A370820 -id:A370820 - cp's OEIS frontend

A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.

2, 6, 10, 22, 34, 42, 62, 82, 118, 134, 166, 218, 230, 254, 314, 358, 382, 390, 422, 482, 554, 566, 662, 706, 734, 798, 802, 862, 922, 1018, 1094, 1126, 1174, 1198, 1234, 1418, 1478, 1546, 1594, 1718, 1754, 1838, 1914, 1934, 1982, 2062, 2126, 2134, 2174, 2306

Offset: 1

Gus Wiseman, Mar 21 2024

nonn

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

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:
     2: {1}
     6: {1,2}
    10: {1,3}
    22: {1,5}
    34: {1,7}
    42: {1,2,4}
    62: {1,11}
    82: {1,13}
   118: {1,17}
   134: {1,19}
   166: {1,23}
   218: {1,29}
   230: {1,3,9}
   254: {1,31}
   314: {1,37}
   358: {1,41}
   382: {1,43}
   390: {1,2,3,6}

Joseph Likar, Table of n, a(n) for n = 1..10000

Partitions of this type are counted by A054973.
The unsorted version is A275700.
These numbers have products A371286, unsorted version A371285.
Squarefree case of A371288, counted by A371284.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000720, A005179, A007416, A034729, A370820, A371131, A371177.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],SameQ[prix[#],Divisors[Last[prix[#]]]]&]

A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

Original entry on oeis.org

1, 2, 6, 4, 10, 12, 42, 8, 36, 20, 22, 24, 390, 84, 60, 16, 34, 72, 798, 40, 252, 44, 230, 48, 100, 780, 216, 168, 1914, 120, 62, 32, 132, 68, 420, 144, 101010, 1596, 2340, 80, 82, 504, 4386, 88, 360, 460, 5170, 96, 1764, 200, 204, 1560, 42294, 432, 220, 336

Offset: 1

Gus Wiseman, Mar 21 2024

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

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}, with divisor sets {{1,2},{1,3},{1,2,4}}, with multiset union {1,1,1,2,2,3,4}, with Heinz number 2520, so a(105) = 2520.
The terms together with their prime indices begin:
          1: {}
          2: {1}
          6: {1,2}
          4: {1,1}
         10: {1,3}
         12: {1,1,2}
         42: {1,2,4}
          8: {1,1,1}
         36: {1,1,2,2}
         20: {1,1,3}
         22: {1,5}
         24: {1,1,1,2}
        390: {1,2,3,6}
         84: {1,1,2,4}
         60: {1,1,2,3}
         16: {1,1,1,1}
         34: {1,7}
         72: {1,1,1,2,2}

Product of A275700 applied to each prime index.
The squarefree case is also A275700.
The sorted version is A371286.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000720, A003963, A005179, A007416, A034729, A054973, A056239, A321898, A370820, A371165, A371181, A371284, A371288.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Join@@Divisors/@prix[n],{n,100}]

If n = prime(x_1)*...*prime(x_k) then a(n) = A275700(x_1)*...*A275700(x_k).

A371286 Products of elements of A275700 (Heinz numbers of divisor sets). Numbers with a (necessarily unique) factorization into elements of A275700.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 60, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 118, 120, 124, 128, 132, 134, 136, 144, 160, 164, 166, 168, 176, 192, 200, 204, 216, 218, 220, 230, 236, 240, 248, 252, 254, 256, 264, 268, 272, 288

