A323022 -id:A323022 - cp's OEIS frontend

A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

1, 2, 60, 1801800, 11657093261814000, 7167827541370578634694420017740000, 291943326350524088652207164949980988754136887856059678357800000

Offset: 0

Matthew Vandermast, Jan 05 2011

a(n) = smallest number m such that A181819(m) = A006939(n).  a(n) belongs to A182855 iff n > 1.
Next term has 105 digits.
Smallest number k with A323022(k) = n, where A323022(m) is the number of distinct multiplicities in the prime signature of m. - Gus Wiseman, Jan 03 2019

			The canonical prime factorization of a(3) = 1801800 is 2^3*3^2*5^2*7*11*13. The prime signature of 1801800 is therefore (3,2,2,1,1,1). Note that (3,2,2,1,1,1) contains exactly one number that appears once (3), one number that appears twice (2), and one number that appears three times (1).

David A. Corneth, Table of n, a(n) for n = 0..12
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A006939, A181819, A182855.

Cf. A056239, A059404, A062770, A071625, A118914, A181821, A182857, A323022.

Table[Product[Times@@Prime[i*(i-1)/2+Ceiling[Range[i*(n-i)]/(n-i)]],{i,n-1}],{n,6}] (* Gus Wiseman, Jan 03 2019 *)

a(n) = if(n == 0, return(1)); my(f = matrix(binomial(n+1,2), 2)); f[, 1] = primes(#f~ )~; f[, 2] = Vecrev(concat(vector(n, i, vector(n+1-i, j, i))))~; factorback(f) \\ David A. Corneth, Jan 03 2019

Partial products of A113511.
log a(n) ~ (1/3) n^3 log n. [Charles R Greathouse IV, Jan 13 2012]
A001222(a(n)) = A000292(n). - Gus Wiseman, Jan 03 2019
a(0) = 1; a(n + 1) = A002110(binomial(n + 2, 2)) * a(n). - David A. Corneth, Jan 03 2019

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}]

A325333 Number of integer partitions of n whose multiplicities all appear the same number of times.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 19, 23, 32, 39, 50, 63, 82, 96, 125, 152, 186, 226, 271, 326, 392, 473, 552, 663, 771, 918, 1065, 1261, 1448, 1710, 1953, 2283, 2608, 3062, 3455, 4013, 4552, 5271, 5974, 6884, 7774, 8937, 10065, 11570, 12953, 14838, 16710, 18979

Offset: 0

Gus Wiseman, May 01 2019

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (321)     (331)
                                (11111)  (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (4111)
                                         (111111)  (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)
For example, the partition (4,3,3,3,2,2,2,1) has multiplicities (1,3,3,1), and since both multiplicities 1 and 3 appear twice, (4,3,3,3,2,2,2,1) is counted under a(20).

Cf. A000041, A007862, A047966, A049988, A098859, A323022, A325329, A325369.

Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]

A325410 Smallest k such that the adjusted frequency depth of k! is n > 2.

Original entry on oeis.org

3, 4, 5, 7, 26, 65, 942, 24147

Offset: 3

Gus Wiseman, Apr 24 2019

If infinite terms were allowed, we would have a(0) = 1, a(1) = 2, a(2) = infinity. It is possible this sequence is finite, or that there are additional gaps.

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.

			Column n is the sequence of images under A181819 starting with a(n)!:
  6  24  120  5040  403291461126605635584000000
  4  10  20   84    11264760
  3  4   6    12    240
     3   4    6     28
         3    4     6
              3     4
                    3

a(n) is the first position of n in A325272.
Cf. A000142, A022559, A181819, A181821, A323023, A325238, A325273, A325274, A325275, A325276, A325277.
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).

fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
dat=Table[fdadj[n!],{n,1000}];
Table[Position[dat,k][[1,1]],{k,3,Max@@dat}]

A307734 Smallest k such that the adjusted frequency depth of k! is n, and 0 if there is no such k.

Original entry on oeis.org

1, 2, 0, 3, 4, 5, 7, 26, 65, 942, 24147

Offset: 0

Gus Wiseman, Apr 25 2019

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.

Conjecture: this sequence has infinitely many nonzero terms.

			Column n is the sequence of images under A181819 starting with a(n)!:
  -  2  -  6  24  120  5040  403291461126605635584000000
           4  10  20   84    11264760
           3  4   6    12    240
              3   4    6     28
                  3    4     6
                       3     4
                             3

Essentially the same as A325410.
a(n) is zero or the first position of n in A325272.
Cf. A000142, A323023, A325238, A325273, A325274, A325275, A325276, A325277, A325416.
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).

