A054973 -id:A054973 - cp's OEIS frontend

A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 11, 9, 13, 8, 0, 0, 10, 17, 0, 19, 0, 0, 0, 14, 15, 23, 0, 0, 0, 12, 0, 29, 16, 25, 21, 31, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 26, 41, 0, 43, 0, 0, 0, 33, 35, 47, 0, 0, 0, 0, 0, 34, 53, 0, 28, 39, 49, 0, 0, 24, 38, 59, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 46, 51, 55, 71, 0, 73

Offset: 1

Omar E. Pol, Mar 12 2018

Essentially the same as the triangle described in the example section of A085790, but with 0's added in empty rows.

Are the records the same as A008578?

			First 24 rows of triangle T(n,k):
-----------------------
. n / k:  1   2   3 ...
-----------------------
| 1|      1;
| 2|      0;
| 3|      2;
| 4|      3;
| 5|      0;
| 6|      5;
| 7|      4;
| 8|      7;
| 9|      0;
|10|      0;
|11|      0;
|12|      6, 11;
|13|      9;
|14|     13;
|15|      8;
|16|      0;
|17|      0;
|18|     10, 17;
|19|      0;
|20|     19;
|21|      0;
|22|      0;
|23|      0;
|24|     14, 15, 23;
...
For n = 23 there are no positive integers whose sum of divisors is 23, so T(23, 1) = 0, which is the only element in the 23rd row of the triangle.
For n = 24 there are three positive integers whose sum of divisors is 24; they are 14, 15 and 23, since sigma(14) = 1 + 2 + 7 + 14 = 24, sigma(15) = 1 + 3 + 5 + 15 = 24 and sigma(23) = 1 + 23 = 24, so the 24th row of the triangle is [14, 15, 23].

Row sums give A258913.
Column 1 gives A051444.
Right border gives A057637.
Positive terms give A085790.
Row n has A054973(n) positive integers.
Positive terms in the first column give A002192.
Indices of the rows that contain a zero give A007369.
Indices of the rows that contain positive terms give A002191.
Cf. A000203, A007368, A007609, A008578.

With[{nn = 74}, ReplacePart[ConstantArray[{0}, nn], PositionIndex@ Array[DivisorSigma[1, #] &, nn]]] // Flatten (* Michael De Vlieger, Mar 16 2018 *)

sigma(T(n,k)) = n, if T(n,k) >= 1.

A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 50, 54, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 108, 118, 124, 126, 128, 134, 136, 144, 160, 162, 164, 166, 168, 176, 192, 200, 216, 218, 230, 236, 242, 248, 250, 252, 254, 256, 268, 272, 288

Offset: 1

Gus Wiseman, Mar 22 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 694782 are {1,2,2,5,5,5,10} with distinct elements {1,2,5,10}, which form the set of divisors of 10, so 694782 is in the sequence.
The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   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}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

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

The squarefree case is A371283, unsorted version A275700.
Partitions of this type are counted by A371284, strict A054973.
Products of squarefree terms are A371286, unsorted version A371285.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A003963, A005179, A007416, A034729, A370820, A371131, A371177.

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

A055486 Number of solutions to sigma(x) = n!.

Original entry on oeis.org

1, 0, 1, 3, 4, 15, 33, 111, 382, 1195, 3366, 14077, 53265, 229603, 910254, 4524029, 18879944, 91336498, 561832582, 2801857644, 14652294729, 78894985156, 408373652461, 2378940665083, 11939275822636, 71931330299023, 392274481206066, 2626331088771946

Offset: 1

Labos Elemer, Jun 28 2000

nonn

			For n = 9, solutions to sigma(x) = n! = 362880 form a set {97440, ..., 361657} of size 382, so a(9) = 382.

R. K. Guy (1981): Unsolved Problems In Number Theory, B39.

Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000142, A000203, A054973, A014197, A055488, A055489, A055506.

with(numtheory): for f from 1 to 9 do fac := f!: k := 0:for n from 1 to fac do if sigma(n)=fac then k := k+1: fi: od: print( k); od:

a(n) = A054973(n!) = Cardinality[{x; A000203(x) = A000142(n)}].

