A316856 -id:A316856 - cp's OEIS frontend

A316854 Number of integer partitions of n whose reciprocal sum is the reciprocal of an integer.

1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 2, 2, 1, 1, 1, 4, 2, 4, 1, 5, 1, 5, 1, 3, 4, 2, 5, 6, 5, 5, 4, 5, 5, 4, 8, 10, 9, 7, 5, 9, 10, 6, 12, 10, 8, 7, 6, 9, 13, 15, 8, 19, 13, 19, 19, 19, 18, 22, 26, 28, 28, 29, 22, 33, 29, 28, 38, 34, 26, 40, 32, 43, 39, 51, 38, 62, 46

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

			The a(36) = 10 partitions:
  (36),
  (30,6), (24,12), (18,18),
  (12,12,12),
  (12,12,6,6),
  (15,10,4,4,3), (12,12,6,3,3), (12,8,8,6,2),
  (6,6,6,6,6,6).

Giovanni Resta, Table of n, a(n) for n = 1..200 (first 100 terms from Robert G. Wilson v)

Cf. A000041, A051908, A058360, A316855, A316856, A316857.

Table[Length[Select[IntegerPartitions[n],IntegerQ[1/Sum[1/m,{m,#}]]&]],{n,30}]
ric[n_, p_, s_] := If[n == 0, If[IntegerQ[1/s], c++], Do[If[s + 1/i <= 1, ric[n - i, Append[p, i], s + 1/i]], {i, Min[p[[-1]], n], 1, -1}]]; a[n_] := (c = 0; Do[ric[n - j, {j}, 1/j], {j, n}]; c); Array[a, 80] (* after Giovanni Resta in A316898, Robert G. Wilson v, Jul 23 2018 *)

a(n)={my(s=0); forpart(p=n, if(frac(1/sum(i=1, #p, 1/p[i]))==0, s++)); s} \\ Andrew Howroyd, Jul 15 2018

a(51)-a(77) from Giovanni Resta, Jul 15 2018

A316855 Heinz numbers of integer partitions whose reciprocal sum is 1.

Original entry on oeis.org

2, 9, 125, 147, 195, 2401, 3185, 4225, 6475, 6591, 7581, 10101, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 107653, 123823, 142805, 161051, 164255, 164983, 171941, 218855, 228085, 267883, 304175, 312785, 333925, 333935, 335405, 343735, 355355, 390963

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

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).

			Sequence of all integer partitions whose reciprocal sum is 1 begins: (1), (2,2), (3,3,3), (4,4,2), (6,3,2), (4,4,4,4), (6,4,4,3), (6,6,3,3), (12,4,3,3), (6,6,6,2), (8,8,4,2).

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316856, A316857.

Select[Range[2,10000],Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1&]

A316857 Heinz numbers of integer partitions whose reciprocal sum 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, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 147, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

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).

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316855, A316856.

Select[Range[2,100],IntegerQ[1/Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]&]

A316888 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.

Original entry on oeis.org

2, 195, 3185, 6475, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 164255, 171941, 218855, 228085, 267883, 312785, 333925, 333935, 335405, 343735, 355355, 414295, 442975, 474513, 527425, 549575, 607475, 633777, 691041, 711321, 722425, 753865, 804837, 822783

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

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).
A partition is aperiodic if its multiplicities are relatively prime.
Does not contain 29888089, which belongs to A316890 and is the Heinz number of a periodic partition.

			The partition (6,4,4,3) with Heinz number 3185 is aperiodic, has relatively prime parts, and 1/6 + 1/4 + 1/4 + 1/3 = 1, so 3185 belongs to the sequence.
The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (6,3,2), (6,4,4,3), (12,4,3,3), (10,5,5,2), (20,5,4,2), (15,10,3,2), (12,12,3,2), (18,9,3,2), (24,8,3,2), (42,7,3,2).

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

Cf. A000837, A002966, A007916, A051908, A100953, A289509, A296150, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,100000],And[GCD@@FactorInteger[#][[All,2]]==1,GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

A316904 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

2, 18, 72, 162, 195, 250, 288, 294, 390, 500, 588, 648, 780, 1125, 1152, 1176, 1458, 1560, 1755, 2000, 2250, 2352, 2592, 2646, 3120, 3185, 3510, 4000, 4500, 4608, 4704, 4802, 5292, 6240, 6370, 6475, 7020, 8450, 9000, 9408, 10125, 10368, 10527, 10584, 12480

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

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).
A partition is aperiodic if its multiplicities are relatively prime.

			The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (221), (22111), (22221), (632), (3331), (2211111), (4421), (6321), (33311), (44211), (2222111).

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

Cf. A000837, A002966, A051908, A058360, A100953, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,20000],And[GCD@@FactorInteger[#][[All,2]]==1,GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

A325620 Number of integer partitions of n whose reciprocal factorial sum is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 14, 14, 15, 16, 18, 19, 20, 22, 24, 25, 26, 28, 31, 33, 34, 36, 39, 41, 43, 45, 49, 52, 54, 57, 61, 65, 68, 71, 76, 80, 84, 88, 93, 98, 103, 107, 113

Offset: 1

Gus Wiseman, May 13 2019

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

			The initial terms count the following partitions:
  1: (1)
  2: (1,1)
  3: (1,1,1)
  4: (2,2)
  4: (1,1,1,1)
  5: (2,2,1)
  5: (1,1,1,1,1)
  6: (2,2,1,1)
  6: (1,1,1,1,1,1)
  7: (2,2,1,1,1)
  7: (1,1,1,1,1,1,1)
  8: (2,2,2,2)
  8: (2,2,1,1,1,1)
  8: (1,1,1,1,1,1,1,1)
  9: (2,2,2,2,1)
  9: (2,2,1,1,1,1,1)
  9: (1,1,1,1,1,1,1,1,1)

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

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316856, A325618, A325621, A325622.

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

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

Original entry on oeis.org

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

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

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

A325621 Heinz numbers of integer partitions whose reciprocal factorial sum is an integer.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 128, 144, 162, 256, 288, 324, 375, 512, 576, 648, 729, 750, 1024, 1152, 1296, 1458, 1500, 2048, 2304, 2592, 2916, 3000, 3375, 4096, 4608, 5184, 5832, 6000, 6561, 6750, 8192, 9216, 10368, 11664, 12000, 13122, 13500

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}
      4: {1,1}
      8: {1,1,1}
      9: {2,2}
     16: {1,1,1,1}
     18: {1,2,2}
     32: {1,1,1,1,1}
     36: {1,1,2,2}
     64: {1,1,1,1,1,1}
     72: {1,1,1,2,2}
     81: {2,2,2,2}
    128: {1,1,1,1,1,1,1}
    144: {1,1,1,1,2,2}
    162: {1,2,2,2,2}
    256: {1,1,1,1,1,1,1,1}
    288: {1,1,1,1,1,2,2}
    324: {1,1,2,2,2,2}
    375: {2,3,3,3}
    512: {1,1,1,1,1,1,1,1,1}

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

Reciprocal factorial sum: A002966, A058360, A316856, A325619, A325620, A325623.

Select[Range[1000],IntegerQ[Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]

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A316854 Number of integer partitions of n whose reciprocal sum is the reciprocal of an integer.

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A316855 Heinz numbers of integer partitions whose reciprocal sum is 1.

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A316857 Heinz numbers of integer partitions whose reciprocal sum is the reciprocal of an integer.

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A316888 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.

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A316904 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is an integer.

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A325620 Number of integer partitions of n whose reciprocal factorial sum is an integer.

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

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

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