A120383 -id:A120383 - cp's OEIS frontend

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.

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]}]

A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Aug 24 2025

The initial interval of a nonnegative integer x is the set {1,...,x}.
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.
We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1,2,3},{1},{1,3},{2}) is not.
This sequence lists all numbers k such that if the prime indices of k are (x1,x2,...,xz), then the sequence of sets (initial intervals) ({1,...,x1},{1,...,x2},...,{1,...,xz}) is not choosable.

			The prime indices of 18 are {1,2,2}, with initial intervals ({1},{1,2},{1,2}), which have choices (1,1,1), (1,1,2), (1,2,1), (1,2,2), and since none of these are strict, 18 is in the sequence.
The prime indices of 85 are {3,7}, with initial intervals {{1,2,3},{1,2,3,4,5,6,7}}, which are choosable, so 85 is in not the sequence.
The prime indices of 90 are {1,2,2,3}, with initial intervals {{1},{1,2},{1,2},{1,2,3}}, which are not choosable, so 90 is in the sequence.
The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

For partitions instead of initial intervals we have A276079, complement A276078.
For prime factors instead of initial intervals we have A355529, complement A368100.
For divisors instead of initial intervals we have A355740, complement A368110.
These are the positions of 0 in A387111, complement A387134.
The complement is A387112.
Partitions of this type are counted by A387118, complement A238873.
For strict partitions instead of initial intervals we have A387137, complement A387176.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
A370585 counts maximal subsets with choosable prime factors.
Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A355747, A370592, A383706, A387110.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Range/@prix[#]],UnsameQ@@#&]=={}&]

A387136 Number of ways to choose a sequence of distinct prime factors, one of each prime index of 2n - 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 30 2025

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 4537 are {6,70}, with choices (2,5), (2,7), (3,2), (3,5), (3,7). Since 4537 = 2 * 2269 - 1, we have a(2269) = 5.

Here we use the version with alternating zeros (put n instead of 2n - 1 in the name).
Twice partitions of this type are counted by A296122.
Positions of zero are A355529, complement A368100.
For divisors instead of prime factors we have A355739.
Allowing repeated choices gives A355741.
For partitions instead of prime factors we have A387110.
For initial intervals instead of prime factors we have A387111.
For strict partitions instead of prime factors we have A387115, disjoint case A383706.
For constant partitions instead of prime factors we have A387120.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A261049, A276078, A276079, A299200, A335433, A335448, A355731, A355745, A367771, A387133, A387135, A387327.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@prix[2n-1]],UnsameQ@@#&]],{n,100}]

A324932 Numerator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 13, 27, 7, 29, 5, 31, 32, 33, 34, 35, 9, 37, 19, 13, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 26, 53, 27, 11, 14, 57, 29, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 23

Offset: 1

Gus Wiseman, Mar 21 2019

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 sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

The denominators are A324933.

Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}]//Numerator

A325031 Numbers divisible by all prime indices of their prime indices.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 16, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 33, 36, 38, 40, 42, 46, 48, 49, 50, 52, 53, 54, 56, 57, 60, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 84, 87, 90, 92, 96, 98, 99, 100, 104, 106, 108, 112, 114, 120, 122, 126, 128

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 55 are {3,5} with prime indices {{2},{3}}. Since 55 is not divisible by 2 or 3, it does not belong to the sequence.

			The sequence of multisets of multisets whose MM-numbers (see A302242) belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}
  33: {{1},{3}}
  36: {{},{},{1},{1}}

Cf. A000720, A003963, A112798, A120383, A302242, A320325.

Cf. A323440, A324846, A324847, A324849, A324850, A324856, A324926, A325032.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@Table[Divisible[#,i],{i,Union@@primeMS/@primeMS[#]}]&]

A330952 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by all parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 14, 20, 25, 32, 42, 54, 69, 87, 109, 137, 172, 215, 269, 331, 409, 499, 612, 751, 917, 1111, 1344, 1626, 1963, 2359, 2834, 3396, 4065, 4849, 5779, 6865, 8146, 9658, 11424, 13483, 15898, 18710, 21999, 25823, 30272, 35417, 41397

Offset: 0

Gus Wiseman, Jan 15 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(9) = 11 partitions:
  1  11  21   211   221    321     2221     3221      621
         111  1111  2111   411     3211     4211      3321
                    11111  2211    4111     22211     22221
                           21111   22111    32111     32211
                           111111  211111   41111     42111
                                   1111111  221111    222111
                                            2111111   321111
                                            11111111  411111
                                                      2211111
                                                      21111111
                                                      111111111

The Heinz numbers of these partitions are given by A120383.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by their product are A324925.
Partitions whose Heinz number is divisible by their sum are A330950.
Cf. A056239, A112798, A324756, A326149, A326155, A330953, A330954, A331383.