More terms from Jud McCranie, Oct 09 2000
a(13)-a(14) from Donovan Johnson, Nov 22 2008
a(15) from Ray Chandler, Jan 13 2009
a(16)-a(28) from Max Alekseyev, Jan 23 2012

A231366 Number of numbers whose sum of non-divisors (A024816) is equal to n.

Original entry on oeis.org

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

Offset: 0

Jaroslav Krizek, Nov 09 2013

nonn

a(n) = frequency of values n in A024816(m), where A024816(m) = sum of non-divisors of m = antisigma(m).
From Charles R Greathouse IV, Nov 11 2013: (Start)
So far all n such that a(n) > 1 correspond to members of A067816:
a(0) = 2 from 1, 2;
a(9) = 2 from 5, 6;
a(36844389) = 2 from 8585, 8586;
a(129894940) = 2 from 16119, 16120;
a(446591224981504) = 2 from 29886159, 29886160.
I checked this, and thus Krizek's conjecture below, up to 4*10^19.
(End)

			a(9) = 2 because there are two numbers m (5, 6) with antisigma(m) = 9.

Antti Karttunen, Table of n, a(n) for n = 0..32001

Cf. A054973 (number of numbers whose divisors sum to n), A231365, A231368, A231367, A231369, A067816.

up_to = 105;
A024816(n) = (n*(n+1)/2-sigma(n));
A231366list(up_to) = { my(v=vector(1+up_to), u); for(n=1, 2+up_to, if((u = A024816(n))<=up_to, v[1+u]++)); (v); };
v231366 = A231366list(up_to);
A231366(n) = v231366[1+n]; \\ Antti Karttunen, Jan 19 2025

Conjecture: max a(n) = 2.
a(A231368(n)) >= 1, a(A231369(n)) = 0.
a(n) = 0 for such n that A231367(n) = 0, a(n) = 0 if A024816(m) = n has no solution.
a(n) >= 1 for such n that A231367(n) = 1, a(n) >= 1 if A024816(m) = n for any m.
Conjecture: a(n) = 2 iff n is number from A225775 (0, 9, 36844389, 129894940, 446591224981504, …)

Data section extended to a(105) by Antti Karttunen, Jan 19 2025

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

Original entry on oeis.org

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[#]]]]&]

A371284 Number of integer partitions of n whose distinct parts form the set of divisors of some number.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 8, 9, 11, 12, 16, 18, 23, 25, 32, 36, 42, 47, 57, 62, 73, 81, 96, 106, 123, 132, 154, 168, 190, 207, 240, 259, 293, 317, 359, 388, 434, 469, 529, 574, 635, 688, 768, 826, 915, 987, 1093, 1181, 1302, 1397, 1540, 1662, 1818, 1959, 2149, 2309

Offset: 0

Gus Wiseman, Mar 22 2024

nonn

The Heinz numbers of these partitions are given by A371288.

			The partition y = (10,5,5,5,2,2,1) has distinct parts {1,2,5,10}, which form the set of divisors of 10, so y is counted under a(30).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (11)  (21)   (31)    (221)    (51)      (331)      (71)
             (111)  (211)   (311)    (2211)    (421)      (3311)
                    (1111)  (2111)   (3111)    (511)      (4211)
                            (11111)  (21111)   (2221)     (5111)
                                     (111111)  (22111)    (22211)
                                               (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

The strict case is A054973, ranks A371283 (unsorted version A275700).
These partitions have ranks A371288.
A000005 counts divisors, row-lengths of A027750.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
Cf. A001221, A002865, A239312, A370803, A371172, A371286, A371285.

