A182850 -id:A182850 - cp's OEIS frontend

A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.

1, 1, 2, 2, 4, 5, 5, 8, 10, 12, 13, 18, 19, 24, 25, 31, 33, 40, 40, 49, 51, 59, 60, 71, 72, 83, 84, 96, 98, 111, 111, 126, 128, 142, 143, 160, 161, 178, 179, 197, 199, 218, 218, 239, 241, 261, 262, 285, 286, 309, 310, 334, 336, 361, 361, 388, 390, 416, 417, 446

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1).

The Heinz numbers of these partitions are given by A325251.

			The a(1) = 1 through a(9) = 12 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)     (9)
       (11)  (21)  (22)   (32)   (33)   (43)    (44)    (54)
                   (31)   (41)   (42)   (52)    (53)    (63)
                   (211)  (221)  (51)   (61)    (62)    (72)
                          (311)  (411)  (322)   (71)    (81)
                                        (331)   (332)   (441)
                                        (511)   (422)   (522)
                                        (3211)  (611)   (711)
                                                (3221)  (3321)
                                                (4211)  (4221)
                                                        (4311)
                                                        (5211)

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325262, A325277.

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

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],normQ[omseq[#]]&]],{n,0,30}]

a(n) + A325262(n) = A000041(n).
Conjectures from Chai Wah Wu, Jan 13 2021: (Start)
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n > 9.
G.f.: (-x^9 - x^8 - x^7 + x^6 - x^5 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)). (End)

A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 6, 7, 12, 18, 29, 38, 58, 77, 110, 145, 198, 257, 345, 441, 576, 733, 942, 1184, 1503, 1875, 2352, 2914, 3620, 4454, 5493, 6716, 8221, 10001, 12167, 14723, 17816, 21459, 25836, 30988, 37139, 44365, 52956, 63022, 74934, 88873, 105296, 124469

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1).

			The a(3) = 1 through a(9) = 18 partitions:
  (111)  (1111)  (2111)   (222)     (421)      (431)       (333)
                 (11111)  (321)     (2221)     (521)       (432)
                          (2211)    (4111)     (2222)      (531)
                          (3111)    (22111)    (3311)      (621)
                          (21111)   (31111)    (5111)      (3222)
                          (111111)  (211111)   (22211)     (6111)
                                    (1111111)  (32111)     (22221)
                                               (41111)     (32211)
                                               (221111)    (33111)
                                               (311111)    (42111)
                                               (2111111)   (51111)
                                               (11111111)  (222111)
                                                           (321111)
                                                           (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325261, A325277, A325285.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325249 (sum).

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],!normQ[omseq[#]]&]],{n,0,30}]

A325416 Least k such that the omega-sequence of k sums to n, and 0 if none exists.

Original entry on oeis.org

1, 2, 0, 4, 8, 6, 32, 30, 12, 24, 48, 96, 60, 120, 240, 480, 960, 1920, 3840, 2520, 5040, 10080, 20160, 40320, 80640

Offset: 0

Gus Wiseman, Apr 25 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1) with sum 13.

			The sequence of terms together with their omega-sequences (n = 2 term not shown) begins:
     1:
     2:  1
     4:  2 1
     8:  3 1
     6:  2 2 1
    32:  5 1
    30:  3 3 1
    12:  3 2 2 1
    24:  4 2 2 1
    48:  5 2 2 1
    96:  6 2 2 1
    60:  4 3 2 2 1
   120:  5 3 2 2 1
   240:  6 3 2 2 1
   480:  7 3 2 2 1
   960:  8 3 2 2 1
  1920:  9 3 2 2 1
  3840: 10 3 2 2 1
  2520:  7 4 3 2 2 1
  5040:  8 4 3 2 2 1

Cf. A056239, A181819, A181821, A304465, A307734, A323023, A325238, A325277, A325280, A325413, A325415.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
da=Table[Total[omseq[n]],{n,10000}];
Table[If[!MemberQ[da,k],0,Position[da,k][[1,1]]],{k,0,Max@@da}]

A328830 The second prime shadow of n: a(1) = 1; for n > 1, a(n) = a(A003557(n)) * prime(A056169(n)) when A056169(n) > 0, otherwise a(n) = a(A003557(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 29 2019

nonn

a(n) depends only on prime signature of n (cf. A025487).