A324206 Numbers with exactly six distinct exponents in their prime factorization, or six distinct parts in their prime signature.

Original entry on oeis.org

5244319080000, 6197831640000, 6857955720000, 7342046712000, 7664774040000, 7866478620000, 8241072840000, 8676964296000, 8740531800000, 9278410680000, 9296747460000, 9578467080000, 9601138008000, 10286933580000, 10329719400000, 10488638160000, 10598658840000, 10705345560000

Offset: 1

David A. Corneth, Feb 17 2019

			6197831640000 = 2^6 * 3^5 * 5^4 * 7^3 * 11 * 13^2 is in the sequence as there are 6 distinct exponents; 1 through 6.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 6
```

A324207 Numbers with exactly seven distinct exponents in their prime factorization, or seven distinct parts in their prime signature.

Original entry on oeis.org

2677277333530800000, 2992251137475600000, 3164055030536400000, 3501054974617200000, 3536296798834800000, 3622198745365200000, 3748188266943120000, 4015916000296200000, 4189151592465840000, 4207150095548400000, 4280780335431600000, 4373290124002800000, 4429677042750960000

Offset: 1

David A. Corneth, Feb 17 2019

			2677277333530800000 = 2^7 * 3^6 * 5^5 * 7^4 * 11^3 * 13^2 * 17 is in the sequence. There are exactly 7 distinct exponents; 1 through 7 in it.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 7
```

A325265 Numbers with sum of omega-sequence > 4.

Original entry on oeis.org

6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

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 omega-sequences begins:
   6: 2 2 1       46: 2 2 1         80: 5 2 2 1       112: 5 2 2 1
  10: 2 2 1       48: 5 2 2 1       81: 4 1           114: 3 3 1
  12: 3 2 2 1     50: 3 2 2 1       82: 2 2 1         115: 2 2 1
  14: 2 2 1       51: 2 2 1         84: 4 3 2 2 1     116: 3 2 2 1
  15: 2 2 1       52: 3 2 2 1       85: 2 2 1         117: 3 2 2 1
  16: 4 1         54: 4 2 2 1       86: 2 2 1         118: 2 2 1
  18: 3 2 2 1     55: 2 2 1         87: 2 2 1         119: 2 2 1
  20: 3 2 2 1     56: 4 2 2 1       88: 4 2 2 1       120: 5 3 2 2 1
  21: 2 2 1       57: 2 2 1         90: 4 3 2 2 1     122: 2 2 1
  22: 2 2 1       58: 2 2 1         91: 2 2 1         123: 2 2 1
  24: 4 2 2 1     60: 4 3 2 2 1     92: 3 2 2 1       124: 3 2 2 1
  26: 2 2 1       62: 2 2 1         93: 2 2 1         126: 4 3 2 2 1
  28: 3 2 2 1     63: 3 2 2 1       94: 2 2 1         128: 7 1
  30: 3 3 1       64: 6 1           95: 2 2 1         129: 2 2 1
  32: 5 1         65: 2 2 1         96: 6 2 2 1       130: 3 3 1
  33: 2 2 1       66: 3 3 1         98: 3 2 2 1       132: 4 3 2 2 1
  34: 2 2 1       68: 3 2 2 1       99: 3 2 2 1       133: 2 2 1
  35: 2 2 1       69: 2 2 1        100: 4 2 1         134: 2 2 1
  36: 4 2 1       70: 3 3 1        102: 3 3 1         135: 4 2 2 1
  38: 2 2 1       72: 5 2 2 1      104: 4 2 2 1       136: 4 2 2 1
  39: 2 2 1       74: 2 2 1        105: 3 3 1         138: 3 3 1
  40: 4 2 2 1     75: 3 2 2 1      106: 2 2 1         140: 4 3 2 2 1
  42: 3 3 1       76: 3 2 2 1      108: 5 2 2 1       141: 2 2 1
  44: 3 2 2 1     77: 2 2 1        110: 3 3 1         142: 2 2 1
  45: 3 2 2 1     78: 3 3 1        111: 2 2 1         143: 2 2 1

Positions of terms > 4 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, 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[#]]>4&]

cp's OEIS Frontend

A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

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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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A325333 Number of integer partitions of n whose multiplicities all appear the same number of times.

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A325410 Smallest k such that the adjusted frequency depth of k! is n > 2.

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A307734 Smallest k such that the adjusted frequency depth of k! is n, and 0 if there is no such k.

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A324206 Numbers with exactly six distinct exponents in their prime factorization, or six distinct parts in their prime signature.

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A324207 Numbers with exactly seven distinct exponents in their prime factorization, or seven distinct parts in their prime signature.

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