A325280 -id:A325280 - cp's OEIS frontend

A353745 Number of runs in the ordered prime signature of n.

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

Offset: 1

Gus Wiseman, May 20 2022

nonn

First differs from A071625 at a(90) = 3.
First differs from A331592 at a(90) = 3.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The prime indices of 630 are {1,2,2,3,4}, with multiplicities {1,2,1,1}, with runs {{1},{2},{1,1}}, so a(630) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are A354233.
A001222 counts prime factors, distinct A001221.
A005361 gives product of prime signature, firsts A353500/A085629.
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/A323014 give frequency depth, counted by A225485/A325280.
Cf. A005811, A097318, A130091, A304678, A333755, A353503, A353507, A353742.
Cf. also A329747.

Table[Length[Split[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i < #lista) && (lista[i]==lista[i+1]), rl++, listput(runs,rl); rl=1)); (runs)));
A353745(n) = #runlengths(runlengths(pis_to_runs(n))); \\ Antti Karttunen, Jan 20 2025

A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 2, 5, 5, 5, 1, 2, 12, 1, 8, 11, 3, 3, 19, 8, 5, 27, 9, 1, 2, 34, 19, 1, 15, 26, 34, 2, 2, 49, 45, 5, 5, 68, 48, 14, 4, 58, 98, 15, 1, 18, 76, 105, 31, 1, 2, 88, 159, 46, 2, 13, 98, 191, 79, 4, 2, 114, 261, 105, 8, 14, 148, 282, 164, 19

Offset: 0

Gus Wiseman, Jun 04 2022

The partition run-sum trajectory is obtained by repeatedly taking the run-sums until a strict partition is reached. For example, the trajectory of y = (3,2,1,1,1) is (3,2,1,1,1) -> (3,3,2) -> (6,2), so y is counted under T(8,2).

			Triangle begins:
   1
   1
   2
   2  1
   4  1
   2  5
   5  5  1
   2 12  1
   8 11  3
   3 19  8
   5 27  9  1
   2 34 19  1
  15 26 34  2
   2 49 45  5
   5 68 48 14
   4 58 98 15  1
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (4211)      (611)
  (41111)     (3311)
  (221111)    (5111)
  (11111111)  (22211)
              (32111)
              (311111)
              (2111111)

Row sums are A000041.
Row-lengths are A003056.
The last part of the same trajectory is A353842.
Column k = 1 is A353845, compositions A353858.
The length of the trajectory is A353846.
The version for compositions is A353856.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with constant run-sums, ranked by A353833/A353834.
A325268 counts partitions by omicron, rank statistic A304465.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
A353932 lists run-sums of standard compositions.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.
Cf. A047966, A237685, A325277, A353841, A353853-A353859.

Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&,#]]==k&]],{n,0,15},{k,0,n}]

A325253 Number of integer partitions of n with adjusted frequency depth ceiling(sqrt(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 6, 8, 17, 26, 25, 44, 53, 63, 83, 128, 168, 212, 273, 344, 429, 525, 662, 796, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 0, 0, 0, 0, 0, 1, 5, 14, 36, 76, 143, 269, 446, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Apr 22 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 a(2) = 1 through a(11) = 26 partitions:
    11  111  22    32  42    43   53    54      433        443
             1111  41  51    52   62    63      442        533
                       321   61   71    72      622        551
                       2211  421  431   81      811        722
                                  521   432     3331       911
                                  3311  531     4222       3332
                                        621     7111       5222
                                        222111  61111      8111
                                                222211     32222
                                                322111     33311
                                                331111     44111
                                                511111     71111
                                                2221111    222221
                                                4111111    322211
                                                22111111   332111
                                                31111111   422111
                                                211111111  611111
                                                           2222111
                                                           3221111
                                                           3311111
                                                           5111111
                                                           22211111
                                                           41111111
                                                           221111111
                                                           311111111
                                                           2111111111

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325252, A325258, A325271, A325278, A325280, A325282.

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

A325267 Number of integer partitions of n with omicron 2.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 7, 12, 17, 24, 33, 44, 57, 76, 100, 129, 168, 214, 282, 355, 462, 586, 755, 937, 1202, 1493, 1900, 2349, 2944, 3621, 4520, 5514, 6813, 8298, 10150, 12240, 14918, 17931, 21654, 25917, 31081, 37029, 44256, 52474, 62405, 73724, 87378, 102887

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A304634.