Offset: 1

Gus Wiseman, Mar 22 2024

nonn

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

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 factorizations and unique factorizations into terms of A275700 begin:
   1 =             = ()
   2 = 2           = (2)
   4 = 2*2         = (2*2)
   6 = 2*3         = (6)
   8 = 2*2*2       = (2*2*2)
  10 = 2*5         = (10)
  12 = 2*2*3       = (2*6)
  16 = 2*2*2*2     = (2*2*2*2)
  20 = 2*2*5       = (2*10)
  22 = 2*11        = (22)
  24 = 2*2*2*3     = (2*2*6)
  32 = 2*2*2*2*2   = (2*2*2*2*2)
  34 = 2*17        = (34)
  36 = 2*2*3*3     = (6*6)
  40 = 2*2*2*5     = (2*2*10)
  42 = 2*3*7       = (42)
  44 = 2*2*11      = (2*22)
  48 = 2*2*2*2*3   = (2*2*2*6)
  60 = 2*2*3*5     = (6*10)
  62 = 2*31        = (62)
  64 = 2*2*2*2*2*2 = (2*2*2*2*2*2)
  68 = 2*2*17      = (2*34)
  72 = 2*2*2*3*3   = (2*6*6)
  80 = 2*2*2*2*5   = (2*2*2*10)
  82 = 2*41        = (82)
  84 = 2*2*3*7     = (2*42)
  88 = 2*2*2*11    = (2*2*22)
  96 = 2*2*2*2*2*3 = (2*2*2*2*6)

Products of elements of A275700.
The squarefree case is A371283.
The unsorted version is A371285.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000720, A003963, A005179, A007416, A034729, A054973, A056239, A370820, A371284, A371288.

nn=100;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1, {{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
s=Table[Times@@Prime/@Divisors[n],{n,nn}];
m=Max@@Table[Select[Range[2,k],prix[#] == Divisors[Last[prix[#]]]&],{k,nn}];
Join@@Position[Table[Length[Select[facs[n], SubsetQ[s,Union[#]]&]],{n,m}],1]

A371132 Number of integer partitions of n with fewer distinct parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 6, 10, 14, 21, 28, 40, 53, 73, 96, 130, 170, 223, 288, 375, 480, 616, 780, 990, 1245, 1567, 1954, 2440, 3024, 3745, 4610, 5674, 6947, 8499, 10349, 12591, 15258, 18468, 22277, 26841, 32238, 38673, 46262, 55278, 65881, 78423, 93136, 110477

Offset: 0

Gus Wiseman, Mar 17 2024

nonn

The Heinz numbers of these partitions are given by A371179.

			The partition (4,3,1,1) has 3 distinct parts {1,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(0) = 0 through a(9) = 14 partitions:
  .  .  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
                  (22)  (32)  (33)   (43)    (44)     (54)
                        (41)  (42)   (52)    (53)     (63)
                              (222)  (61)    (62)     (72)
                              (411)  (322)   (332)    (81)
                                     (4111)  (422)    (333)
                                             (431)    (432)
                                             (611)    (441)
                                             (2222)   (522)
                                             (41111)  (621)
                                                      (3222)
                                                      (4311)
                                                      (6111)
                                                      (411111)

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
The complement counting all parts on the LHS is A371172, ranks A371165.
Counting all parts on the LHS gives A371173, ranks A371168.
The complement is counted by A371178, ranks A371177.
These partitions are ranked by A371179.
The strict case is A371180, complement A371128.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A003963, A239312, A319055, A355740, A370802, A370803, A370808, A370813, A371130, A371171.

Table[Length[Select[IntegerPartitions[n],Length[Union[#]] < Length[Union@@Divisors/@#]&]],{n,0,30}]

A371180 Number of strict integer partitions of n with fewer parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 2, 4, 4, 7, 8, 10, 12, 15, 19, 22, 29, 33, 40, 47, 57, 68, 81, 95, 110, 129, 152, 178, 207, 240, 277, 317, 365, 422, 486, 558, 632, 723, 824, 940, 1067, 1210, 1371, 1544, 1751, 1977, 2233, 2508, 2820, 3162, 3555, 3983, 4465, 4990, 5571, 6224

Offset: 0

Gus Wiseman, Mar 18 2024

nonn

			The strict partition (6,4,2,1) has 4 parts and 5 distinct divisors of parts {1,2,3,4,5}, so is counted under a(13).
The a(2) = 1 through a(11) = 10 partitions:
  (2)  (3)  (4)  (5)    (6)    (7)    (8)      (9)      (10)     (11)
                 (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)
                 (4,1)         (5,2)  (6,2)    (6,3)    (7,3)    (7,4)
                               (6,1)  (4,3,1)  (7,2)    (8,2)    (8,3)
                                               (8,1)    (9,1)    (9,2)
                                               (4,3,2)  (5,3,2)  (10,1)
                                               (6,2,1)  (5,4,1)  (5,4,2)
                                                        (6,3,1)  (6,3,2)
                                                                 (6,4,1)
                                                                 (8,2,1)

