Displaying 1-10 of 53 results found.
Number of integer partitions of n having a unique mode.
+10
57
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
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 of {a,a,b,b,b,c,d,d,d} are {b,d}.
FORMULA
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
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Commonest[#]]==1&]], {n, 0, 30}]
PROG
(PARI) 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
CROSSREFS
For parts instead of multiplicities we have A000041(n-1), ranks A102750. These partitions have ranks A356862. A275870 counts collapsible partitions.
A359893 counts partitions by median.
Number of modes in the prime factorization of n.
+10
56
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
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 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
EXAMPLE
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.
MATHEMATICA
Table[x=Last/@If[n==1, 0, FactorInteger[n]]; Count[x, Max@@x], {n, 100}]
PROG
(Python)
from sympy import factorint
def A362611(n): return list(v:=factorint(n).values()).count(max(v, default=0)) # Chai Wah Wu, May 08 2023 (PARI) a(n) = if(n==1, 0, my(f=factor(n)[, 2], m=vecmax(f)); #select(v->v==m, f)) \\ Andrew Howroyd, May 08 2023
CROSSREFS
Positions of first appearances are A002110. This statistic (mode-count) has triangular form A362614. A027746 lists prime factors (with multiplicity).
A362606 ranks partitions with more than one co-mode, counted by A362609.
Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.
+10
46
1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 10, 4, 1, 0, 13, 7, 2, 0, 16, 11, 3, 0, 23, 14, 4, 1, 0, 30, 19, 6, 1, 0, 35, 29, 11, 2, 0, 50, 34, 14, 3, 0, 61, 46, 23, 5, 0, 73, 69, 27, 6, 1, 0, 95, 81, 44, 10, 1, 0, 123, 105, 53, 14, 2
COMMENTS
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}.
EXAMPLE
Triangle begins:
1
0 1
0 2
0 2 1
0 4 1
0 5 2
0 7 3 1
0 10 4 1
0 13 7 2
0 16 11 3
0 23 14 4 1
0 30 19 6 1
0 35 29 11 2
0 50 34 14 3
0 61 46 23 5
0 73 69 27 6 1
0 95 81 44 10 1
Row n = 8 counts the following partitions:
(8) (53) (431)
(44) (62) (521)
(332) (71)
(422) (3221)
(611) (3311)
(2222) (4211)
(5111) (32111)
(22211)
(41111)
(221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
comsi[ms_]:=Select[Union[ms], Count[ms, #]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[comsi[#]]==k&]], {n, 0, 15}, {k, 0, Floor[(Sqrt[1+8n]-1)/2]}]
CROSSREFS
Removing columns 0 and 1 and taking sums gives A362609, ranks A362606. This statistic (co-mode count) is ranked by A362613. A008284 counts partitions by length.
A275870 counts collapsible partitions.
Number of integer partitions of n having a unique part of least multiplicity.
+10
45
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
COMMENTS
Alternatively, these are partitions with a part appearing fewer times than each of the others.
FORMULA
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
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]==1&]], {n, 0, 30}]
PROG
(PARI) 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
CROSSREFS
These partitions have ranks A359178. A275870 counts collapsible partitions.
A359893 counts partitions by median.
Cf. A008284, A053263, A098859, A237984, A240219, A304442, A327472, A353863, A353864, A353865, A362612.
Numbers with a unique largest prime exponent.
+10
44
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, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104
COMMENTS
If the prime factorization of k has a unique largest exponent, then k is a term.
Numbers whose multiset of prime factors (with multiplicity) has a unique mode. - Gus Wiseman, May 12 2023 Disjoint union of A246655 and A376250. The asymptotic density of this sequence, 0.3660366524547281232052..., is equal to the density of A376250 since the prime powers have a zero density. - Amiram Eldar, Sep 17 2024
EXAMPLE
Prime powers ( A246655) are in the sequence, since they have only one prime exponent in their prime factorization, hence a unique largest exponent. 144 is in the sequence, since 144 = 2^4 * 3^2 and there is the unique largest exponent 4.
225 is not in the sequence, since 225 = 3^2 * 5^2 and the largest exponent 2 is not unique, but rather it is the exponent of both the prime factor 3 and of the prime factor 5.
