A323014 -id:A323014 - cp's OEIS frontend

A325282 Maximum adjusted frequency depth among integer partitions of n.

0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (A325280).
Run lengths are A325258, i.e., first differences of Levine's sequence A011784 (except at n = 1).

Cf. A011784, A032741, A127002, A181819, A225486, A275870, A323014, A323023, A325245, A325254, A325258, A325278, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Max@@fdadj/@IntegerPartitions[n],{n,0,30}]

a(0) = 0; a(1) = 1; a(n > 1) = A225486(n).

A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 3, 7, 8, 11, 11, 17, 17, 23, 23, 30, 31, 39, 38, 48, 49, 58, 58, 70, 70, 82, 82, 95, 96, 110, 109, 125, 126, 141, 141, 159, 159, 177, 177, 196, 197, 217, 216, 238, 239, 260, 260, 284, 284, 308, 308, 333, 334, 360, 359, 387, 388, 415, 415, 445

Offset: 1

Clark Kimberling, Jan 01 2007

From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n - 4 of the form a+b, a+a+b, or a+a+b+c, ignoring ordering. A bijection can be constructed from the partitions described in the name by subtracting one from all parts and deleting zeros. These are also partitions with adjusted frequency depth (A323014, A325280) equal to their length plus one, and their Heinz numbers are given by A325281. For example, the a(7) = 1 through a(13) = 11 partitions are:
  (21)  (31)   (32)   (42)   (43)    (53)    (54)
        (211)  (41)   (51)   (52)    (62)    (63)
               (221)  (411)  (61)    (71)    (72)
               (311)         (322)   (332)   (81)
                             (331)   (422)   (441)
                             (511)   (611)   (522)
                             (3211)  (3221)  (711)
                                     (4211)  (3321)
                                             (4221)
                                             (4311)
                                             (5211)
(End)

			a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.
a(11) counts {1,1,2,7}, {1,1,3,6}, {1,1,4,5}, {2,2,1,6}, {2,2,3,4}, {3,3,1,4}, {4,4,1,2}
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(7) = 1 through a(13) = 11 partitions of the form a+a+b+c are the following. The Heinz numbers of these partitions are given by A085987.
  (3211)  (3221)  (3321)  (5221)  (4322)  (4332)  (4432)
          (4211)  (4221)  (5311)  (4331)  (4431)  (5332)
                  (4311)  (6211)  (4421)  (5322)  (5422)
                  (5211)          (5411)  (5331)  (5521)
                                  (6221)  (6411)  (6322)
                                  (6311)  (7221)  (6331)
                                  (7211)  (7311)  (6511)
                                          (8211)  (7411)
                                                  (8221)
                                                  (8311)
                                                  (9211)
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1)

Cf. A000041, A008284, A085987, A090858, A116608, A117571, A183558, A325242, A325244, A325280, A325281.

R:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0] cat Coefficients(R!( x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, May 30 2019

g:=sum(sum(sum(x^(i+j+k)*(x^i+x^j+x^k),i=1..j-1),j=2..k-1),k=3..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..65); # Emeric Deutsch, Jan 05 2007
isA127002 := proc(p) local s; if nops(p) = 4 then s := convert(p,set) ; if nops(s) = 3 then RETURN(1) ; else RETURN(0) ; fi ; else RETURN(0) ; fi ; end:
A127002 := proc(n) local part,res,p; part := combinat[partition](n) ; res := 0 ; for p from 1 to nops(part) do res := res+isA127002(op(p,part)) ; od ; RETURN(res) ; end:
for n from 1 to 200 do print(A127002(n)) ; od ; # R. J. Mathar, Jan 07 2007

Table[Length[Select[IntegerPartitions[n],Sort[Length/@Split[#]]=={1,1,2}&]],{n,70}] (* Gus Wiseman, Apr 19 2019 *)
Rest[CoefficientList[Series[x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)), {x,0,70}], x]] (* G. C. Greubel, May 30 2019 *)

my(x='x+O('x^70)); concat(vector(6), Vec(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)))) \\ G. C. Greubel, May 30 2019

a=(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 70).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 30 2019

G.f.: x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) - Vladeta Jovovic, Jan 03 2007
G.f.: Sum_{k>=3} Sum_{j=2..k-1} Sum_{m=1..j-1} x^(m+j+k)*(x^m +x^j +x^k). - Emeric Deutsch, Jan 05 2007
a(n) = binomial(floor((n-1)/2),2) - floor((n-1)/3) - floor((n-1)/4) + floor(n/4). - Mircea Merca, Nov 23 2013
a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - R. J. Mathar, Nov 23 2013

A353426 Number of integer partitions of n that are empty or a singleton or whose multiplicities are a sub-multiset that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 3, 3, 5, 4, 6, 5, 6, 6, 7, 8, 10, 12, 12, 14, 13, 13, 18, 15, 16, 19, 20, 20, 32, 37, 53, 74, 105

Offset: 0

Gus Wiseman, May 16 2022

a(n) is number of integer partitions of n whose Heinz number belongs to A353393, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(n) partitions for selected n (A..M = 10..22):
  n=1: n=4:  n=14:     n=16:     n=17:     n=18:        n=22:
------------------------------------------------------------------
  (1)  (4)   (E)       (G)       (H)       (I)          (M)
       (22)  (5522)    (4444)    (652211)  (7722)       (9922)
             (532211)  (6622)    (742211)  (752211)     (972211)
                       (642211)  (832211)  (842211)     (A62211)
                       (732211)            (932211)     (B52211)
                                           (333222111)  (C42211)
                                                        (D32211)

The non-recursive version is A325702, ranked by A325755.
The version for compositions is A353391, non-recursive A353390.
These partitions are ranked by A353393, nonprime A353389.
A047966 counts uniform partitions, compositions A329738.
A239455 counts Look-and-Say partitions, ranked by A351294.
Cf. A000041, A002033, A074761, A181819, A181821, A323014, A325534, A333755.

oosQ[y_]:=Length[y]<=1||MemberQ[Subsets[Sort[y],{Length[Union[y]]}],Sort[Length/@Split[y]]]&&oosQ[Sort[Length/@Split[y]]];
Table[Length[Select[IntegerPartitions[n],oosQ]],{n,0,30}]

A325245 Number of integer partitions of n with adjusted frequency depth 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 11, 19, 17, 25, 29, 37, 37, 56, 53, 75, 80, 99, 103, 145, 143, 181, 199, 247, 255, 336, 339, 426, 459, 548, 590, 738, 759, 916, 999, 1192, 1259, 1529, 1609, 1915, 2083, 2406, 2589, 3085, 3267, 3809, 4134, 4763, 5119, 5964

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014.

			The a(3) = 1 through a(10) = 11 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)      (64)
              (41)  (51)    (52)   (62)    (63)      (73)
                    (321)   (61)   (71)    (72)      (82)
                    (2211)  (421)  (431)   (81)      (91)
                                   (521)   (432)     (532)
                                   (3311)  (531)     (541)
                                           (621)     (631)
                                           (222111)  (721)
                                                     (3322)
                                                     (4321)
                                                     (4411)

Column k = 3 of A225485 and A325280.

Cf. A008284, A047966, A116608, A127002, A181819, A182850, A323014, A323023, A325239, A325246, A325254, A325268, A325280.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==3&]],{n,0,30}]

A325258 a(1) = 1; otherwise, first differences of Levine's sequence A011784.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 28, 171, 2624, 172613, 139584150, 6837485347187, 266437138079023501057, 508009471379222384299345337895696, 37745517525533091954228691786161750063795478326636142, 5347426383812697233786139576220412396732847744407175515852823296919414647252347610750

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

a(n) is the number of nonnegative integers k such that the maximum adjusted frequency depth among integer partitions of k is n. For example, the a(5) = 7 numbers are 7, 8, 9, 10, 11, 12, and 13.

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is A323014(n). The maximum adjusted frequency depth for partitions of n is A325282(n).

Run lengths of A325282.

Cf. A011784, A181819, A181821, A182850, A182857, A225485, A225486, A323014, A325238, A325254, A325278, A325280, A325283.

grw[q_]:=Join@@Table[ConstantArray[i,q[[Length[q]-i+1]]],{i,Length[q]}];
ReplacePart[Differences[Last/@NestList[grw,{1,1},9]],2->1]

A325334 Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 1, 1, 0, 4, 0, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 3, 0, 0, 2, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 4, 0, 0, 2, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 2, 0, 0, 3, 0, 1, 1, 0, 0, 4, 0, 0, 1, 0, 0, 5, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 4

Offset: 0

Gus Wiseman, May 01 2019

nonn

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The Heinz numbers of these partitions are given by A325374.

			The first 30 terms count the following partitions:
   3: (21)
   6: (321)
   6: (2211)
   9: (222111)
  10: (4321)
  12: (332211)
  12: (22221111)
  15: (54321)
  15: (2222211111)
  18: (333222111)
  18: (222222111111)
  20: (44332211)
  21: (654321)
  21: (22222221111111)
  24: (333322221111)
  24: (2222222211111111)
  27: (222222222111111111)
  28: (7654321)
  30: (5544332211)
  30: (444333222111)
  30: (333332222211111)
  30: (22222222221111111111)

Antti Karttunen, Table of n, a(n) for n = 0..100000

Column k = 3 of A325336.

Cf. A007862, A181819, A182850, A320348, A323014, A325245, A325280, A325326, A325335, A325374.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
unifQ[m_]:=SameQ@@Length/@Split[m];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&!SameQ@@#&&unifQ[#]&]],{n,0,30}]

A007862(n) = sumdiv(n, d, ispolygonal(d, 3));
A325334(n) = if(!n,n,A007862(n)-1); \\ Antti Karttunen, Jan 17 2025

a(n) = A007862(n) - 1.

Data section extended to a(105) by Antti Karttunen, Jan 17 2025

A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

In each case, 2 is the fixed point that is reached (1 is the other fixed point of the x -> A181819(x) map).
Includes all integers whose prime signature a) contains two or more distinct numbers, and b) contains no number that occurs the same number of times as any other number. The first member of this sequence that does not fit that description is 75675600, whose prime signature is (4,3,2,2,1,1).
A full characterization is: Numbers whose prime signature (1) has not all equal multiplicities but (2) the numbers of distinct parts appearing with each distinct multiplicity are all equal. For example, the prime signature of 2520 is {1,1,2,3}, which satisfies (1) but fails (2), as the numbers of distinct parts appearing with each distinct multiplicity are 1 (with multiplicity 2, the part being 1) and 2 (with multiplicity 1, the parts being 2 and 3). Hence the sequence does not contain 2520. - Gus Wiseman, Jan 02 2019

			1. 180 requires exactly five iterations under the x -> A181819(x) map to reach a fixed point (namely, 2).  A181819(180) = 18;  A181819(18) = 6; A181819(6) = 4; A181819(4) = 3;  A181819(3) = 2 (and A181819(2) = 2).
2. The prime signature of 180 (2^2*3^2*5) is (2,2,1).
a. Two distinct numbers appear in (2,2,1) (namely, 1 and 2).
b. Neither 1 nor 2 appears in (2,2,1) the same number of times as any other number that appears there.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

Numbers n such that A182850(n) = 5. See also A182853, A182854.
Subsequence of A059404 and A182851.  Includes A085987 and A179642 as subsequences.
Cf. A001221, A001222, A006939, A062770, A071625, A118914, A181819, A181821, A182857, A323014, A323022, A323023.

