A323014 -id:A323014 - cp's OEIS frontend

A325759 Number of distinct frequencies in the frequency span of n.

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

Offset: 1

Gus Wiseman, May 19 2019

nonn

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer n is the frequency span of its prime indices (row n of A296150).

Row lengths of A325758.
Number of distinct entries in row n of A325757.
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A323014, A325249, A325277, A325760.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[Length[Union[freqspan[primeMS[n]]]],{n,100}]

A325760 Heinz number of the frequency span of n.

Original entry on oeis.org

1, 2, 3, 12, 5, 72, 7, 40, 27, 120, 11, 864, 13, 168, 180, 112, 17, 1296, 19, 1440, 252, 264, 23, 2880, 75, 312, 135, 2016, 29, 1200, 31, 352, 396, 408, 420, 972, 37, 456, 468, 4800, 41, 1680, 43, 3168, 3240, 552, 47, 8064, 147, 3600, 612, 3744, 53, 6480, 660

Offset: 1

Gus Wiseman, May 19 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer n is the frequency span of its prime indices (row n of A296150).

Row-products of A325277.
The prime indices of a(n) are row n of A325757.
The unsorted prime signature of a(n) is row n of A325758.
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A323014, A324843, A325248, A325249, A325759.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[Times@@Prime/@freqspan[primeMS[n]],{n,30}]

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

Original entry on oeis.org

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

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}]

A325261 Numbers whose omega-sequence does not cover an initial interval of positive integers.

Original entry on oeis.org

8, 16, 24, 27, 30, 32, 36, 40, 42, 48, 54, 56, 64, 66, 70, 72, 78, 80, 81, 88, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 125, 128, 130, 135, 136, 138, 144, 152, 154, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195, 196, 200

Offset: 1

Gus Wiseman, Apr 23 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).

			The sequence of terms together with their omega sequences begins:
    8: 3->1           108: 5->2->2->1        189: 4->2->2->1
   16: 4->1           110: 3->3->1           190: 3->3->1
   24: 4->2->2->1     112: 5->2->2->1        192: 7->2->2->1
   27: 3->1           114: 3->3->1           195: 3->3->1
   30: 3->3->1        120: 5->3->2->2->1     196: 4->2->1
   32: 5->1           125: 3->1              200: 5->2->2->1
   36: 4->2->1        128: 7->1              208: 5->2->2->1
   40: 4->2->2->1     130: 3->3->1           210: 4->4->1
   42: 3->3->1        135: 4->2->2->1        216: 6->2->1
   48: 5->2->2->1     136: 4->2->2->1        222: 3->3->1
   54: 4->2->2->1     138: 3->3->1           224: 6->2->2->1
   56: 4->2->2->1     144: 6->2->2->1        225: 4->2->1
   64: 6->1           152: 4->2->2->1        230: 3->3->1
   66: 3->3->1        154: 3->3->1           231: 3->3->1
   70: 3->3->1        160: 6->2->2->1        232: 4->2->2->1
   72: 5->2->2->1     162: 5->2->2->1        238: 3->3->1
   78: 3->3->1        165: 3->3->1           240: 6->3->2->2->1
   80: 5->2->2->1     168: 5->3->2->2->1     243: 5->1
   81: 4->1           170: 3->3->1           246: 3->3->1
   88: 4->2->2->1     174: 3->3->1           248: 4->2->2->1
   96: 6->2->2->1     176: 5->2->2->1        250: 4->2->2->1
  100: 4->2->1        180: 5->3->2->2->1     252: 5->3->2->2->1
  102: 3->3->1        182: 3->3->1           255: 3->3->1
  104: 4->2->2->1     184: 4->2->2->1        256: 8->1
  105: 3->3->1        186: 3->3->1           258: 3->3->1

Complement of A325251.
Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325260, A325262, 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).

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],!normQ[omseq[#]]&]

A325266 Numbers whose adjusted frequency depth equals their number of prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 24, 25, 29, 30, 31, 37, 40, 41, 42, 43, 47, 49, 53, 54, 56, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 121, 127, 130, 131, 135, 136, 137, 138, 139, 149

Offset: 1

Gus Wiseman, Apr 17 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(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. 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 whose adjusted frequency depth is equal to their length. The enumeration of these partitions by sum is given by A325246.

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
   2:       {1} (1)
   3:       {2} (1)
   4:     {1,1} (2,1)
   5:       {3} (1)
   7:       {4} (1)
   9:     {2,2} (2,1)
  11:       {5} (1)
  13:       {6} (1)
  17:       {7} (1)
  19:       {8} (1)
  23:       {9} (1)
  24: {1,1,1,2} (4,2,2,1)
  25:     {3,3} (2,1)
  29:      {10} (1)
  30:   {1,2,3} (3,3,1)
  31:      {11} (1)
  37:      {12} (1)
  40: {1,1,1,3} (4,2,2,1)
  41:      {13} (1)
  42:   {1,2,4} (3,3,1)

Cf. A056239, A112798, A118914, A181819, A225485, A323023, A325246, A325258, A325277, A325278, A325281, A325283.

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).

fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
Select[Range[100],fdadj[#]==PrimeOmega[#]&]

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&]

cp's OEIS Frontend

A325759 Number of distinct frequencies in the frequency span of n.

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A325760 Heinz number of the frequency span of n.

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

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

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A325261 Numbers whose omega-sequence does not cover an initial interval of positive integers.

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A325266 Numbers whose adjusted frequency depth equals their number of prime factors counted with multiplicity.

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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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