A325248 -id:A325248 - cp's OEIS frontend

A325247 Numbers whose omega-sequence is strict (no repeated parts).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

First differs from A323306 in having 216.
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).
Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). 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}
     3: {2}
     4: {1,1}
     5: {3}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    11: {5}
    13: {6}
    16: {1,1,1,1}
    17: {7}
    19: {8}
    23: {9}
    25: {3,3}
    27: {2,2,2}
    29: {10}
    31: {11}
    32: {1,1,1,1,1}
    36: {1,1,2,2}

Positions of squarefree numbers in A325248.
Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.
Omega-sequence statistics: A001221 (second omega), A001222 (first omega), A071625 (third omega), A304465 (second-to-last omega), A182850 or A323014 (depth), A323022 (fourth omega), A325248 (Heinz number).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#1]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],UnsameQ@@omseq[#]&]

A325264 Numbers whose omega-sequence sums to 7.

Original entry on oeis.org

30, 36, 42, 64, 66, 70, 78, 100, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 196, 222, 225, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, Apr 18 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 prime indices and omega-sequences begins:
   30: {1,2,3} (3,3,1)
   36: {1,1,2,2} (4,2,1)
   42: {1,2,4} (3,3,1)
   64: {1,1,1,1,1,1} (6,1)
   66: {1,2,5} (3,3,1)
   70: {1,3,4} (3,3,1)
   78: {1,2,6} (3,3,1)
  100: {1,1,3,3} (4,2,1)
  102: {1,2,7} (3,3,1)
  105: {2,3,4} (3,3,1)
  110: {1,3,5} (3,3,1)
  114: {1,2,8} (3,3,1)
  130: {1,3,6} (3,3,1)
  138: {1,2,9} (3,3,1)
  154: {1,4,5} (3,3,1)
  165: {2,3,5} (3,3,1)
  170: {1,3,7} (3,3,1)
  174: {1,2,10} (3,3,1)
  182: {1,4,6} (3,3,1)
  186: {1,2,11} (3,3,1)
  190: {1,3,8} (3,3,1)
  195: {2,3,6} (3,3,1)
  196: {1,1,4,4} (4,2,1)

Positions of 7's in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325265, 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).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],Total[omseq[#]]==7&]

A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129, 132

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also numbers whose adjusted frequency depth is one plus their number of prime factors counted with multiplicity. The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one 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 whose adjusted frequency depth is equal to their length plus 1. The enumeration of these partitions by sum is given by A127002.

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
   6:     {1,2} (2,2,1)
  10:     {1,3} (2,2,1)
  12:   {1,1,2} (3,2,2,1)
  14:     {1,4} (2,2,1)
  15:     {2,3} (2,2,1)
  18:   {1,2,2} (3,2,2,1)
  20:   {1,1,3} (3,2,2,1)
  21:     {2,4} (2,2,1)
  22:     {1,5} (2,2,1)
  26:     {1,6} (2,2,1)
  28:   {1,1,4} (3,2,2,1)
  33:     {2,5} (2,2,1)
  34:     {1,7} (2,2,1)
  35:     {3,4} (2,2,1)
  38:     {1,8} (2,2,1)
  39:     {2,6} (2,2,1)
  44:   {1,1,5} (3,2,2,1)
  45:   {2,2,3} (3,2,2,1)
  46:     {1,9} (2,2,1)
  50:   {1,3,3} (3,2,2,1)
  51:     {2,7} (2,2,1)
  52:   {1,1,6} (3,2,2,1)
  55:     {3,5} (2,2,1)
  57:     {2,8} (2,2,1)
  58:    {1,10} (2,2,1)
  60: {1,1,2,3} (4,3,2,2,1)

Cf. A056239, A112798, A118914, A127002, A181819, A323023, A325246, A325259, A325266, A325270, A325277, A325282.

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).

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

A325414 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 24 2019

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) with sum 13, so (32211) is counted under T(9,13).

