A002966 -id:A002966 - cp's OEIS frontend

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

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]!]]]&]

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}]
```

A003167 Number of n-dimensional cuboids with integral edge lengths for which volume = surface area.

Original entry on oeis.org

2, 10, 108, 2892, 270332

Offset: 2

mjzerger(AT)adams.edu

For n>1 it is always true that a(n) > 0 because for dimension n we always have the n-dimensional cuboid with all edge lengths = 2n = A062971(n) having hypervolume (2n)^n equal to "surface hyper-area". - Jonathan Vos Post, Mar 15 2006

Number of nondecreasing tuples (x_1, x_2, ..., x_n) such that 1/2 = 1/x_1 + 1/x_2 + ... + 1/x_n. - Lewis Chen, Dec 20 2019

			From _Joseph Myers_, Feb 24 2004: (Start)
For n=2 the cuboids are 3 X 6 and 4 X 4.
For n=3 the cuboids are 3 X 7 X 42, 3 X 8 X 24, 3 X 9 X 18, 3 X 10 X 15, 3 X 12 X 12, 4 X 5 X 20, 4 X 6 X 12, 4 X 8 X 8, 5 X 5 X 10, 6 X 6 X 6. (End)
For n=4 see the Marcus link.

Gerald E. Gannon, Martin V. Bonsangue and Terrence J. Redfern, One Good Problem Leads to Another and Another and..., Math. Teacher, 90 (#3, 1997), pp. 188-191.
Michel Marcus, Cuboids for n=4, after Joseph Myers.

Cf. A002966.

a(5)-a(6) from Joseph Myers, Feb 24 2004

A115855 Number of partitions of 1 into fractions i/j with 1<=i

Original entry on oeis.org

0, 1, 3, 6, 12, 21, 35, 58, 106, 188, 243, 493, 593, 1062, 3275, 5507, 5803, 12426, 12915, 42410, 131772, 167587, 168841, 428012, 839367, 1015501, 1968161, 5787286, 5791850, 15163758, 15170599, 28838712, 75983559, 82753547, 486356262, 1158442726, 1158464362

Offset: 1

Reinhard Zumkeller, Feb 01 2006

nonn

			a(4) = #{1/2+1/2, 1/2+1/4+1/4, 1/3+2/3, 1/3+1/3+1/3, 1/4+3/4, 1/4+1/4+1/4+1/4} = 6.

Cf. A000041, A002966, A038566, A038567, A115856, A116084.

Table[Length@ Select[Flatten[Map[IntegerPartitions[1, {#}, Rest@ Union[Flatten@ TensorProduct[#, 1/#] &@ Range@ n /. {Integer -> 0, k /; k > 1 -> 0}]] &, Range@ n], 1], Total@ # == 1 &], {n, 25}] (* Michael De Vlieger, Jul 15 2016 *) (* or *)
a[n_] := Sum[ Length@ IntegerPartitions[1, {k}, Union@ Flatten[ Table[i/j, {j, n}, {i, j-1}]]], {k, n}]; Array[a, 20] (* Giovanni Resta, Jun 15 2017 *)

A115856(n) = a(n+1) - a(n).

a(21)-a(28) from Michael De Vlieger, Jul 15 2016

More terms from Jinyuan Wang, Dec 11 2024

A227611 Number of ways 2/n can be expressed as the sum of three distinct unit fractions: 2/n = 1/x + 1/y + 1/z with 0 < x < y < z.

Original entry on oeis.org

0, 1, 5, 6, 9, 15, 14, 22, 21, 30, 22, 45, 17, 36, 72, 62, 22, 69, 29, 84, 77, 56, 39, 142, 48, 53, 82, 124, 30, 178, 34, 118, 94, 67, 176, 191, 29, 74, 151, 274, 37, 227, 37, 145, 220, 87, 57, 342, 80, 146, 138, 162, 39, 216, 214, 322, 134, 100, 73, 461, 31, 84, 316, 257, 197, 304, 47, 199, 166, 435, 69, 508, 34, 79, 317

Offset: 1

Robert G. Wilson v, Jul 17 2013

nonn

See A073101 for the 4/n conjecture due to Erdős and Straus.

Cf. A002966, A073546.

Cf. A227610 (1/n), A075785 (3/n), A073101 (4/n), A075248 (5/n), A227612.

f[n_] := Length@ Solve[2/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Array[f, 75]

A227612 Table read by antidiagonals: Number of ways m/n can be expressed as the sum of three distinct unit fractions, i.e., m/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and read by antidiagonals.

Original entry on oeis.org

1, 0, 6, 0, 1, 15, 0, 1, 5, 22, 0, 0, 1, 6, 30, 0, 0, 1, 3, 9, 45, 0, 0, 1, 1, 7, 15, 36, 0, 0, 0, 2, 2, 6, 14, 62, 0, 0, 0, 1, 1, 5, 6, 22, 69, 0, 0, 0, 1, 1, 1, 5, 16, 21, 84, 0, 0, 0, 0, 1, 1, 3, 6, 15, 30, 56, 0, 0, 0, 0, 1, 4, 1, 5, 4, 15, 22, 142, 0, 0, 0, 0, 0, 1, 1, 3, 9, 9, 13, 45, 53

Offset: 1

Robert G. Wilson v, Jul 17 2013

The main diagonal is 1, 1, 1, 1, 1, 1, 1, ..., ; i.e., 1 = 1/2 + 1/3 + 1/6.

			  m\n| 1  2   3   4   5   6   7   8   9  10  11   12  13   14   15
  ---+------------------------------------------------------------
   1 | 1  6  15  22  30  45  36  62  69  84  56  142  53  124  178  A227610
   2 | 0  1   5   6   9  15  14  22  21  30  22   45  17   36   72  A227611
   3 | 0  1   1   3   7   6   6  16  15  15  13   22   8   27   30  A075785
   4 | 0  0   1   1   2   5   5   6   4   9   7   15   4   14   33  A073101
   5 | 0  0   1   2   1   1   3   5   9   6   3   12   5   18   15  A075248
   6 | 0  0   0   1   1   1   1   3   5   7   5    6   1    6    9  n/a
   7 | 0  0   0   1   1   4   1   2   2   2   2    9   6    6    7  n/a
   8 | 0  0   0   0   1   1   1   1   1   2   0    5   3    5   15  n/a
