A316857 -id:A316857 - 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&]

A316856 Heinz numbers of integer partitions whose reciprocal sum is an integer.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 147, 162, 195, 250, 256, 288, 294, 324, 390, 500, 512, 576, 588, 648, 729, 780, 1000, 1024, 1125, 1152, 1176, 1296, 1323, 1458, 1560, 1755, 2000, 2048, 2250, 2304, 2352, 2401, 2592, 2646, 2916, 3120

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).

			195 is the Heinz number of (6,3,2), which has reciprocal sum 1/6 + 1/3 + 1/2 = 1, which is an integer, so 195 belongs to the sequence.
The sequence of all integer partitions whose reciprocal sum is an integer begins: (), (1), (11), (111), (22), (1111), (221), (11111), (2211), (111111), (22111), (2222).

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316855, A316857.

Select[Range[1000],IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,If[#==1,{},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[#]}]]]&]

A325622 Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 5, 4, 4, 3, 3, 4, 6, 3, 4, 5, 5, 5, 6, 3, 7, 6, 5, 6, 6, 6, 5, 6, 8, 5, 7, 5, 4, 8, 7, 7, 7, 7, 9, 9, 9, 10, 12, 6, 12, 8, 10, 7, 14, 10, 8, 11, 11, 12, 11, 10, 10, 12, 14, 11, 10, 9, 10, 12, 10, 15, 14, 11, 10

Offset: 1

Gus Wiseman, May 13 2019

nonn

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: (2)
   3: (3)
   4: (4)
   4: (2,2)
   5: (5)
   6: (6)
   6: (3,3)
   7: (7)
   8: (8)
   8: (4,4)
   9: (9)
   9: (5,4)
   9: (3,3,3)
  10: (10)
  10: (5,5)
  11: (11)
  11: (4,4,3)
  11: (3,3,3,2)
  12: (12)
  12: (6,6)
  12: (4,4,4)

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

Reciprocal factorial sum: A002966, A316854, A316857, A325618, A325620, A325623.

f:= proc(n) nops(select(proc(t) local i; (1/add(1/i!,i=t))::integer end proc, combinat:-partition(n))) end proc:
map(f, [$1..70]); # Robert Israel, May 09 2024

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

a(n) = my(c=0); forpart(v=n, if(numerator(sum(i=1, #v, 1/v[i]!))==1, c++)); c; \\ Jinyuan Wang, Feb 25 2025

a(61)-a(70) from Robert Israel, May 09 2024

a(71)-a(80) from Jinyuan Wang, Feb 25 2025

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

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

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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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A316856 Heinz numbers of integer partitions whose reciprocal sum is 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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A325622 Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of 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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