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
The version for equality is A371128.
The non-strict version is A371132, ranks A371179.
The non-strict complement is A371178, ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A003963, A239312, A319055, A355529, A370803, A370808, A370813, A371130 (A370802), A371171, A371172 (A371165), A371173 (A371168).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Union[#]] < Length[Union@@Divisors/@#]&]],{n,0,30}]

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.

A371287 Numbers whose product of prime indices has exactly two distinct prime factors.

Original entry on oeis.org

13, 15, 26, 29, 30, 33, 35, 37, 39, 43, 45, 47, 51, 52, 55, 58, 60, 61, 65, 66, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 86, 87, 89, 90, 91, 93, 94, 95, 99, 101, 102, 104, 105, 107, 110, 111, 116, 117, 119, 120, 122, 123, 129, 130, 132, 135, 137, 138, 139, 140

Offset: 1

Gus Wiseman, Mar 21 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:
  13: {6}
  15: {2,3}
  26: {1,6}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  43: {14}
  45: {2,2,3}
  47: {15}
  51: {2,7}
  52: {1,1,6}
  55: {3,5}
  58: {1,10}
  60: {1,1,2,3}

Positions of 2's in A303975 (positions of 1's are A320698).
Counting divisors (not factors) gives A371127, positions of 2's in A370820.
A000005 counts divisors.
A000961 lists powers of primes, of prime index A302596.
A001221 counts distinct prime factors.
A001358 lists semiprimes, squarefree A006881.
A003963 gives product of prime indices.
A027746 lists prime factors, indices A112798, length A001222.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000079, A007416, A056239, A302540, A319899, A336101, A355739, A370802.

Select[Range[100],2==PrimeNu[Times @@ PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A001221(A003963(a(n))) = A303975(a(n)) = 2.

A387119 Numbers whose prime indices all have exactly 2 divisors in common.

Original entry on oeis.org

3, 5, 9, 11, 17, 21, 25, 27, 31, 39, 41, 57, 59, 63, 65, 67, 81, 83, 87, 91, 109, 111, 115, 117, 121, 125, 127, 129, 147, 157, 159, 171, 179, 183, 185, 189, 191, 203, 211, 213, 235, 237, 241, 243, 247, 261, 267, 273, 277, 283, 289, 299, 301, 303, 305, 319, 321

Offset: 1

Gus Wiseman, Aug 21 2025

All terms are odd.

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 87 are {2,10}, with divisors {{1,2},{1,2,5,10}}, with intersection {1,2}, so 87 is in the sequence.
The prime indices of 91 are {4,6}, with divisors {{1,2,4},{1,2,3,6}}, with intersection {1,2}, so 91 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    5: {3}
    9: {2,2}
   11: {5}
   17: {7}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   31: {11}
   39: {2,6}
   41: {13}
   57: {2,8}
   59: {17}
   63: {2,2,4}
   65: {3,6}
   67: {19}
   81: {2,2,2,2}

For initial intervals instead of divisors we have A016945.
Positions of 1 are A289509, complement A318978.
Positions of 2 in A387114, for prime factors or indices A387135.
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, A055396, A061395, A335433, A355731, A355733, A355739, A355740, A370820.

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

cp's OEIS Frontend

A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.

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A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

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A371286 Products of elements of A275700 (Heinz numbers of divisor sets). Numbers with a (necessarily unique) factorization into elements of A275700.

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A371132 Number of integer partitions of n with fewer distinct parts than distinct divisors of parts.

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A371180 Number of strict integer partitions of n with fewer parts than distinct divisors of parts.

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

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A371287 Numbers whose product of prime indices has exactly two distinct prime factors.

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A387119 Numbers whose prime indices all have exactly 2 divisors in common.

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