			For n = 30 = 2 * 3 * 5, there are three unitary prime factors, while A003557(30) = 1, which terminates the recursion, thus a(30) = prime(3) = 5.
For n = 60060 = 2^2 * 3 * 5 * 7 * 11 * 13, there are 5 unitary prime factors, while in A003557(60060) = 2 there is only one, thus a(60060) = prime(5) * prime(1) = 11 * 2 = 22.
The number 1260 = 2^2*3^2*5*7 has prime exponents (2,2,1,1) so its prime shadow is prime(2)*prime(2)*prime(1)*prime(1) = 36.  Next, 36 = 2^2*3^2 has prime exponents (2,2) so its prime shadow is prime(2)*prime(2) = 9. In fact, the term a(1260) = 9 is the first appearance of 9 in the sequence. - _Gus Wiseman_, Apr 28 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Column 2 of A353510.
Cf. A001221, A001222, A003557, A008578, A056169, A071625, A323022.
Differs from A182860 for the first time at a(30) = 5, while A182860(30) = 4.
Cf. A182863 for the first appearances.
A005361 gives product of prime exponents.
A112798 gives prime indices, sum A056239.
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.
Cf. A109297, A182850, A353393, A353507.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ From A056169
A328830(n) = if(1==n,n, my(u=A056169(n)); if(0==u,1,prime(u)) * A328830(A003557(n)));

a(1) = 1; for n > 1, a(n) = A008578(1+A056169(n)) * a(A003557(n)).
A001221(a(n)) = A323022(n).
A001222(a(n)) = A071625(n).
a(n) = A181819(A181819(n)). - Gus Wiseman, Apr 27 2022

Added Gus Wiseman's new name to the front of the definition. - Antti Karttunen, Apr 27 2022

A332576 Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 31, 35, 51, 59, 80, 97, 130, 153, 204, 244, 308, 376, 475, 564, 708, 851, 1043, 1247, 1533, 1816, 2216, 2633, 3174, 3766, 4526, 5324, 6376, 7520, 8917, 10479, 12415, 14524, 17134, 20035, 23489, 27423, 32091, 37286, 43512

Offset: 0

Gus Wiseman, Mar 05 2020

nonn

First differs from A317491 at a(11) = 31, A317491(11) = 30.

			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)

The narrow version is A317081.
Heinz numbers of these partitions first differ from A317492 in having 420.
Not counting constant-1 sequences gives A317081.
Dominated by A332295.
Cf. A000009, A001462, A181819, A182850, A317245, A317491, A329746, A329747, A332272, A332277.

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

a(n > 1) = A317081(n) + 1.

A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 3, 1, 3, 4, 3, 7, 5, 9, 8, 12, 15, 15, 20, 21, 25, 31, 33, 38, 42, 46, 56, 61, 67, 78, 76, 96, 100, 114, 131, 130, 157, 157, 185, 200, 214, 236, 253, 275, 302, 333, 351, 386, 408, 440, 486, 515, 564, 596, 633, 691, 734, 800, 854, 899, 964

Offset: 0

Gus Wiseman, May 15 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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) = 9 partitions (A..D = 10..13):
  (53)  (72)    (73)    (B)     (75)     (D)      (B3)
        (621)   (532)   (A1)    (651)    (B2)     (752)
        (4221)  (631)   (4331)  (732)    (A21)    (761)
                (4411)          (6321)   (43321)  (A31)
                                (6411)   (44311)  (C11)
                                (43221)           (6521)
                                (44211)           (9221)
                                                  (54221)
                                                  (64211)

The LHS (prime shadow) is A181819, with an inverse A181821.
The RHS (product of prime shadows) is A353394, first appearances A353397.
These partitions are ranked by A353395.
A related comparison is A353398, ranked by A353399.
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, product A005361.
A239455 counts Look-and-Say partitions, ranked by A351294.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000005, A002033, A143773, A182850, A316428, A325131, A325702, A325755, A353389, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==red[Times@@Prime/@#]&]],{n,0,15}]

A304647 Smallest term of A304636 that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

5, 8, 30, 360, 1801800, 2746644314348614680000, 13268350773236509446586539974366689358164301703214270074935844483572035447570761114173070859428708074413696096366645684575600000000

Offset: 0

Gus Wiseman, May 15 2018

nonn

The first entry 5 is optional so has offset 0.

