A051908 -id:A051908 - cp's OEIS frontend

A318585 Number of integer partitions of n whose sum of reciprocals squared is an integer.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 12, 12, 13, 14, 16, 16, 18, 19, 21, 23, 26, 27, 29, 30, 34, 35, 39, 43, 48, 51, 55, 57, 63, 67, 74, 78, 84, 89, 99, 103, 112, 119, 132, 139, 148, 156, 170, 182, 199

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

From David A. Corneth, Sep 03 2018: (Start)
Let a valid tuple be a tuple of positive integers whose sum of reciprocals squared is an integer. Initially one only needs to consider tuples of positive integers where each element is > 1. After that some ones could be prepended to a valid tuple to find new valid tuples.
One could define a prime tuple as a valid tuple where no proper part with elements is a valid tuple. So (1) would be a prime tuple as no proper part of (1) has elements and is a valid tuple. Other examples of prime tuples are (2, 2, 2, 2) and (2, 2, 2, 3, 3, 6).
The list of distinct elements in a tuple could be whittled down by finding for each positive integer m the least sum of a prime tuple in which that integer is. For each m, that sum is at most m^3. (End)

			The a(26) = 7 integer partitions:
  (6332222222)
  (44442221111)
  (63322211111111)
  (22222222222211)
  (222222221111111111)
  (2222111111111111111111)
  (11111111111111111111111111)

Giovanni Resta, Table of n, a(n) for n = 1..130

Cf. A000041, A002865, A051908, A058360, A289506, A289507, A316854, A316856.

Cf. A316854, A318584, A318586, A318587, A318588, A318589.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Total[#^(-2)]]&]],{n,30}]

a(61)-a(70) from Giovanni Resta, Sep 03 2018

A318586 Number of integer partitions of n whose sum of reciprocals squared is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 3, 3, 1, 4, 1, 3, 1, 2, 1, 5, 2, 1, 4, 5, 1, 5, 1, 6, 3, 2, 4, 8, 2, 4, 2, 6, 3, 9, 2, 4, 7, 5, 4, 11, 8, 7, 8, 9, 5, 12, 5, 16, 5, 10, 5, 25, 10, 9, 13, 18, 12, 18, 6, 11, 14, 22, 9, 24, 11, 21, 22, 25, 24, 23, 28, 32

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

			The a(42) = 9 integer partitions:
  (42)
  (21,14,7)
  (18,9,9,6)
  (18,9,9,3,3)
  (20,10,4,4,4)
  (12,12,12,4,2)
  (10,5,5,5,5,5,5,2)
  (12,6,6,4,4,4,2,2,2)
  (6,6,4,4,4,4,3,3,3,3,2)

Giovanni Resta, Table of n, a(n) for n = 1..150

Cf. A000041, A051908, A058360, A289506, A289507, A316854, A316857.

Cf. A318584, A318585, A318587, A318588, A318589.

Table[Length[Select[IntegerPartitions[n],IntegerQ[1/Total[#^(-2)]]&]],{n,30}]

a(61)-a(80) from Giovanni Resta, Sep 03 2018

A325623 Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 221, 223, 227, 229

Offset: 1

Gus Wiseman, May 13 2019

nonn

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

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316857, A325619, A325621, A325622.

Select[Range[100],IntegerQ[1/Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]

A318573 Numerator of the reciprocal sum of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 5, 1, 5, 5, 4, 1, 2, 1, 7, 3, 6, 1, 7, 2, 7, 3, 9, 1, 11, 1, 5, 7, 8, 7, 3, 1, 9, 2, 10, 1, 7, 1, 11, 4, 10, 1, 9, 1, 5, 9, 13, 1, 5, 8, 13, 5, 11, 1, 17, 1, 12, 5, 6, 1, 17, 1, 15, 11, 19, 1, 4, 1, 13, 7, 17, 9, 5, 1, 13, 2, 14, 1, 11, 10, 15, 3, 16, 1, 7, 5, 19, 13, 16, 11, 11, 1, 3

Offset: 1

Gus Wiseman, Aug 29 2018

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Positions of 1's are A316857.

Cf. A051908, A056239, A058360, A112798, A289506, A289507, A296150, A316854, A316855, A316856, A318574, A325704.

Table[Sum[pr[[2]]/PrimePi[pr[[1]]],{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]//Numerator

A318573(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/primepi(f[i, 1]))); }; \\ Antti Karttunen, Nov 17 2019

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) is the numerator of Sum y_i/x_i.

