]); if(#e < 2, 0, e = vecsort(e); e[#e-1] == e[#e]);} \\ Amiram Eldar, Jan 20 2024

from sympy import factorint
def ok(n): return n>1 and (e:=list(factorint(n).values())).count(max(e))>1
print([k for k in range(155) if ok(k)]) # Michael S. Branicky, May 06 2023

cp's OEIS Frontend

A376250 Numbers with a unique largest prime exponent (A356862) that are not prime powers (A246655).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A362608 Number of integer partitions of n having a unique mode.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A362611 Number of modes in the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A362610 Number of integer partitions of n having a unique part of least multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A362613 Number of co-modes in the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A359178 Numbers with a unique smallest prime exponent.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs