A182850 -id:A182850 - cp's OEIS frontend

A332276 Heinz numbers of widely totally normal integer partitions.

1, 2, 4, 6, 8, 12, 16, 18, 30, 32, 60, 64, 90, 128, 150, 180, 210, 256, 300, 360, 450, 512, 540, 600, 630, 1024, 1050, 1350, 1500, 2048, 2100, 2250, 2310, 2520, 2940, 3150, 3780, 4096, 4200, 4410, 5880, 8192, 8820, 9450, 10500, 11550, 12600, 13230, 14700

Offset: 1

Gus Wiseman, Feb 12 2020

nonn

First differs from A317246 in having 630.
A sequence of positive integers is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  150: {1,2,3,3}
  180: {1,1,2,2,3}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}
  300: {1,1,2,3,3}
  360: {1,1,1,2,2,3}
For example, starting with (4,3,2,2,1), the partition with Heinz number 630, and repeatedly taking run-lengths gives (4,3,2,2,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1). These are all normal and the last is all 1's, so 630 belongs to the sequence.

Contains all powers of two A000079 and the primorials A002110.
Heinz numbers of normal integer partitions are A055932.
The case of reversed integer partitions is A332276 (this sequence).
The enumeration of these partitions by sum is A332277.
The enumeration of the generalization to compositions is A332279.
The co-strong version is A332290.
The strong version is A332291.
Cf. A007097, A056239, A133808, A181819, A182850, A305732, A317081, A317089, A317090, A317246, A317492, A329747, A332295, A332296.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
gnaQ[y_]:=Or[y=={},Union[y]=={1},And[Union[y]==Range[Max[y]],gnaQ[Length/@Split[y]]]];
Select[Range[1000],gnaQ[primeMS[#]]&]

A332338 Number of alternately co-strong compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 24, 39, 72, 125, 224, 387, 697, 1205, 2141, 3736, 6598, 11516, 20331, 35526, 62507, 109436, 192200, 336533, 590582, 1034187

Offset: 0

Gus Wiseman, Feb 17 2020

A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.

			The a(1) = 1 through a(5) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (112)   (41)
                    (121)   (113)
                    (1111)  (131)
                            (212)
                            (221)
                            (1112)
                            (1121)
                            (11111)
For example, starting with the composition y = (1,6,2,2,1,1,1,1) and repeatedly taking run-lengths and reversing gives (1,6,2,2,1,1,1,1) -> (4,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2). All of these have weakly increasing run-lengths and the last is a singleton, so y is counted under a(15).

The case of partitions is A317256.
The recursive (rather than alternating) version is A332274.
The total (rather than alternating) version is (also) A332274.
The strong version is this same sequence.
The case of reversed partitions is A332339.
The normal version is A332340(n) + 1 for n > 1.
Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.

tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tniQ]],{n,0,10}]

A332339 Number of alternately co-strong reversed integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 18, 20, 29, 28, 40, 45, 54, 59, 82, 81, 108, 118, 141, 154, 204, 204, 255, 285, 339, 363, 458, 471, 580, 632, 741, 806, 983, 1015, 1225, 1341, 1562, 1667, 2003, 2107, 2491, 2712, 3101, 3344, 3962, 4182, 4860, 5270, 6022, 6482

Offset: 0

Gus Wiseman, Feb 17 2020

nonn

A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.

Also the number of alternately strong integer partitions of n.

			The a(1) = 1 through a(8) = 12 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (122)    (33)      (34)       (35)
                            (11111)  (123)     (124)      (44)
                                     (222)     (133)      (125)
                                     (1122)    (1222)     (134)
                                     (111111)  (1111111)  (233)
                                                          (1133)
                                                          (2222)
                                                          (11222)
                                                          (11111111)
For example, starting with the composition y = (1,2,3,3,4,4,4) and repeatedly taking run-lengths and reversing gives (1,2,3,3,4,4,4) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2) -> (1). All of these have weakly increasing run-lengths and the last is equal to (1), so y is counted under a(21).

The total (instead of alternating) version is A316496.
Alternately strong partitions are A317256.
The case of ordinary (not reversed) partitions is (also) A317256.
The generalization to compositions is A332338.
Cf. A100883, A181819, A182850, A317257, A329744, A329746, A332275, A332289, A332292, A332340.

tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Sort/@IntegerPartitions[n],tniQ]],{n,0,30}]

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[#]&]

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 6, 5, 4, 4, 6, 6, 8, 8, 13, 16, 13, 16, 18, 16, 20, 21, 27, 30, 27, 33, 41, 44, 51, 48, 58, 61, 66, 66, 74, 83, 86, 99, 102, 111, 115, 126, 137, 147, 156

Offset: 0

Gus Wiseman, May 17 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.

			The a(8) = 1 through a(14) = 4 partitions (A = 10, B = 11):
  3311  711     61111  521111   5511      B11       A1111
        321111         3221111  9111      721111    731111
                                531111    811111    33221111
                                3321111   5221111   422111111
                                22221111  43111111
                                42111111

The LHS (product of multiplicities) is A005361, counted by A266477.
The RHS (product of prime shadows) is A353394, first appearances A353397.
A related comparison is A353396, ranked by A353395.
These partitions are ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
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.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, counted by A325702.
A339095 counts partitions by product (or factorizations by sum).
Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353393, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==Times@@Length/@Split[#]&]],{n,0,30}]

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

Members m of A025487 such that A181819(m) is also a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.
Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - Gus Wiseman, May 21 2022

			The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From _Gus Wiseman_, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Conjugate Partition.

Intersection of A025487 and A179983.
Subsequence of A129912 and A181826.
Includes all members of A182862.
Positions of first appearances in A353742, unordered version A238747.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];
Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

A304687 Start with the multiset of prime multiplicities of n. Given a multiset, take the multiset of its multiplicities. Repeat until a constant multiset {k,k,...,k} is reached, and set a(n) to the sum of this multiset (k times the length).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 16 2018

nonn

			The following are examples showing the reduction of a multiset starting with the multiset of prime multiplicities of n.
         a(60) = 2: {1,1,2} -> {1,2} -> {1,1}.
        a(360) = 3: {1,2,3} -> {1,1,1}.
       a(1260) = 4: {1,1,2,2} -> {2,2}.
a(21492921450) = 6: {1,1,2,2,3,3} -> {2,2,2}.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A001221, A001222, A071625, A112798, A181819, A182850, A182857, A304465, A304634, A304636.

a:= proc(n) map(i-> i[2], ifactors(n)[2]);
      while nops({%[]})>1 do [coeffs(add(x^i, i=%))] od;
      add(i, i=%)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 17 2018

Table[If[n==1,0,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],!SameQ@@#&]//Total],{n,360}]

A319151 Heinz numbers of superperiodic integer partitions.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

First differs from A061345 at a(1) = 2 and next at a(98) = 441.
A number n is in the sequence iff n = 2 or the prime indices of n have a common divisor > 1 and the Heinz number of the multiset of prime multiplicities of n, namely A181819(n), is already in the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (2), (3), (4), (2,2), (5), (6), (7), (8), (9), (3,3), (2,2,2), (10), (11), (12), (13), (14), (15), (4,4), (16), (17), (18), (19), (20), (21), (22), (2,2,2,2).

Vincenzo Librandi, Table of n, a(n) for n = 1..2326

Cf. A001597, A056239, A072774, A098859, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319152, A319157.

supperQ[n_]:=Or[n==2,And[GCD@@PrimePi/@FactorInteger[n][[All,1]]>1,supperQ[Times@@Prime/@FactorInteger[n][[All,2]]]]];
Select[Range[500],supperQ]

A332279 Number of widely totally normal compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 12, 22, 29, 62, 119, 208, 368, 650, 1197, 2173, 3895, 7022, 12698, 22940, 41564

Offset: 0

Gus Wiseman, Feb 12 2020

A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(7) = 22 compositions:
  (1)  (11)  (12)   (112)   (122)    (123)     (1123)
             (21)   (121)   (212)    (132)     (1132)
             (111)  (211)   (221)    (213)     (1213)
                    (1111)  (1121)   (231)     (1231)
                            (1211)   (312)     (1312)
                            (11111)  (321)     (1321)
                                     (1212)    (2113)
                                     (1221)    (2122)
                                     (2112)    (2131)
                                     (2121)    (2212)
                                     (11211)   (2311)
                                     (111111)  (3112)
                                               (3121)
                                               (3211)
                                               (11221)
                                               (12112)
                                               (12121)
                                               (12211)
                                               (21121)
                                               (111211)
                                               (112111)
                                               (1111111)
For example, starting with y = (3,2,1,1,2,2,2,1,2,1,1,1,1) and repeatedly taking run-lengths gives y -> (1,1,2,3,1,1,4) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1). These are all normal and the last is all 1's, so y is counted under a(20).

Normal compositions are A107429.
Constantly recursively normal partitions are A332272.
The case of partitions is A332277.
The case of reversed partitions is (also) A332277.
The narrow version is A332296.
The strong version is A332337.
The co-strong version is (also) A332337.
Cf. A001462, A181819, A182850, A317081, A317245, A317491, A329744, A332276, A332289, A332292, A332295, A332297, A332336, A332340.

recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],recnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],recnQ]],{n,0,10}]

For n > 1, a(n) = A332296(n) - 1.

A332295 Number of widely recursively normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 30, 34, 48, 54, 74, 86, 113, 132, 169, 200, 246, 293, 360, 422, 512, 599, 726, 840, 1009, 1181, 1401, 1631, 1940, 2240, 2636, 3069, 3567, 4141, 4846, 5556, 6470, 7505, 8627, 9936, 11523, 13176, 15151, 17430, 19935, 22846

Offset: 0

Gus Wiseman, Feb 16 2020

nonn

A sequence is widely recursively normal if either it is all 1's (wide) or its run-lengths cover an initial interval of positive integers (normal) and are themselves a widely recursively normal sequence.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (43)       (53)
             (111)  (211)   (41)     (51)      (52)       (62)
                    (1111)  (221)    (321)     (61)       (71)
                            (311)    (411)     (322)      (332)
                            (11111)  (111111)  (331)      (422)
                                               (421)      (431)
                                               (511)      (521)
                                               (3211)     (611)
                                               (1111111)  (3221)
                                                          (4211)
                                                          (11111111)
For example, starting with y = (4,3,2,2,1) and repeatedly taking run-lengths gives (4,3,2,2,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1), all of which have normal run-lengths, so y is widely recursively normal. On the other hand, starting with y and repeatedly taking multiplicities gives (4,3,2,2,1) -> (2,1,1,1) -> (3,1), so y is not fully normal (A317491).
Starting with y = (5,4,3,3,2,2,2,1,1) and repeatedly taking run-lengths gives (5,4,3,3,2,2,2,1,1) -> (1,1,2,3,2) -> (2,1,1,1) -> (1,3), so y is not widely recursively normal. On the other hand, starting with y and repeatedly taking multiplicities gives (5,4,3,3,2,2,2,1,1) -> (3,2,2,1,1) -> (2,2,1) -> (2,1) -> (1,1), so y is fully normal (A317491).

The narrow version is A000012.
Partitions with normal multiplicities are A317081.
The Heinz numbers of these partitions are a proper superset of A317492.
Accepting any constant sequence instead of just 1's gives A332272.
The total (instead of recursive) version is A332277.
The case of reversed partitions is this same sequence.
The alternating (instead of recursive) version is this same sequence.
Dominated by A332576.
Cf. A000009, A001462, A181819, A182850, A317245, A317491, A329746, A329747, A332276.

recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[Length/@Split[ptn]]==Range[Max[Length/@Split[ptn]]],recnQ[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

cp's OEIS Frontend

A332276 Heinz numbers of widely totally normal integer partitions.

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A332338 Number of alternately co-strong compositions of n.

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A332339 Number of alternately co-strong reversed integer partitions of n.

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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.

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A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

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A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

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A304687 Start with the multiset of prime multiplicities of n. Given a multiset, take the multiset of its multiplicities. Repeat until a constant multiset {k,k,...,k} is reached, and set a(n) to the sum of this multiset (k times the length).

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Maple

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A319151 Heinz numbers of superperiodic integer partitions.

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A332279 Number of widely totally normal compositions of n.

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