			The list of multisets with Heinz numbers in the sequence is the following. The number of k's in row n + 1 is equal to the k-th largest term of row n.
                     5: {3}
                     8: {1,1,1}
                    30: {1,2,3}
                   360: {1,1,1,2,2,3}
               1801800: {1,1,1,2,2,3,3,4,5,6}
2746644314348614680000: {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,7,8,9,10}

Cf. A001221, A001222, A001462, A012257, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A275870, A304464, A304465, A304634, A304636.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{Reverse[#][[i]]}],{i,Length[#]}]&,{3},6]

A317493 Heinz numbers of integer partitions that are not fully normal.

Original entry on oeis.org

9, 24, 25, 27, 36, 40, 48, 49, 54, 56, 72, 80, 81, 88, 96, 100, 104, 108, 112, 120, 121, 125, 135, 136, 144, 152, 160, 162, 168, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 264, 270, 272, 280, 288, 289, 296, 297, 304, 312

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

			Sequence of all integer partitions that are not fully normal begins: (22), (2111), (33), (222), (2211), (3111), (21111), (44), (2221), (4111), (22111), (31111), (2222), (5111), (211111), (3311).

Cf. A055932, A056239, A181819, A182850, A296150, A305733, A317089, A317090, A317245, A317246, A317491, A317492.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Select[Range[100],!fulnrmQ[Reverse[primeMS[#]]]&]

A319152 Nonprime Heinz numbers of superperiodic integer partitions.

Original entry on oeis.org

9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1331, 1369, 1521, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3249, 3481, 3721, 4225, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7569, 7921, 8281, 9261, 9409, 10201

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A subsequence of A001597.
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: (22), (33), (222), (44), (2222), (55), (333), (66), (22222), (77), (444), (88), (4422), (99), (3333), (222222).

Cf. A001597, A056239, A072774, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319151.

supperQ[n_]:=Or[n==2,And[GCD@@PrimePi/@FactorInteger[n][[All,1]]>1,supperQ[Times@@Prime/@FactorInteger[n][[All,2]]]]];
Select[Range[10000],And[!PrimeQ[#],supperQ[#]]&]

A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 3, 3, 5, 8, 6, 13, 12, 14, 17, 22, 17, 28, 29, 30, 38, 50, 46, 67, 64, 75, 81, 104, 99, 127, 128, 150, 155, 201, 189, 236, 244, 293, 302, 363, 372, 437, 457, 548, 547, 638, 671, 754, 809, 922, 947, 1074, 1144, 1290, 1342, 1515, 1574

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 A325387.

			The a(4) = 1 through a(10) = 5 partitions:
  (211)  (221)   (21111)  (2221)    (22211)    (22221)     (222211)
         (2111)           (22111)   (221111)   (2211111)   (322111)
                          (211111)  (2111111)  (21111111)  (2221111)
                                                           (22111111)
                                                           (211111111)

Column k = 4 of A325336.

Cf. A000009, A007862, A181819, A182850, A317081, A320348, A323014, A325280, A325326, A325334, A325387.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==4&]],{n,0,30}]

cp's OEIS Frontend

A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.

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A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

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A325416 Least k such that the omega-sequence of k sums to n, and 0 if none exists.

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A328830 The second prime shadow of n: a(1) = 1; for n > 1, a(n) = a(A003557(n)) * prime(A056169(n)) when A056169(n) > 0, otherwise a(n) = a(A003557(n)).

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A332576 Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.

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A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

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A304647 Smallest term of A304636 that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

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A317493 Heinz numbers of integer partitions that are not fully normal.

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A319152 Nonprime Heinz numbers of superperiodic integer partitions.

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