   9 | 0  0   0   0   0   1   1   3   1   1   0    3   1    2    7  n/a
  10 | 0  0   0   0   0   1   0   2   2   1   0    1   1    3    5  n/a
.
Antidiagonals are {1}, {0, 6}, {0, 1, 15}, {0, 1, 5, 22}, {0, 0, 1, 6, 30}, {0, 0, 1, 3, 9, 45}, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Christian Elsholtz, Sums Of k Unit Fractions, Trans. Amer. Math. Soc. 353 (2001), 3209-3227.
David Eppstein, Algorithms for Egyptian Fractions
David Eppstein, Ten Algorithms for Egyptian Fractions, Wolfram Library Archive.
Ron Knott Egyptian Fractions
Oakland University, The Erdős Number Project
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A002966, A073546, A227610 (1/n), A227611 (2/n), A075785 (3/n), A073101 (4/n), A075248 (5/n).

f[m_, n_] := Length@ Solve[m/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Table[ f[n, m - n + 1], {m, 12}, {n, m, 1, -1}]

A325704 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 1, 7, 1, 5, 1, 25, 2, 4, 1, 2, 1, 13, 13, 121, 1, 7, 1, 721, 3, 49, 1, 5, 1, 5, 61, 5041, 5, 3, 1, 40321, 361, 19, 1, 37, 1, 241, 7, 362881, 1, 9, 1, 4, 2521, 1441, 1, 5, 7, 73, 20161, 3628801, 1, 8, 1, 39916801, 25, 6, 121, 181, 1

Offset: 1

Gus Wiseman, May 18 2019

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the numerator of 1/i_1! + ... + 1/i_k!.

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.

Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325703.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Numerator

A325704(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/(primepi(f[i, 1])!))); }; \\ Antti Karttunen, Nov 17 2019

a(n) = A318573(A325709(n)).

A347566 Number of ways 1/n can be expressed as the sum of five distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t, with 0 < p < q < r < s < t.

Original entry on oeis.org

72, 2293, 15304, 47314, 112535, 190665, 368474, 577623, 925336, 1164976, 1478492, 2051830, 2240745, 3789424, 4989958, 4672559, 4467275, 7104589, 6548335, 13844524, 13580094, 10633142, 10451326, 20262957, 16621976, 18697914, 25613090, 27523671

Offset: 1

Jud McCranie, Sep 06 2021

nonn

Cf. A002966, A073546, A227610, A241883, A347569.

A342267 Number of partitions of 1/n into n reciprocals of positive integers.

Original entry on oeis.org

1, 2, 21, 694, 118995, 132891609

Offset: 1

Ilya Gutkovskiy, Mar 07 2021

			a(2) = 2 because we have 1/2 = 1/4 + 1/4 = 1/3 + 1/6.

Cf. A002966, A003167, A004194, A018892, A020327, A020328, A343074, A343214, A369470.

a(6) from Jud McCranie, Sep 02 2021

A347569 Number of ways 1/n can be expressed as the sum of six distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t + 1/u, with 0 < p < q < r < s < t < u.

Original entry on oeis.org

2320, 244817, 3421052, 18420699, 64025680, 131223239, 431008820, 681922142

Offset: 1

Jud McCranie, Sep 06 2021

Cf. A002966, A073546, A227610, A241883, A347566.

a(7) and a(8) from Jud McCranie, Oct 15 2021

cp's OEIS Frontend

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

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

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A003167 Number of n-dimensional cuboids with integral edge lengths for which volume = surface area.

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A115855 Number of partitions of 1 into fractions i/j with 1<=i

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A227611 Number of ways 2/n can be expressed as the sum of three distinct unit fractions: 2/n = 1/x + 1/y + 1/z with 0 < x < y < z.

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A227612 Table read by antidiagonals: Number of ways m/n can be expressed as the sum of three distinct unit fractions, i.e., m/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and read by antidiagonals.

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A325704 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

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A347566 Number of ways 1/n can be expressed as the sum of five distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t, with 0 < p < q < r < s < t.

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A342267 Number of partitions of 1/n into n reciprocals of positive integers.

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Examples

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Extensions

A347569 Number of ways 1/n can be expressed as the sum of six distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t + 1/u, with 0 < p < q < r < s < t < u.

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