Select[Range[1000],With[{sig=Sort[Last/@FactorInteger[#]]},And[!SameQ@@Length/@Split[sig],SameQ@@Length/@Union/@GatherBy[sig,Length[Position[sig,#]]&]]]&] (* Gus Wiseman, Jan 02 2019 *)

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 3, 1, 3, 7, 10, 17, 27, 38, 1, 4, 8, 17, 31, 52, 83, 122, 181, 257, 361, 499, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 1, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754, 2570, 3742, 5269

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The Heinz numbers of these partitions are given by A325283.
The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014. The maximum adjusted frequency depth for integer partitions of n is given by A325282.
Essentially, the last numbers of rows of the array in A225485. - Clark Kimberling, Sep 13 2022

			The a(1) = 1 through a(11) = 17 partitions:
  1  11  21  211  221   411    3211  3221   3321    5221     4322
                  311   3111         4211   4221    5311     4331
                  2111  21111        32111  4311    6211     4421
                                            5211    32221    5411
                                            32211   33211    6221
                                            42111   42211    6311
                                            321111  43111    7211
                                                    52111    33221
                                                    421111   42221
                                                    3211111  43211
                                                             52211
                                                             53111
                                                             62111
                                                             431111
                                                             521111
                                                             4211111
                                                             32111111

Cf. A011784, A181819, A182850, A182857, A225486, A323014, A323023, A325246, A325258, A325278, A325281, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

nn=30;
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==mfds[[n]]&]],{n,0,nn}]

A323024 Numbers with exactly three distinct exponents in their prime factorization, or three distinct parts in their prime signature.

Original entry on oeis.org

360, 504, 540, 600, 720, 756, 792, 936, 1008, 1176, 1188, 1200, 1224, 1350, 1368, 1400, 1404, 1440, 1500, 1584, 1620, 1656, 1836, 1872, 1960, 2016, 2052, 2088, 2160, 2200, 2232, 2250, 2268, 2352, 2400, 2448, 2484, 2520, 2600, 2646, 2664, 2736, 2800, 2880, 2904

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

Positions of 3's in A071625.
Numbers k such that A001221(A181819(k)) = 3.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} r(n)/((n-1)*psi(n)) = 0.030575..., where psi is the Dedekind psi function (A001615), and r(n) = Sum_{d|n, 1Amiram Eldar, Oct 18 2020

			1500 = 2^2 * 3^1 * 5^3 has three distinct exponents {1, 2, 3}, so belongs to the sequence.
52500 = 2^2 * 3^1 * 5^4 * 7^1 has three distinct exponents {1, 2, 4}, so belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

Cf. A001221, A001222, A001615, A006939, A033992, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323025.

tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[1000],tom[#]==3&]

is(n) = #Set(factor(n)[, 2]) == 3 \\ David A. Corneth, Jan 02 2019

A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

Original entry on oeis.org

9, 36, 125, 225, 441, 1089, 1260, 1521, 1980, 2340, 2401, 2601, 2772, 3060, 3249, 3276, 3420, 4140, 4284, 4761, 4788, 5148, 5220, 5580, 5796, 6660, 6732, 7308, 7380, 7524, 7569, 7740, 7812, 7956, 8460, 8649, 8892, 9108, 9324, 9540, 10332, 10620, 10764, 10836

Offset: 1

Gus Wiseman, May 15 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

Said differently, these are nonprime numbers > 1 whose prime shadow is a divisor that is either a prime number or a number already in the sequence.

			The initial terms and their prime indices:
     9: {2,2}
    36: {1,1,2,2}
   125: {3,3,3}
   225: {2,2,3,3}
   441: {2,2,4,4}
  1089: {2,2,5,5}
  1260: {1,1,2,2,3,4}
  1521: {2,2,6,6}
  1980: {1,1,2,2,3,5}

The first term that is not a perfect power A001597 is 1260.
Without the recursion we have A325755 (a superset), counted by A325702.
Before removing the primes we had A353393.
These partitions are counted by A353426 minus one.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A182850 and A323014 give frequency depth, counted by A225485 and A325280.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A005117, A316413, A316428, A324850, A353394, A353395, A353397, A353399.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,2000],suQ[#]&&!PrimeQ[#]&]

cp's OEIS Frontend

A325282 Maximum adjusted frequency depth among integer partitions of n.

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A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

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A353426 Number of integer partitions of n that are empty or a singleton or whose multiplicities are a sub-multiset that is already counted.

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A325245 Number of integer partitions of n with adjusted frequency depth 3.

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A325258 a(1) = 1; otherwise, first differences of Levine's sequence A011784.

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A325334 Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.

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A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

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A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

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