A275700 -id:A275700 - cp's OEIS frontend

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

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]

A054973 Number of numbers whose divisors sum to n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, May 16 2000

nonn

a(n) = frequency of values n in A000203(m), where A000203(m) = sum of divisors of m. a(n) >= 1 for such n that A175192(n) = 1, a(n) >= 1 if A000203(m) = n for any m. a(n) = 0 for such n that A175192(n) = 0, a(n) = 0 if A000203(m) = n has no solution. - Jaroslav Krizek, Mar 01 2010
First occurrence of k: 2, 1, 12, 24, 96, 72, ..., = A007368. - Robert G. Wilson v, May 14 2014
a(n) is also the number of positive terms in the n-th row of triangle A299762. - Omar E. Pol, Mar 14 2018
Also the number of integer partitions of n whose parts form the set of divisors of some number (necessarily the greatest part). The Heinz numbers of these partitions are given by A371283. For example, the a(24) = 3 partitions are: (23,1), (15,5,3,1), (14,7,2,1). - Gus Wiseman, Mar 22 2024

			a(12) = 2 since 11 has factors 1 and 11 with 1 + 11 = 12 and 6 has factors 1, 2, 3 and 6 with 1 + 2 + 3 + 6 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203 (sum-of-divisors function).
For partial sums see A074753.
Cf. A002191, A007609, A308039.
The non-strict version is A371284, ranks A371288.
These partitions have ranks A371283, unsorted version A275700.
A000005 counts divisors, row-lengths of A027750.
A000041 counts integer partitions, strict A000009.
Cf. A001221, A002865, A008289, A371286, A371285.

nn = 105; t = Table[0, {nn}]; k = 1; While[k < 6 nn^(3/2)/Pi^2, d = DivisorSigma[1, k]; If[d < nn + 1, t[[d]]++]; k++]; t (* Robert G. Wilson v, May 14 2014 *)
Table[Length[Select[IntegerPartitions[n],#==Reverse[Divisors[Max@@#]]&]],{n,30}] (* Gus Wiseman, Mar 22 2024 *)

a(n)=v = vector(0); for (i = 1, n, if (sigma(i) == n, v = concat(v, i));); #v; \\ Michel Marcus, Oct 22 2013

a(n)=sum(k=1,n,sigma(k)==n) \\ Charles R Greathouse IV, Nov 12 2013

first(n)=my(v=vector(n),t); for(k=1,n, t=sigma(n); if(t<=n, v[t]++)); v \\ Charles R Greathouse IV, Mar 08 2017

A054973(n)=#invsigma(n) \\ See Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308039. - Amiram Eldar, Dec 23 2024

Incorrect comment deleted by M. F. Hasler, Nov 21 2019

A328166 Heinz number of the run-lengths of the divisors of n.

Original entry on oeis.org

2, 3, 4, 6, 4, 10, 4, 12, 8, 12, 4, 28, 4, 12, 16, 24, 4, 40, 4, 36, 16, 12, 4, 112, 8, 12, 16, 48, 4, 120, 4, 48, 16, 12, 16, 224, 4, 12, 16, 144, 4, 120, 4, 48, 64, 12, 4, 448, 8, 48, 16, 48, 4, 160, 16, 144, 16, 12, 4, 832, 4, 12, 64, 96, 16, 160, 4, 48, 16

Offset: 1

Gus Wiseman, Oct 07 2019

nonn

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

			Splitting the divisors of 30 into runs gives {{1, 2, 3}, {5, 6}, {10}, {15}, {30}}, and the Heinz number of {1, 1, 1, 2, 3} is 120, so a(30) = 120.
More examples from _Antti Karttunen_, Dec 09 2021: (Start)
Splitting the divisors of 1 into runs gives {1}, and the Heinz number of that is 2.
Splitting the divisors of 2 into runs gives {1, 2}, and the Heinz number of that is 3. [one run of length 2, therefore a(2) = prime(2)^1].
Splitting the divisors of 3 into runs gives {1} and {3}, and the Heinz number of that is 4. [two runs of length 1, therefore a(3) = prime(1)^2].
Splitting the divisors of 4 into runs gives {1, 2} and {4}, and the Heinz number of that is 6. [one run of length 1, and other run of length 2, therefore a(4) = prime(1)*prime(2)].
Splitting the divisors of 5 into runs gives {1} and {5}, and the Heinz number of that is 4. [two runs of length 1, therefore a(5) = prime(1)^2].
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Wikipedia, Run (cards)
Index entries for sequences related to Heinz numbers

The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the set of divisors of n is A275700(n).
Numbers whose divisors do not have weakly decreasing run-lengths are A328165.
Cf. A000005, A056239, A060680, A060775, A119313, A137921, A181063, A199970.

Table[Times@@Prime/@Length/@Split[Divisors[n],#2==#1+1&],{n,30}]

A328166(n) = { my(rl=0,pd=0,v=vector(numdiv(n)),m=1); fordiv(n, d, if(d>(1+pd), v[rl]++; rl=0); pd=d; rl++); v[rl]++; for(i=1,#v, m *= prime(i)^v[i]); (m); }; \\ Antti Karttunen, Dec 09 2021

A001222(a(n)) = A137921(n).

A056239(a(n)) = A000005(n).

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[#]]&]

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).

A328337 The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).

Original entry on oeis.org

0, 0, 0, 2, 0, 6, 0, 10, 4, 18, 0, 46, 0, 66, 20, 138, 0, 294, 0, 538, 68, 1026, 0, 2222, 16, 4098, 260, 8266, 0, 16950, 0, 32906, 1028, 65538, 80, 133422, 0, 262146, 4100, 524954, 0, 1056870, 0, 2098186, 16660, 4194306, 0, 8423598, 64, 16777746, 65540

Offset: 1

Gus Wiseman, Oct 15 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The nontrivial divisors of 18 are {2, 3, 6, 9}, so a(18) = 2^1 + 2^2 + 2^5 + 2^8 = 294.

Removing zeros gives binary indices of rows of A163870.
The version for all divisors is A034729.
The version for proper divisors is A247146.
Cf. A000005, A000120, A027750, A029931, A033676, A048793, A060775, A070824, A070939, A275700, A326031.

Table[Total[(2^DeleteCases[Divisors[n],1|n])/2],{n,100}]

from sympy import divisors
def A328337(n): return sum(1<<(d-1) for d in divisors(n,generator=True) if 1Chai Wah Wu, Jul 15 2022

A000120(a(n)) = A070824(n).
A070939(a(n)) = A032742(n).
A001511(a(n)) = A107286(n).

cp's OEIS Frontend

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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A054973 Number of numbers whose divisors sum to n.

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A328166 Heinz number of the run-lengths of the divisors of n.

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

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

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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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A328337 The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).

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