Table[Length[Select[IntegerPartitions[n], Union[#]==Divisors[Max[#]]&]],{n,0,30}]

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]

A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 3, 2, 4, 2, 4, 3, 3, 4, 2, 4, 2, 4, 3, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 3, 4, 4, 3, 5, 3, 5, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 4, 3, 6, 2, 6, 3, 5, 5, 3, 6, 3, 5, 4, 4, 4, 4, 4, 4, 4, 5, 3, 6, 3, 5, 5, 4, 6, 3, 7, 3, 6, 5, 5, 6, 5, 6, 5, 6, 6, 5, 6, 6

Offset: 1

Labos Elemer, Aug 28 2001

This sequence appears to be growing. However, for 704338 < n < 756839, a(n) is 0. See A078426 for the n such that a(n) = 0. - T. D. Noe, Oct 12 2006

Numbers k such that sigma(k) = 2^n. - Juri-Stepan Gerasimov, Mar 08 2017

			n = 50 = 2 + 5 + 7 + 17 + 19 = 2 + 17 + 31 = 19 + 31, so a(50) = 3. The first numbers for which the number of these Mersenne-exponent partitions is k = 0, 1, 2, 3, 4, 5, 6, 7, 8 are 1, 2, 5, 20, 22, 39, 66, 92, 107, respectively.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000043, A046528, A048947, A054784, A054973, A063889.

Numbers k such that a(k) = m: A078426 (m = 0), A283160 (m = 1).

N:= 500: # to get the first N terms
G:= mul(1+x^i,i=select(t -> numtheory:-mersenne(t)::integer, [$1..N])):
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=1..N); # Robert Israel, Sep 22 2016

exponents[n_] := Reap[For[k = 1, k <= n, k++, If[PrimeQ[2^k-1], Sow[k]]]][[2, 1]]; r[n_] := Module[{ee, x, xx}, ee = exponents[n]; xx = Array[x, Length[ee]]; Reduce[And @@ (0 <= # <= 1 & /@ xx) && xx.ee == n, xx, Integers]]; a[n_] := Which[rn = r[n]; Head[rn] === Or, Length[rn],  Head[rn] === And, 1, Head[rn] === Equal, 1, rn === False, 0, True, Print["error ", rn]]; a[1] = 0; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 05 2014 *)
With[{e = MersennePrimeExponent[Range[10]]}, Rest@ CoefficientList[Product[1 + x^e[[i]], {i, 1, Length[e]}], x]] (* Amiram Eldar, Dec 23 2024 *)

first(lim)=my(M=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667], x='x); if(lim>M[#M], error("Need more Mersenne exponents to compute further")); M=select(p->p<=lim, M); Vec(prod(i=1, #M, 1+x^M[i], O(x^(lim\1+1))+1)) \\ Charles R Greathouse IV, Mar 08 2017

a(n) = sum(k=1, 2^n+1, sigma(k)==2^n); \\ Michel Marcus, Mar 07 2017

a(n) = A054973(2^n). - Michel Marcus, Mar 08 2017

A070242 a(n) = Card( k>0 : sigma(k)=sigma(n) ).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, May 09 2002

Antti Karttunen, Table of n, a(n) for n = 1..16384
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A054973, A066412.

for(n=1,150,print1(sum(i=1,10*n,if(sigma(n)-sigma(i),0,1)),","))

A070242(n) = { my(s=sigma(n)); length(select(i->sigma(i) == s, vector(s, i, i))); } \\ Antti Karttunen, Nov 07 2017

A070242(n) = { my(s=sigma(n)); sum(k=1, s, (sigma(k)==s)); }; \\ Antti Karttunen, Nov 07 2017

a(n) = invsigmaNum(sigma(n)); \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

a(n) = A054973(A000203(n)). - Antti Karttunen, Nov 07 2017

cp's OEIS Frontend

A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

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A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

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A055486 Number of solutions to sigma(x) = n!.

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Maple

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A231366 Number of numbers whose sum of non-divisors (A024816) is equal to n.

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PARI

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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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Mathematica

A371284 Number of integer partitions of n whose distinct parts form the set of divisors of some number.

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Mathematica

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

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Mathematica

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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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