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. We define the omicron of an integer partition to be 0 if the partition is empty, 1 if it is a singleton, and otherwise the second-to-last part of its omega-sequence. 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), and its omicron is 2.

			The a(1) = 1 through a(8) = 17 partitions:
  (11)  (21)  (22)   (32)    (33)     (43)      (44)
              (31)   (41)    (42)     (52)      (53)
              (211)  (221)   (51)     (61)      (62)
                     (311)   (411)    (322)     (71)
                     (2111)  (2211)   (331)     (332)
                             (3111)   (511)     (422)
                             (21111)  (2221)    (611)
                                      (3211)    (3221)
                                      (4111)    (3311)
                                      (22111)   (4211)
                                      (31111)   (5111)
                                      (211111)  (22211)
                                                (32111)
                                                (41111)
                                                (221111)
                                                (311111)
                                                (2111111)

Cf. A056239, A112798, A181819, A304634, A304636, A323014, A323023, A325250, A325273, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==2&]],{n,0,30}]

A325271 Number of integer partitions of n with frequency depth round(sqrt(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 6, 8, 11, 11, 19, 44, 53, 63, 83, 113, 124, 171, 190, 344, 429, 525, 662, 796, 981, 1182, 1442, 1709, 2096, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754

Offset: 0

Gus Wiseman, Apr 23 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 a(2) = 1 through a(10) = 11 partitions:
  (2)  (111)  (22)    (11111)  (33)      (43)   (53)    (54)      (64)
              (1111)           (222)     (52)   (62)    (63)      (73)
                               (111111)  (61)   (71)    (72)      (82)
                                         (421)  (431)   (81)      (91)
                                                (521)   (432)     (532)
                                                (3311)  (531)     (541)
                                                        (621)     (631)
                                                        (222111)  (721)
                                                                  (3322)
                                                                  (4321)
                                                                  (4411)

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325272, A325253, A325258, A325278, A325280, A325282.

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

A325285 Number of integer partitions of n whose omega-sequence has repeated parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 13, 17, 26, 36, 54, 66, 98, 125, 164, 214, 285, 354, 468, 585, 745, 945, 1195, 1477, 1864, 2317, 2867, 3544, 4383, 5348, 6589, 8028, 9778, 11885, 14403, 17362, 20992, 25212, 30239, 36158, 43242, 51408, 61240, 72568, 85989, 101607, 120027

Offset: 0

Gus Wiseman, Apr 24 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), which has repeated parts, so (32211) is counted under a(9).

The Heinz numbers of these partitions are given by A325411.

			The a(3) = 1 through a(8) = 17 partitions:
  (21)  (31)   (32)    (42)     (43)      (53)
        (211)  (41)    (51)     (52)      (62)
               (221)   (321)    (61)      (71)
               (311)   (411)    (322)     (332)
               (2111)  (3111)   (331)     (422)
                       (21111)  (421)     (431)
                                (511)     (521)
                                (2221)    (611)
                                (3211)    (3221)
                                (4111)    (4211)
                                (22111)   (5111)
                                (31111)   (22211)
                                (211111)  (32111)
                                          (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

Cf. A047966, A181819, A323014, A323023, A325247, A325250, A325260, A325262, A325411.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

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

A325387 Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.

Original entry on oeis.org

12, 18, 24, 48, 54, 72, 96, 108, 144, 162, 192, 288, 324, 360, 384, 432, 486, 540, 576, 600, 648, 720, 768, 864, 972, 1152, 1200, 1260, 1350, 1440, 1458, 1500, 1536, 1620, 1728, 1944, 2100, 2160, 2250, 2304, 2400, 2592, 2880, 2916, 2940, 3072, 3150, 3240, 3456

Offset: 1

Gus Wiseman, May 02 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with adjusted frequency depth 4 whose parts cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325335.

