A355737 -id:A355737 - cp's OEIS frontend

A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92

Offset: 1

Gus Wiseman, Mar 16 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
     1: {}       22: {1,5}      42: {1,2,4}    63: {2,2,4}
     2: {1}      23: {9}        43: {14}       65: {3,6}
     3: {2}      25: {3,3}      45: {2,2,3}    66: {1,2,5}
     5: {3}      26: {1,6}      46: {1,9}      67: {19}
     6: {1,2}    28: {1,1,4}    47: {15}       69: {2,9}
     7: {4}      29: {10}       49: {4,4}      70: {1,3,4}
     9: {2,2}    30: {1,2,3}    51: {2,7}      71: {20}
    10: {1,3}    31: {11}       52: {1,1,6}    73: {21}
    11: {5}      33: {2,5}      53: {16}       74: {1,12}
    13: {6}      34: {1,7}      55: {3,5}      75: {2,3,3}
    14: {1,4}    35: {3,4}      57: {2,8}      76: {1,1,8}
    15: {2,3}    37: {12}       58: {1,10}     77: {4,5}
    17: {7}      38: {1,8}      59: {17}       78: {1,2,6}
    19: {8}      39: {2,6}      61: {18}       79: {22}
    21: {2,4}    41: {13}       62: {1,11}     82: {1,13}

The complement is A370348, counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The strict version is A371168 counted by A371173.
The opposite version is A371169.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A319899, A355737, A355739, A355741, A370808, A371127.

Select[Range[100],PrimeOmega[#]<=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A354911 Number of factorizations of n into relatively prime prime-powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 1, 1, 0, 5, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

A355736 Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.

Original entry on oeis.org

1, 3, 7, 13, 21, 37, 39, 89, 133, 117, 111, 273, 351, 259, 267, 333, 453, 793, 669, 623, 999, 777, 843, 1491, 1157, 1561, 2863, 1443, 1963, 2331, 1977, 1869, 2899, 2529, 3207, 4107, 3171, 5073, 4329, 3653, 4667, 3471, 7399, 4613, 7587, 5931, 7269, 5889, 7483

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355735.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
     3: {2}
     7: {4}
    13: {6}
    21: {2,4}
    37: {12}
    39: {2,6}
    89: {24}
   133: {4,8}
   117: {2,2,6}
   111: {2,12}
   273: {2,4,6}
   351: {2,2,2,6}
For example, the choices for a(12) = 273 are:
  {1,1,1}  {1,2,2}  {2,2,2}
  {1,1,2}  {1,2,3}  {2,2,3}
  {1,1,3}  {1,2,6}  {2,2,6}
  {1,1,6}  {1,4,6}  {2,4,6}

Allowing any choice of divisors gives A355732, firsts of A355731.
Choosing a multiset instead of sequence gives A355734, firsts of A355733.
Positions of first appearances in A355735.
The case of prime factors instead of divisors is counted by A355745.
The decreasing version is counted by A355749.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A355737, A355739, A355740, A355741, A355744.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],LessEqual@@#&]],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]

A355749 Number of ways to choose a weakly decreasing sequence of divisors, one of each prime index of n (with multiplicity, taken in weakly increasing order).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 3, 1, 3, 1, 3, 1, 4, 1, 4, 1, 2, 1, 2, 1, 3, 1, 6, 1, 3, 1, 2, 1, 4, 1, 3, 1, 4, 1, 6, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 6, 1, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 1, 2, 1, 3

Offset: 1

Gus Wiseman, Jul 18 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(2) = 1 through a(19) = 4 choices:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
     2      3      2       21      5       2      21        7       2
                   4       22              3                        4
                                           6                        8

Wikipedia, Cartesian product.

Allowing any choice of divisors gives A355731, firsts A355732.
Choosing a multiset instead of sequence gives A355733, firsts A355734.
The reverse version is A355735, firsts A355736, only primes A355745.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061395 selects the maximum prime index.
Cf. A000720, A076610, A120383, A316524, A324850, A355737, A355739, A355740, A355741, A355742, A355744.