More terms from Antti Karttunen, Nov 17 2019

A318584 Number of integer partitions of n whose sum of reciprocals squared is 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 0, 0, 2, 0, 2, 1, 2, 2, 2, 1, 1, 2, 3, 0, 1, 1, 6, 2, 3, 2, 6, 2, 2, 3, 2, 6, 7, 2, 4, 3, 9, 4, 7, 5, 8, 8, 7, 9, 9, 11, 12, 7, 9, 11, 17, 9, 13, 12, 17, 16, 13, 15, 20, 26, 27, 18, 23

Offset: 0

Gus Wiseman, Aug 29 2018

nonn

The a(16) = 1 integer partition:
  (6,3,3,2,2,2)
The a(48) = 2 integer partitions:
  (18,9,9,3,3,2,2,2)
  (6,6,6,6,3,3,3,3,3,3,3,3)
The a(56) = 3 integer partitions:
  (12,6,6,4,4,4,4,4,4,4,2,2)
  (10,6,5,5,5,5,5,5,3,3,2,2)
  (6,6,4,4,4,4,4,4,4,4,3,3,3,3)
The a(60) = 6 integer partitions:
  (12,12,12,12,3,3,2,2,2)
  (8,8,8,8,6,4,4,4,3,3,2,2)
  (6,6,6,6,6,6,6,6,6,2,2,2)
  (12,12,12,4,3,3,3,3,3,3,2)
  (10,5,5,5,5,5,5,4,4,4,4,2,2)
  (6,4,4,4,4,4,4,4,4,4,4,4,4,3,3)

Cf. A000041, A051908, A058360, A289506, A289507, A316854, A316855.

Cf. A318585, A318586, A318587, A318588, A318589.

Table[Length[Select[IntegerPartitions[n],Total[#^(-2)]==1&]],{n,30}]

a(61)-a(100) from Alois P. Heinz, Aug 30 2018

A318587 Heinz numbers of integer partitions whose sum of reciprocals squared is 1.

Original entry on oeis.org

2, 81, 8775, 64827, 950625, 1953125, 7022925, 9055935, 21781575, 36020025, 50124555, 51883209, 57909033, 102984375, 118978125, 760816875, 816747435, 981059625, 1206902781, 1265058675, 1387132263, 2359670625, 3902169375, 4868424351, 5222768733, 5430160125

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

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

			The sequence of integer partitions with Heinz numbers in this sequence begins: (1), (2222), (633222), (4444222), (66333322).

Cf. A001222, A051908, A056239, A112798, A289506, A289507, A316855, A316856, A316857.

Cf. A318584, A318585, A318586, A318588, A318589.

Select[Range[100000],Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]==1&]

a(6)-a(26) from Alois P. Heinz, Aug 30 2018

A318589 Heinz numbers of integer partitions whose sum of reciprocals squared is the reciprocal of an integer.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

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

Cf. A001222, A051908, A056239, A058360, A112798, A289506, A289507, A316854, A316855.

Cf. A318584, A318585, A318586, A318587, A318588.

Select[Range[2,1000],IntegerQ[1/Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]]&]

A325624 a(n) = prime(n)^(n!).

Original entry on oeis.org

2, 9, 15625, 191581231380566414401, 92709068817830061978520606494193845859707401497097037749844778027824097442147966967457359038488841338006006032592594389655201

Offset: 1

Gus Wiseman, May 13 2019

nonn

A subsequence of A325619 (numbers whose prime indices have reciprocal factorial sum equal to 1).

Cf. A056239, A112798.
Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A051908, A316855, A325618, A325619.

Mathematica
```
Table[Prime[n]^n!,{n,5}]
```

A051907 Number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 0, 4, 1, 3, 4, 0, 2, 0, 6, 0, 1, 2, 1, 3, 0, 4, 2, 1, 5, 5, 3, 2, 3, 3, 5, 5, 5, 2, 1, 12, 5, 4, 11, 4, 5, 2, 11, 3, 5

Offset: 1

Jud McCranie, Dec 16 1999

nonn

			1 = 1/2+1/4+1/9+1/12+1/18 = 1/2+1/5+1/6+1/12+1/20. The sum of the denominators of each of these is 45, these are the only 2 with sum of denominators = 45, so a(45)=2.

Seiichi Manyama, Table of n, a(n) for n = 1..150
Index entries for sequences related to Egyptian fractions

A051882 lists n such that a(n)=0.

Cf. A051908.

R. L. Graham showed that a(n)>0 for n>77.

A318574 Denominator of the reciprocal sum of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 6, 1, 7, 1, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 6, 11, 1, 10, 7, 12, 1, 12, 8, 3, 3, 13, 4, 14, 5, 3, 9, 15, 2, 2, 3, 14, 6, 16, 2, 15, 4, 8, 10, 17, 6, 18, 11, 4, 1, 2, 10, 19, 7, 18, 12, 20, 1, 21, 12, 6, 8, 20, 3, 22

Offset: 1

Gus Wiseman, Aug 29 2018

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Positions of 1's are A316856.

Cf. A051908, A056239, A058360, A112798, A289506, A289507, A296150, A316854, A316855, A316857, A318573.

Table[Sum[pr[[2]]/PrimePi[pr[[1]]],{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]//Denominator

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) is the denominator of Sum y_i/x_i.

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A318585 Number of integer partitions of n whose sum of reciprocals squared is an integer.

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A318586 Number of integer partitions of n whose sum of reciprocals squared is the reciprocal of an integer.

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A325623 Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

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A318573 Numerator of the reciprocal sum of the integer partition with Heinz number n.

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A318584 Number of integer partitions of n whose sum of reciprocals squared is 1.

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A318587 Heinz numbers of integer partitions whose sum of reciprocals squared is 1.

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A318589 Heinz numbers of integer partitions whose sum of reciprocals squared is the reciprocal of an integer.

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A325624 a(n) = prime(n)^(n!).

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A051907 Number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.

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