MATHEMATICA
Select[Range[2, 100], Count[(e = FactorInteger[#][[;; , 2]]), Max[e]] == 1 &] (* Amiram Eldar, Sep 01 2022 *)
PROG
(Python)
from sympy import factorint
from collections import Counter
def ok(k):
c = Counter(factorint(k)).most_common(2)
return not (len(c) > 1 and c[0][1] == c[1][1])
print([k for k in range(2, 105) if ok(k)])
(Python)
from sympy import factorint
from itertools import count, islice
def A356862_gen(startvalue=2): # generator of terms >= startvalue return filter(lambda n:len(f:=sorted(factorint(n).values(), reverse=True))==1 or f[0]!=f[1], count(max(startvalue, 2)))
(PARI) isok(k) = if (k>1, my(f=factor(k), m=vecmax(f[, 2]), w=select(x->(f[x, 2] == m), [1..#f~])); #w == 1); \\ Michel Marcus, Sep 01 2022
CROSSREFS
Subsequence of A319161 (which has additional terms 1, 180, 252, 300, 396, 450, 468, ...). For factors instead of exponents we have A102750. Partitions of this type are counted by A362608.
Number of co-modes in the prime factorization of n.
+10
44
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
COMMENTS
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
EXAMPLE
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.
MATHEMATICA
Table[x=Last/@If[n==1, 0, FactorInteger[n]]; Count[x, Min@@x], {n, 100}]
PROG
(Python)
from sympy import factorint
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 (PARI) a(n) = if(n==1, 0, my(f=factor(n)[, 2], m=vecmin(f)); #select(v->v==m, f)) \\ Andrew Howroyd, May 08 2023
CROSSREFS
Positions of first appearances are A002110. Counting partitions by this statistic (co-mode count) gives A362615. A027746 lists prime factors (with multiplicity).
Number of integer partitions of n with more than one mode.
+10
36
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
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 of {a,a,b,b,b,c,d,d,d} are {b,d}.
FORMULA
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
EXAMPLE
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)
MAPLE
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):
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Commonest[#]]>1&]], {n, 0, 30}]
PROG
(PARI)
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))))}
CROSSREFS
For parts instead of multiplicities we have A002865. These partitions have ranks A362605. A359893 counts partitions by median.
Number of integer partitions of n such that the greatest part is the unique mode.
+10
35
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
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 of {a,a,b,b,b,c,d,d,d} are {b,d}.
FORMULA
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
EXAMPLE
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
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Commonest[#]=={Max[#]}&]], {n, 0, 30}]
PROG
(PARI)
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))}
CROSSREFS
These partitions have ranks A362616. A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization.
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A238478, A238479, A327472, A362609, A362610.
Number of integer partitions of n with more than one part of least multiplicity.
+10
33
0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 19, 26, 42, 51, 74, 103, 136, 174, 246, 303, 411, 523, 674, 844, 1114, 1364, 1748, 2174, 2738, 3354, 4247, 5139, 6413, 7813, 9613, 11630, 14328, 17169, 20958, 25180, 30497, 36401, 44025, 52285, 62834, 74626, 89111, 105374, 125662
COMMENTS
These are partitions where no part appears fewer times than all of the others.
EXAMPLE
The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1 (namely 1 and 4), so is counted under a(9).
The a(3) = 1 through a(9) = 14 partitions:
(21) (31) (32) (42) (43) (53) (54)
(41) (51) (52) (62) (63)
(321) (61) (71) (72)
(2211) (421) (431) (81)
(3211) (521) (432)
(3221) (531)
(3311) (621)
(4211) (3321)
(32111) (4221)
(4311)
(5211)
(42111)
(222111)
(321111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]>1&]], {n, 0, 30}]
CROSSREFS
These partitions have ranks A362606. For mode complement instead of co-mode we have A362608, ranks A356862. A275870 counts collapsible partitions.
A359893 counts partitions by median.
Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.
+10
32
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
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 of {a,a,b,b,b,c,d,d,d} are {b,d}.
EXAMPLE
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}
MAPLE
q:= n-> (l-> nops(l)>1 and l[-1]=l[-2])(sort(map(i-> i[2], ifactors(n)[2]))):
MATHEMATICA
Select[Range[100], Count[Last/@FactorInteger[#], Max@@Last/@FactorInteger[#]]>1&]
PROG
(Python)
from sympy import factorint
def ok(n): return n>1 and (e:=list(factorint(n).values())).count(max(e))>1
(PARI) is(n) = {my(e = factor(n)[, 2]); if(#e < 2, 0, e = vecsort(e); e[#e-1] == e[#e]); } \\ Amiram Eldar, Jan 20 2024
CROSSREFS
The first term with bigomega n appears to be A166023(n). Partitions of this type are counted by A362607. These are the positions of terms > 1 in A362611.
Search completed in 0.029 seconds
|