A058360 -id:A058360 - cp's OEIS frontend

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

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

A318588 Heinz numbers of integer partitions whose sum of reciprocals squared is an integer.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 81, 128, 162, 256, 324, 512, 648, 1024, 1296, 2048, 2592, 4096, 5184, 6561, 8192, 8775, 10368, 13122, 16384, 17550, 20736, 26244, 32768, 35100, 41472, 52488, 64827, 65536, 70200, 82944, 104976, 129654, 131072, 140400, 165888, 209952

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).

			Sequence of integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (11), (111), (1111), (11111), (111111), (2222), (1111111), (22221), (11111111), (222211), (111111111), (2222111).

Cf. A001222, A056239, A058360, A112798, A289506, A289507, A316855, A316857.

Cf. A318584, A318585, A318586, A318587, A318589.

Select[Range[10000],IntegerQ[Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]]&]

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]]]]&]

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]!]]]&]

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.

A067540 Number of partitions of n in which the sum of reciprocal of parts divides number of parts.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 5, 7, 4, 8, 5, 7, 5, 10, 5, 16, 8, 20, 9, 20, 10, 25, 13, 30, 21, 34, 32, 46, 35, 58, 48, 66, 59, 97, 65, 109, 94, 125, 112, 154, 133, 191, 154, 224, 195, 265, 236, 308, 289, 388, 353, 475, 424, 575, 528, 693, 661, 835

Offset: 1

Naohiro Nomoto, Jan 27 2002

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..147 (n = 1..80 from Sean A. Irvine)

Cf. A000041.

a(n)=#{n=p_1+p_2+...+p_r : r/sum_{i=1..r} 1/p_i is an integer} where n=p_1+p_2+...+p_r runs over all partitions of n. - Sean A. Irvine, Mar 24 2013

a(n) <= A058360(n) <= A000041(n). - Charles R Greathouse IV, Dec 15 2020

More terms from Sean A. Irvine, Mar 24 2013

A316890 Heinz numbers of 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).
Includes 29888089, which is the first perfect power in the sequence and is absent from A316888.

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

Cf. A000837, A002966, A051908, A058360, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

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

cp's OEIS Frontend

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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A318588 Heinz numbers of integer partitions whose sum of reciprocals squared is an integer.

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

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

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A067540 Number of partitions of n in which the sum of reciprocal of parts divides number of parts.

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