Table[Length[Select[Tuples[Divisors/@primeMS[n]], GreaterEqual@@#&]],{n,100}]

A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 62, 63, 64, 66, 68, 72, 75, 80, 81, 82, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 118, 120, 121, 124, 125, 126, 128, 132, 134, 135

Offset: 1

Gus Wiseman, Mar 16 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    22: {1,5}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    32: {1,1,1,1,1}
    34: {1,7}
    36: {1,1,2,2}

The strict version is A370348 counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The complement is A371168, counted by A371173.
The opposite version is A371170.
The version for prime factors instead of divisors on the RHS is A319899.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A355737, A355739, A355741, A370808, A370813, A370814, A371127.

Select[Range[100],PrimeOmega[#]>=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A370816 Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 2, 4, 4, 7, 2, 7, 2, 7, 4, 4, 2, 11, 3, 4, 5, 7, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 11, 2, 8, 2, 7, 7, 4, 2, 17, 3, 7, 4, 7, 2, 11, 4, 11, 4, 4, 2, 15, 2, 4, 7, 14, 4, 8, 2, 7, 4, 8, 2, 20, 2, 4, 7, 7, 4, 8, 2, 17, 7, 4, 2

Offset: 1

Gus Wiseman, Mar 06 2024

nonn

			For the factorizations of 12 we have the following choices:
  (2*2*3): {{1,1,1},{1,1,2},{1,1,3},{1,2,2},{1,2,3},{2,2,3}}
    (2*6): {{1,1},{1,2},{1,3},{1,6},{2,2},{2,3},{2,6}}
    (3*4): {{1,1},{1,2},{1,3},{1,4},{2,3},{3,4}}
     (12): {{1},{2},{3},{4},{6},{12}}
So a(12) = 7.

Max Alekseyev, Table of n, a(n) for n = 1..10000

The version for partitions is A370808, for just prime factors A370809.
For just prime factors we have A370817.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A368413 counts non-choosable factorizations, complement A368414.
A370813 counts non-divisor-choosable factorizations, complement A370814.
Cf. A000792, A048249, A066739, A319055, A319057, A355737, A355739, A355740, A355741, A368110, A370803.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@facs[n]],{n,100}]

A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 74, 79, 89, 91, 95, 97, 101, 103, 106, 107, 111, 113, 122, 131, 137, 139, 141, 142, 143, 145, 149, 151, 159, 161, 163, 167, 169, 173, 178, 181, 183, 185, 193, 197, 199, 203, 209, 213, 214, 215, 219, 221, 223, 226

Offset: 1

Gus Wiseman, Mar 14 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     7: {4}       101: {26}      163: {38}      223: {48}
    13: {6}       103: {27}      167: {39}      226: {1,30}
    19: {8}       106: {1,16}    169: {6,6}     227: {49}
    23: {9}       107: {28}      173: {40}      229: {50}
    29: {10}      111: {2,12}    178: {1,24}    233: {51}
    37: {12}      113: {30}      181: {42}      239: {52}
    43: {14}      122: {1,18}    183: {2,18}    247: {6,8}
    47: {15}      131: {32}      185: {3,12}    251: {54}
    53: {16}      137: {33}      193: {44}      257: {55}
    61: {18}      139: {34}      197: {45}      259: {4,12}
    71: {20}      141: {2,15}    199: {46}      262: {1,32}
    73: {21}      142: {1,20}    203: {4,10}    263: {56}
    74: {1,12}    143: {5,6}     209: {5,8}     265: {3,16}
    79: {22}      145: {3,10}    213: {2,20}    267: {2,24}
    89: {24}      149: {35}      214: {1,28}    269: {57}
    91: {4,6}     151: {36}      215: {3,14}    271: {58}
    95: {3,8}     159: {2,16}    219: {2,21}    281: {60}
    97: {25}      161: {4,9}     221: {6,7}     293: {62}

The RHS is A370820, for prime factors instead of divisors A303975.
For (equal to) instead of (less than) we have A371165, counted by A371172.
For (greater than) instead of (less than) we have A371167.
For prime factors on the LHS we get A371168, counted by A371173.
Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
Other inequalities: A370348 (A371171), A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355737, A355739, A355741, A368100, A370808, A371127.

Select[Range[100],Length[Divisors[#]] < Length[Union@@Divisors/@PrimePi/@First/@FactorInteger[#]]&]

A000005(a(n)) < A370820(a(n)).