			Triangle begins:
  1
  0 1
  0 1 0 1
  0 1 0 0 1 1
  0 1 0 1 0 2 0 0 1
  0 1 0 0 0 2 1 0 2 1
  0 1 0 1 1 2 0 3 1 1 1
  0 1 0 0 0 3 0 1 4 2 2 1 1
  0 1 0 1 0 4 0 3 3 2 2 2 3 1
  0 1 0 0 1 4 0 3 3 3 4 1 6 3 1
  0 1 0 1 0 4 1 6 4 4 1 4 5 8 2 1
Row n = 9 counts the following partitions:
  9  333  54  432  441  3222    22221      411111  3321     32211     321111
          63  531  522  6111    33111              4221     42111
          72  621  711  222111  51111              4311     21111111
          81                    111111111          5211
                                                   2211111
                                                   3111111

Row sums are A000041.
Row lengths are A325413(n) + 1 (because k starts at 0).
Number of nonzero terms in row n is A325415(n).
Cf. A181819, A225486, A323014, A323023, A325248, A325249, A325277, A325412, A325415, A325416.
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],Total[omseq[#]]==k&]],{n,0,10},{k,0,Max[Total/@omseq/@IntegerPartitions[n]]}]

A325415 Number of distinct sums of omega-sequences of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 10, 11, 13, 12, 15, 14, 16, 18, 18, 18, 21, 20, 23, 23, 24, 24, 27, 27, 28, 29, 30, 30, 34, 32, 34, 35, 36, 37, 39, 38, 40, 41, 43, 42, 45, 44, 46, 48, 48, 48, 51, 50, 53, 53, 54, 54, 57, 57, 58, 59, 60, 60, 64

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) with sum 13.

			The partitions of 9 organized by sum of omega sequence (first column) are:
   1: (9)
   4: (333)
   5: (81) (72) (63) (54)
   7: (621) (531) (432)
   8: (711) (522) (441)
   9: (6111) (3222) (222111)
  10: (51111) (33111) (22221) (111111111)
  11: (411111)
  12: (5211) (4311) (4221) (3321) (3111111) (2211111)
  13: (42111) (32211) (21111111)
  14: (321111)
There are a total of 11 distinct sums {1,4,5,7,8,9,10,11,12,13,14}, so a(9) = 11.

Number of nonzero terms in row n of A325414.
Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325249, A325277, A325412, A325413, A325416.
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[Union[Total/@omseq/@IntegerPartitions[n]]],{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}]

A325412 Number of distinct omega-sequences of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 10, 9, 14, 15, 20, 21, 33, 30, 39, 45, 54, 54, 69, 68, 85, 90, 100, 104, 128, 127, 141, 153, 172, 175, 205, 203, 229, 240, 257, 274, 308, 309, 335, 356, 390, 395, 437, 444, 481, 506, 530, 549, 602, 609, 648, 672, 710, 727, 777, 798, 848, 871

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).

			The a(1) = 1 through a(9) = 15 omega-sequences:
  (1)  (1)   (1)    (1)     (1)     (1)     (1)      (1)      (1)
       (21)  (31)   (21)    (51)    (21)    (71)     (21)     (31)
             (221)  (41)    (221)   (31)    (221)    (41)     (91)
                    (221)   (3221)  (61)    (331)    (81)     (221)
                    (3221)  (4221)  (221)   (3221)   (221)    (331)
                                    (331)   (4221)   (331)    (621)
                                    (421)   (5221)   (421)    (3221)
                                    (3221)  (6221)   (3221)   (4221)
                                    (4221)  (43221)  (4221)   (5221)
                                    (5221)           (5221)   (6221)
                                                     (6221)   (7221)
                                                     (7221)   (8221)
                                                     (43221)  (43221)
                                                     (53221)  (53221)
                                                              (63221)

Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325277, A325413, A325415.

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[Union[omseq/@IntegerPartitions[n]]],{n,0,30}]

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}]

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[#]&]

cp's OEIS Frontend

A325247 Numbers whose omega-sequence is strict (no repeated parts).

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A325264 Numbers whose omega-sequence sums to 7.

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A325281 Numbers of the form a*b, a*a*b, or a*a*b*c where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

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A325414 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.

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A325415 Number of distinct sums of omega-sequences of integer partitions of n.

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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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A325412 Number of distinct omega-sequences of integer partitions of n.

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A325760 Heinz number of the frequency span of n.

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

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A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).