A325238 -id:A325238 - cp's OEIS frontend

A325275 Heinz number of the omega-sequence of n!.

1, 1, 2, 18, 126, 990, 850, 11970, 19530, 25830, 4606, 73458, 92862, 116298, 43134, 229086, 275418, 366894, 440946, 515394, 568062, 613206, 769158, 963378, 1060254, 1135602, 6108570, 6431490, 6915870, 8923590, 9398610, 10191870, 11352510, 3139866, 16458210

Offset: 0

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

A001222(a(n)) = A325272.
A055396(a(n)/2) = A325273.
A056239(a(n)) = A325274.
Row n of A325276 is row a(n) of A112798.
Cf. A000142, A006939, A056239, A112798, 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&]];
Table[Times@@Prime/@omseq[n!],{n,30}]

A331200 Least number with each record number of factorizations into distinct factors > 1.

Original entry on oeis.org

1, 6, 12, 24, 48, 60, 96, 120, 180, 240, 360, 480, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 8640, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 90720, 100800, 110880, 120960, 151200, 181440, 221760

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A330997 in lacking 64.

			Strict factorizations of the initial terms:
  ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)
      (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)
             (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)
                    (2*12)   (3*16)   (4*15)    (4*24)    (4*30)
                    (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)
                             (2*3*8)  (6*10)    (8*12)    (6*20)
                             (2*4*6)  (2*5*6)   (2*6*8)   (8*15)
                                      (3*4*5)   (3*4*8)   (10*12)
                                      (2*3*10)  (2*3*16)  (3*5*8)
                                                (2*4*12)  (4*5*6)
                                                          (2*3*20)
                                                          (2*4*15)
                                                          (2*5*12)
                                                          (2*6*10)
                                                          (3*4*10)
                                                          (2*3*4*5)

Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

A subset of A330997.
All terms belong to A025487.
This is the strict version of highly factorable numbers A033833.
The corresponding records are A331232(n) = A045778(a(n)).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n)
Cf. A045783, A325238, A330972, A330973, A331023/A331024, A331201.

nn=1000;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]],{n,nn}];
Table[Position[qv,i][[1,1]],{i,Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]}]

a(37) and beyond from Giovanni Resta, Jan 17 2020

A331232 Record numbers of factorizations into distinct factors > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 16, 18, 25, 34, 38, 57, 59, 67, 70, 91, 100, 117, 141, 161, 193, 253, 296, 306, 426, 552, 685, 692, 960, 1060, 1067, 1216, 1220, 1589, 1591, 1912, 2029, 2157, 2524, 2886, 3249, 3616, 3875, 4953, 5147, 5285, 5810, 6023, 6112, 6623, 8129

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

			Representatives for the initial records and their strict factorizations:
  ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)
      (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)
             (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)
                    (2*12)   (3*16)   (4*15)    (4*24)    (4*30)
                    (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)
                             (2*3*8)  (6*10)    (8*12)    (6*20)
                             (2*4*6)  (2*5*6)   (2*6*8)   (8*15)
                                      (3*4*5)   (3*4*8)   (10*12)
                                      (2*3*10)  (2*3*16)  (3*5*8)
                                                (2*4*12)  (4*5*6)
                                                          (2*3*20)
                                                          (2*4*15)
                                                          (2*5*12)
                                                          (2*6*10)
                                                          (3*4*10)
                                                          (2*3*4*5)

Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

The non-strict version is A272691.
The first appearance of a(n) in A045778 is at index A331200(n).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n).
Cf. A033833, A045783, A325238, A330972, A330973, A330997, A331023/A331024, A331201.

nn=1000;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]],{n,nn}];
Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]

def fact(num):
    ret = []
    temp = num
    div = 2
    while temp > 1:
        while temp % div == 0:
            ret.append(div)
            temp //= div
        div += 1
    return ret
def all_partitions(lst):
    if lst:
        x = lst[0]
        for partition in all_partitions(lst[1:]):
            yield [x] + partition
            for i, _ in enumerate(partition):
                partition[i] *= x
                yield partition
                partition[i] //= x
    else:
        yield []
best = 0
terms = [0]
q = 2
while len(terms) < 100:
    total_set = set()
    factors = fact(q)
    total_set = set(tuple(sorted(x)) for x in all_partitions(factors) if len(x) == len(set(x)))
    if len(total_set) > best:
        best = len(total_set)
        terms.append(best)
        print(q,best)
    q += 2#only check evens
print(terms)
#  David Consiglio, Jr., Jan 14 2020

a(n) = A045778(A331200(n)).

a(26)-a(37) from David Consiglio, Jr., Jan 14 2020

a(38) and beyond from Giovanni Resta, Jan 17 2020

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&]

A325413 Largest sum of the omega-sequence of an integer partition of n.

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

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.

Appears to contain all nonnegative integers except 2, 4, 6, 7, and 11.

			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)
The largest term in the first column is 14, so a(9) = 14.

Row lengths of A325414.
Cf. A181819, A225486, A323014, A323023, A325238, A325249, 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[Max[Total/@omseq/@IntegerPartitions[n]],{n,0,30}]

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}]

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}]

A331049 Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 4, 7, 5, 7, 9, 12, 7, 11, 11, 16, 11, 19, 16, 21, 15, 29, 11, 12, 26, 30, 15, 31, 38, 22, 21, 47, 26, 29, 52, 45, 36, 57, 26, 64, 19, 30, 52, 77, 52, 36, 57, 98, 21, 67, 38, 74, 97, 66, 105, 47, 42, 36, 109, 118, 98, 92, 109, 52, 171, 30

Offset: 1

Gus Wiseman, Jan 10 2020

nonn

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. Factorizations are counted by A001055.

The unsorted prime signature of A055932(n) is given by row n of A124829.

			The a(1) = 1 through a(11) = 7 factorizations:
  {}  2  4    6    8      12     16       18     24       30     32
         2*2  2*3  2*4    2*6    2*8      2*9    3*8      5*6    4*8
                   2*2*2  3*4    4*4      3*6    4*6      2*15   2*16
                          2*2*3  2*2*4    2*3*3  2*12     3*10   2*2*8
                                 2*2*2*2         2*2*6    2*3*5  2*4*4
                                                 2*3*4           2*2*2*4
                                                 2*2*2*3         2*2*2*2*2

The sorted-signature version is A050322.
This sequence has range A045782.
Factorizations are A001055.
Cf. A025487, A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length@*facs/@First/@GatherBy[Range[1500],If[#==1,{},Last/@FactorInteger[#]]&]

a(n) = A001055(A055932(n)).

cp's OEIS Frontend

A325275 Heinz number of the omega-sequence of n!.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A331200 Least number with each record number of factorizations into distinct factors > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A331232 Record numbers of factorizations into distinct factors > 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A325264 Numbers whose omega-sequence sums to 7.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325413 Largest sum of the omega-sequence of an integer partition of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author