Table[Length[Select[IntegerPartitions[n],And@@Table[Divisible[Times@@Prime/@#,i],{i,#}]&]],{n,0,30}]

A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 3, 0, 4, 3, 4, 0, 8, 0, 10, 0, 11, 12, 19, 0, 0, 22, 0, 0, 46, 23, 56, 0, 64, 66, 86, 0, 125, 104, 135, 0, 196, 111, 230, 0, 0, 274, 353, 0, 0, 0, 563, 0, 687, 0, 974, 0, 1039, 1052, 1290, 0, 1473, 1511, 0, 0, 2707, 1614, 2664, 0

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. The Heinz numbers of these partitions are squarefree numbers divisible by the sum of their prime indices.

			The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
  321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
       421        4321  542      643   6431  6432      764       865
                        5321     652   7421  9321      872       874
                                 6421        54321     971       982
                                                       7532      A81
                                                       7541      8542
                                                       7631      8632
                                                       74321     8641
                                                                 8731
                                                                 85321

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of zeros are 2 and A013929.
The non-strict version is A330950 (A324851) q.v.
A000009 counts strict partitions.
A003963 multiplies together prime indices.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A057568 counts partitions whose product is divisible by their sum (A326149).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A112798 lists the prime indices of each positive integer.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A324925 counts partitions whose Heinz number is divisible by their product.
A326842 counts partitions whose parts and length all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A330952 counts partitions whose Heinz number is divisible by all parts.
A340828 counts strict partitions with length divisible by maximum.
A340830 counts strict partitions with parts divisible by length.
Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@Prime/@#,n]&]],{n,30}]

A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, Aug 29 2025

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.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

			The prime indices of 50 are {1,3,3}, and {(1),(3),(2,1)} is a valid choice of distinct strict partitions, so 50 is in the sequence.

The version for all partitions appears to be A276078, counted by A052335.
The complement for all partitions appears to be A276079, counted by A387134.
The complement for divisors is A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The version for divisors is A368110, counted by A239312.
The version for initial intervals is A387112, counted by A238873, see A387111.
The complement for initial intervals is A387113, counted by A387118.
These are the positions of nonzero terms in A387115.
The complement is A387176.
Partitions of this type are counted by A387178, complement A387137.
The complement for constant partitions is A387180, counted by A387329, see A387120.
The version for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A120383, A270995, A299200, A335433, A335448, A357978, A357982, A383706, A387110.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Select[Range[100],Select[Tuples[strptns/@prix[#]],UnsameQ@@#&]!={}&]

A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204

Offset: 1

Gus Wiseman, Aug 30 2025

First differs from A276079 in having 125.
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 numbers n with at least one prime index k such that the multiplicity of prime(k) in the prime factorization of n exceeds the number of divisors of k.

			The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
  ((1),(1),(2),(3))
  ((1),(1),(2),(1,1,1))
  ((1),(1),(1,1),(3))
  ((1),(1),(1,1),(1,1,1))
Since none of these is strict, 60 is in the sequence.
The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
  ((1),(2),(1,1),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(3))
  ((1),(1,1),(2),(1,1,1))
So 90 is not in the sequence.

For prime factors instead of constant partitions we have A355529, counted by A370593.
For divisors instead of constant partitions we have A355740, counted by A370320.
The complement for prime factors is A368100, counted by A370592.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873.
For initial intervals instead of partitions we have A387113, counted by A387118.
These are the positions of zero in A387120.
For strict instead of constant partitions we have A387176, counted by A387137.
The complement for strict partitions is A387177, counted by A387178.
Twice-partitions of this type are counted by A387179, constant-block case of A296122.
The complement is A387181 (nonzeros of A387120), counted by A387330.
Partitions of this type are counted by A387329.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A355739, A367771, A387111, A387115.
Cf. A000005, A052335, A063834, A276079, A299200, A299201, A335433, A335448, A355731, A383706, A387110.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]

A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 32, 36, 48, 54, 56, 64, 72, 78, 84, 96, 108, 112, 128, 144, 152, 156, 162, 168, 192, 196, 216, 224, 234, 252, 256, 288, 304, 312, 324, 336, 384, 392, 432, 444, 448, 456, 468, 486, 504, 512, 576, 588, 608, 624, 648, 672, 702

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))

A subsequence of A120383.
Cf. A000081, A007097, A112798, A279861, A290689, A290760, A290822, A318186.
Cf. A324704, A324736, A324748, A324753, A324843, A324847, A324848, A324854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[Divisible[n,x],{x,primeMS[n]}],And@@qaQ/@primeMS[n]];
Select[Range[1000],qaQ]

cp's OEIS Frontend

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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A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.

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A387136 Number of ways to choose a sequence of distinct prime factors, one of each prime index of 2n - 1.

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A324932 Numerator in the division of n by the product of prime indices of n.

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A325031 Numbers divisible by all prime indices of their prime indices.

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A330952 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by all parts.

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A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

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A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

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A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

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