			The sequence of terms together with their prime indices begins:
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   192: {1,1,1,1,1,1,2}
   288: {1,1,1,1,1,2,2}
   324: {1,1,2,2,2,2}
   360: {1,1,1,2,2,3}
   384: {1,1,1,1,1,1,1,2}
   432: {1,1,1,1,2,2,2}
   486: {1,2,2,2,2,2}
   540: {1,1,2,2,2,3}
   576: {1,1,1,1,1,1,2,2}
   600: {1,1,1,2,3,3}

Cf. A055932, A056239, A112798, A181819,, A323014, A325280, A325326, A325335, A325336, A325374.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#1]&,n,!PrimeQ[#1]&]]];
Select[Range[10000],normQ[#]&&fdadj[#]==4&]

A325252 Number of integer partitions of n with frequency depth floor(sqrt(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 8, 11, 11, 19, 17, 25, 29, 83, 113, 124, 171, 190, 242, 289, 368, 399, 796, 981, 1182, 1442, 1709, 2096, 2469, 2990, 3545, 4276, 5037, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 33856, 40515, 48508, 57826, 68982, 81493, 446, 738

Offset: 0

Gus Wiseman, Apr 22 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 a(2) = 1 through a(12) = 19 partitions (A = 10, B = 11):
  2  3  22    11111  33      1111111  44        54      64    65    75
        1111         222              2222      63      73    74    84
                     111111           11111111  72      82    83    93
                                                81      91    92    A2
                                                432     532   A1    B1
                                                531     541   542   543
                                                621     631   632   642
                                                222111  721   641   651
                                                        3322  731   732
                                                        4321  821   741
                                                        4411  5321  831
                                                                    921
                                                                    4422
                                                                    5421
                                                                    5511
                                                                    6321
                                                                    332211
                                                                    333111
                                                                    22221111

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325253, A325258, A325271, A325278, A325280, A325282.

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

A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 5, 6, 9, 10, 18, 18, 31, 34, 48, 57, 80, 86, 122, 138, 183, 211, 275, 311, 402, 461, 576, 663, 825, 942, 1163, 1334, 1621, 1865, 2248, 2566, 3084, 3532, 4193, 4794, 5674, 6472, 7617, 8685, 10153, 11576, 13483, 15320, 17790, 20200, 23342

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A325241.

For example, the partition (44111) has two multiplicities (2 and 3) which differ by 1, so is counted under a(11).

			The a(4) = 1 through a(11) = 18 partitions:
  (211)  (221)  (411)  (322)    (332)    (441)    (433)      (443)
         (311)         (331)    (422)    (522)    (442)      (533)
                       (511)    (611)    (711)    (622)      (551)
                       (3211)   (3221)   (3321)   (811)      (722)
                       (22111)  (4211)   (4221)   (5221)     (911)
                                (22211)  (4311)   (5311)     (4322)
                                         (5211)   (6211)     (4331)
                                         (32211)  (33211)    (4421)
                                         (33111)  (42211)    (5411)
                                                  (2221111)  (6221)
                                                             (6311)
                                                             (7211)
                                                             (33221)
                                                             (33311)
                                                             (43211)
                                                             (44111)
                                                             (52211)
                                                             (2222111)

Cf. A047966, A062770, A071625, A098859, A117571, A127002, A325242, A325244, A325245, A325280.

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

A354233 Least number with n runs in ordered prime signature.

Original entry on oeis.org

1, 2, 12, 90, 2100, 48510, 3303300, 139369230, 18138420300, 1157182716690, 278261505822300, 30168910606824990, 9894144362523521100, 1693350783450479863710, 715178436956287675671300, 147157263134197051595990130, 83730945863531292204568790100

Offset: 0

Gus Wiseman, May 20 2022

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The prime indices of 90 are {1,2,2,3}, with multiplicities {1,2,1}, with runs {{1},{2},{1}}, and this is the first case of 3 runs, so a(3) = 90.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Positions of first appearances in A353745.
A001222 counts prime factors with multiplicity, distinct A001221.
A005361 gives product of 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. A005811, A097318, A304678, A325755, A353507, A353742.

Table[Product[Prime[i]^If[EvenQ[n-i],1,2],{i,n}],{n,0,15}]

cp's OEIS Frontend

A353745 Number of runs in the ordered prime signature of n.

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A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

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A325253 Number of integer partitions of n with adjusted frequency depth ceiling(sqrt(n)).

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A325267 Number of integer partitions of n with omicron 2.

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A325271 Number of integer partitions of n with frequency depth round(sqrt(n)).

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A325285 Number of integer partitions of n whose omega-sequence has repeated parts.

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A325387 Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.

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A325252 Number of integer partitions of n with frequency depth floor(sqrt(n)).

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A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

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