A051908 -id:A051908 - cp's OEIS frontend

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!.

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)).

A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 5, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 5, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 7, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 2, 3

Offset: 1

Lars Blomberg, Feb 14 2022

nonn

By symmetry, if (i, j) is a solution then so is (j, i). When j=i we get n = 3i, corresponding to the solution 1/2 + 1/2 = 1. Therefore, when 3|n, a(n) > 0 and odd, otherwise a(n) >= 0 and even.
For n < 10^6, the largest term is a(720720) = 285, and 483188 terms are 0.
Other record terms: a(1081080) = 369, a(2162160) = 457, a(3243240) = 481, a(4324320) = 533, a(5405400) = 559, a(6126120) = 597. Record terms with index >= 360360 appear to occur at indices that are multiples of 180180. - Chai Wah Wu, Feb 15 2022

			For n = 3: (i, j) = (1, 1), so a(3) = 1. (1/2 + 1/2 = 1)
For n = 18: (i, j) = (3, 8), (6, 6), (8, 3), so a(18) = 3. (3/15 + 8/10 = 1/5 + 4/5 = 1)
For n = 20: (i, j) = (5, 8), (8, 5), so a(20) = 2.
For n = 36: (i, j) = (6, 16), (8, 15), (12, 12), (15, 8), (16, 6), so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A018892, A016017, A051908, A073546, A227610, A238795, A241883, A297895, A303400.

Cf. A351694, A351695 for records.

a(n)={my(x=n^2, y=2*n); sum(i=1,(n-1)/2, x-=2*n; y-=3; if(x%y==0,1,0))}

from sympy.abc import i, j
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A351532(n):
    return sum(1 for d in diop_quadratic(n**2+3*i*j-2*n*(i+j)) if 0 < d[0] < n and 0 < d[1] < n) # Chai Wah Wu, Feb 15 2022

The original equation can be solved for j giving j = (n(n - 2i)) / (2n - 3i). Varying i from 1 to n-1, a(n) is given by the number of integer values of j, 0 < j < n.

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A270599 Number of ways to express 1 as the sum of unit fractions with odd denominators such that the sum of those denominators is n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0

Offset: 1

Seiichi Manyama, Mar 26 2016

nonn

Number of partitions of n into such odd parts that the sum of their reciprocals is one. - Antti Karttunen, Jul 23 2018

It would be nice to know whether nonzero values may occur only on n of the form 8k+1.

			1 = 1/3 + 1/3 + 1/3, the sum of denominators is 9, this is the only expression of 1 as unit fractions with odd denominators that sum to 9, so a(9)=1.
1 = 1/15 + 1/5 + 1/5 + 1/5 + 1/3 = 1/9 + 1/9 + 1/9 + 1/3 + 1/3 are the only solutions with odd denominators that sum to 33, thus a(33) = 2. - _Antti Karttunen_, Jul 24 2018

David A. Corneth, Table of n, a(n) for n = 1..370 (terms 1..150 from Seiichi Manyama, terms 150..273 from Antti Karttunen)
David A. Corneth, Tuples up to n = 370.
Index entries for sequences related to Egyptian fractions

Cf. A000009, A051908.

Cf. also A201644, A201646, A201647, A201648, A201649.

Array[Count[IntegerPartitions[#, All, Range[1, #, 2]], ?(Total[1/#] == 1 &)] &, 70] (* _Michael De Vlieger, Jul 26 2018 *)

A270599(n,maxfrom=n,fracsum=0) = if(!n,(1==fracsum),my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599(n-k,min(k,n-k),tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 23 2018

\\ More verbose version for computing values of a(n) for large n:
A270599(n) = if(!(n%2), 0, my(s=0); forstep(k = n, 1, -2, print("A270599(", n, ") at toplevel, k=", k, " s=", s); s += A270599aux(n-k, min(k, n-k), 1/k)); (s));
A270599aux(n,maxfrom,fracsum) = if(!n,(1==fracsum),my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599aux(n-k,min(k,n-k),tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 24 2018

def f(n)
  n - 1 + n % 2
end
def partition(n, min, max)
  return [[]] if n == 0
  [f(max), f(n)].min.step(min, -2).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A270599(n)
  ary = [1]
  (2..n).each{|m|
    cnt = 0
    partition(m, 2, m).each{|ary|
      cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1
    }
    ary << cnt
  }
  ary
end

a(2*k) = 0. - David A. Corneth, Jul 24 2018

Name corrected by Antti Karttunen, Jul 23 2018 at the suggestion of David A. Corneth

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]&]

A316898 Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 4, 1, 3, 1, 1, 1, 3, 1, 8, 3, 1, 1, 9, 2, 11, 3, 3, 3, 5, 2, 7, 6, 4, 7, 12, 5, 14, 6, 11, 12, 25, 11, 27, 17, 15, 19, 25, 9, 37, 20, 21, 19, 31, 19, 38, 33, 26, 37, 38, 36, 64, 39, 46, 53, 63, 39, 80, 63, 65, 66, 94, 59, 105

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

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

Records: 1, 2, 4, 8, 9, 11, 12, 14, 25, 27, 37, 38, 64, 80, 94, 105, 119, 154, 184, ..., . - Robert G. Wilson v, Jul 18 2018

			The a(37) = 8 partitions: (20,12,5), (15,12,10), (24,8,3,2), (15,10,6,6), (20,5,4,4,4), (15,10,6,3,3), (14,7,7,7,2), (10,10,10,5,2).