A355748 Number of ways to choose a sequence of divisors, one of each part of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 2, 2, 2, 1, 2, 3, 4, 2, 4, 4, 4, 2, 3, 2, 4, 2, 2, 2, 2, 1, 4, 2, 6, 3, 4, 4, 4, 2, 6, 4, 8, 4, 4, 4, 4, 2, 2, 3, 4, 2, 4, 4, 4, 2, 3, 2, 4, 2, 2, 2, 2, 1, 2, 4, 4, 2, 6, 6, 6, 3, 6, 4, 8, 4, 4, 4, 4, 2, 4, 6, 8, 4, 8, 8, 8

Offset: 0

Gus Wiseman, Jul 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition number 152 in standard order is (3,1,4), and the a(152) = 6 choices are: (1,1,1), (1,1,2), (1,1,4), (3,1,1), (3,1,2), (3,1,4).

Positions of 1's are A000079 (after the first).
The anti-run case is A354578, zeros A354904, firsts A354905.
An unordered version (using prime indices) is A355731:
- firsts A355732,
- resorted A355733,
- weakly increasing A355735,
- relatively prime A355737,
- strict A355739.
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A005811 counts runs in binary expansion.
A029837 adds up standard compositions, lengths A000120.
A066099 lists the compositions in standard order.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
Cf. A124767, A175413, A238279, A274174, A326841, A333381, A333755, A353847, A353849, A355747.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Times@@Length/@Divisors/@stc[n],{n,0,100}]

A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

1, 2, 6, 9, 15, 49, 35, 27, 45, 98, 63, 105, 171, 117, 81, 135, 245, 343, 273, 549, 189, 1083, 315, 5618, 741, 686, 507, 513, 351, 243, 405, 7467, 6419, 5575, 735, 6859, 1813, 3231, 1183, 1197, 3537, 819, 1647, 567, 945, 2197, 8397, 3211, 1715, 3249, 3367

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355737.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     6: {1,2}
     9: {2,2}
    15: {2,3}
    49: {4,4}
    35: {3,4}
    27: {2,2,2}
    45: {2,2,3}
    98: {1,4,4}
    63: {2,2,4}
   105: {2,3,4}
   171: {2,2,8}
   117: {2,2,6}
    81: {2,2,2,2}
   135: {2,2,2,3}
For example, the choices for a(12) = 105 are:
  (1,1,1)  (1,3,2)  (2,1,4)
  (1,1,2)  (1,3,4)  (2,3,1)
  (1,1,4)  (2,1,1)  (2,3,2)
  (1,3,1)  (2,1,2)  (2,3,4)

Wikipedia, Coprime integers.

Not requiring coprimality gives A355732, firsts of A355731.
Positions of first appearances in A355737.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A007359, A051424, A076610, A302696, A302698, A355733, A355735, A355739, A355741, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}];
Table[Position[az+1,k][[1,1]],{k,mnrm[az+1]}]

A387114 Number of divisors in common to all prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 2, 1, 4, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 4, 1, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Aug 19 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also the number of divisors of the greatest common divisor of the prime indices of n.

			The prime indices of 703 are {8,12}, with divisors {{1,2,4,8},{1,2,3,4,6,12}}, with {1,2,4} in common, so a(703) = 3.

For initial interval instead of divisors we have A055396.
Positions of 1 are A289509, complement A318978.
Positions of 2 are A387119.
For prime factors or indices instead of divisors we have A387135, see A010055 or A069513.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives greatest common divisor of prime indices.
Cf. A000720, A061395, A326841, A335433, A335448, A355731, A355733, A355734, A355737, A355739, A355740, A370820.

Table[If[n==1,0,Length[Divisors[GCD@@PrimePi/@First/@FactorInteger[n]]]],{n,100}]

a(1) = 0; a(n) = A000005(A289508(n)) for n > 1.

cp's OEIS Frontend

A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A354911 Number of factorizations of n into relatively prime prime-powers.

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A355736 Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.

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A355749 Number of ways to choose a weakly decreasing sequence of divisors, one of each prime index of n (with multiplicity, taken in weakly increasing order).

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A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A370816 Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.

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A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

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A355748 Number of ways to choose a sequence of divisors, one of each part of the n-th composition in standard order.

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A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

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