Robert G. Wilson v, Table of n, a(n) for n = 1..150
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888, A316889, A316890, A316891, A316892, A316893, A316904.

f[n_] := Block[{k = 1, s = 0, lmt = 1 + PartitionsP@ n}, While[k < lmt, s += Length[ Select[ IntegerPartitions[n, {k, k}], GCD @@ # == 1 && IntegerQ[1/Sum[1/m, {m, #}]] &]]; k++]; s]; Array[f, 50] (* slightly modified by Robert G. Wilson v, Jul 17 2018 *) (* or *)
ric[n_,p_,s_] := If[n==0, If[IntegerQ[1/s] && GCD @@ p == 1, 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, 50] (* Giovanni Resta, Jul 18 2018 *)

a(51)-a(91) from Robert G. Wilson v, Jul 17 2018

A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 1, 1, 6, 120, 2, 720, 24, 3, 1, 5040, 1, 40320, 6, 24, 120, 362880, 2, 3, 720, 2, 24, 3628800, 3, 39916800, 1, 120, 5040, 24, 1, 479001600, 40320, 720, 6, 6227020800, 24, 87178291200, 120, 6, 362880, 1307674368000, 2, 12, 3, 5040, 720

Offset: 1

Gus Wiseman, May 18 2019

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

Robert Israel, Table of n, a(n) for n = 1..3168
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

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, A325704.

f:= proc(n) local F,t;
    F:= ifactors(n)[2];
    denom(add(t[2]/numtheory:-pi(t[1])!,t=F))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024

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

a(n) = A318574(A325709(n)).

A339627 Number of partitions of n into distinct parts such that the sum of reciprocals of parts is an integer.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 4, 5, 4, 7, 4, 2, 2, 6, 6, 1, 3, 3, 4, 3, 4

Offset: 0

Ilya Gutkovskiy, Dec 10 2020

nonn

Also the number of ways to express an integer as the sum of distinct unit fractions such that the sum of the denominators is n.

			a(31) = 2 because we have 2 + 4 + 5 + 20 = 1 + 2 + 3 + 10 + 15 = 31 and 1/2 + 1/4 + 1/5 + 1/20 = 1, 1/1 + 1/2 + 1/3 + 1/10 + 1/15 = 2 are integers.

Cf. A051907, A051908, A058360, A339628, A339629.

A339628 Number of compositions (ordered partitions) of n into distinct parts such that the sum of reciprocals of parts is an integer.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 120, 0, 0, 0, 0, 24, 144, 144, 120, 0, 0, 0, 24, 240, 720, 0, 0, 0, 120, 720, 240, 1440, 0, 0, 0, 840, 5760, 720, 5160, 744, 240, 720, 1440, 10080, 720, 5760, 5160, 1560, 5760, 2280, 16560, 6000, 7560

Offset: 0

Ilya Gutkovskiy, Dec 10 2020

nonn

Also the number of ordered ways to express an integer as the sum of distinct unit fractions such that the sum of the denominators is n.

			a(11) = 6 because we have 2 + 3 + 6 = 2 + 6 + 3 = 3 + 2 + 6 = 3 + 6 + 2 = 6 + 2 + 3 = 6 + 3 + 2 = 11 and 1/2 + 1/3 + 1/6 = 1 is an integer.

Cf. A051907, A051908, A058360, A339627, A339629.

A339629 Number of compositions (ordered partitions) of n into distinct parts such that the sum of reciprocals of parts is 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 24, 24, 24, 0, 0, 0, 0, 24, 120, 0, 0, 0, 0, 120, 0, 240, 0, 0, 0, 0, 840, 0, 720, 120, 24, 120, 0, 1440, 0, 720, 720, 120, 840, 0, 2280, 720, 960, 1080, 0, 840, 0, 11760, 0, 5040, 1440, 720

Offset: 0

Ilya Gutkovskiy, Dec 10 2020

nonn

Also the number of ordered ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.

			a(11) = 6 because we have 2 + 3 + 6 = 2 + 6 + 3 = 3 + 2 + 6 = 3 + 6 + 2 = 6 + 2 + 3 = 6 + 3 + 2 = 11 and 1/2 + 1/3 + 1/6 = 1.

Cf. A051907, A051908, A058360, A339627, A339628.

A316113 a(n) is the least sum of a tuple containing n as an element where the denominator of the sum of reciprocals isn't divisible by any of the prime factors of n.

Original entry on oeis.org

1, 4, 9, 10, 20, 11, 35, 22, 30, 22, 66, 22, 91, 37, 28, 46, 119, 32, 152, 31, 41, 68, 184, 37, 110, 106, 93, 53, 261, 51, 279, 94, 69, 121, 59, 60, 370, 192, 97, 64, 451, 54, 430, 94, 65, 186, 517, 74, 259, 112, 122, 97, 583, 95, 92, 92, 158, 263, 767, 77, 671

Offset: 1

David A. Corneth, Jul 22 2018

nonn

			a(12) = 22 because the tuple [3, 3, 4, 12] has the sum of reciprocals 1/3 + 1/3 + 1/4 + 1/12 = 1 of which the denominator is 1 and has no common prime factors with n = 12.
a(25) = 110 because the tuple [5, 5, 25, 25, 50] has the sum of reciprocals 1/5 + 1/5 + 1/25 + 1/25 + 1/50 = 1/2 of which the denominator is 2 and has no common prime factors with n = 25.

Ray Chandler, Table of n, a(n) for n = 1..100

Cf. A051908, A201644, A201646, A201647, A201648, A201649.

More terms from Ray Chandler, Oct 02 2018

cp's OEIS Frontend

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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A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

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A270599 Number of ways to express 1 as the sum of unit fractions with odd denominators such that the sum of those denominators is n.

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A316890 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is 1.

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A316898 Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

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A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

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A339627 Number of partitions of n into distinct parts such that the sum of reciprocals of parts is an integer.

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A339628 Number of compositions (ordered partitions) of n into distinct parts such that the sum of reciprocals of parts is an integer.