A002966 -id:A002966 - cp's OEIS frontend

A369470 a(n) = number of integer solutions to 1 <= x1 <= x2 <= ... <= xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

Original entry on oeis.org

1, 1, 2, 35, 455, 13624, 1176579

Offset: 1

Max Alekseyev, Jan 23 2024

For any n, a(n) >= A369469(n) >= 1 (see A369607).

Max Muller et al., The diophantine equation \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right), MathOverflow, 2024.

Cf. A002966, A007850, A054377, A085098, A118086, A158649, A164014, A342267, A348625, A369469, A369607.

A181700 Smallest positive integer that can be represented as the sum of n of its distinct divisors in the maximum number of ways, or a(n)=0 if no such number exists.

Original entry on oeis.org

1, 0, 6, 2520, 48348686786400, 10543141534556403817127800577537146514577188497111149855093902265479066128013109211427715400552367011213513440000

Offset: 1

William Rex Marshall, Nov 06 2010

A006585(n) gives the number of representations of a(n) as the sum of n of its distinct divisors.

For n >= 4, a(n) appears to coincide with A357765(n). - Max Alekseyev, Oct 09 2022

			For n=4, a(4)=2520 has the six solutions: 60+360+840+1260, 105+315+840+1260, 126+504+630+1260, 140+280+840+1260, 168+252+840+1260, 210+420+630+1260.

Max Alekseyev, Table of n, a(n) for n = 1..7

Cf. A002966, A006585, A357765.

For n >= 3, a(n) = LCM of all denominators of Egyptian fractions enumerated by A006585(n). - Max Alekseyev, Oct 12 2022

Edited by Max Alekseyev, Oct 09 2022

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)).

A348536 Number of partitions of n into 3 parts that divide n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0

Offset: 1

Wesley Ivan Hurt, Oct 21 2021

From David A. Corneth, Oct 08 2022: (Start)
Proof of formula: suppose we have a partition d_1 + d_2 + d_3 = n where 0 < d_1 <= d_2 <= d_3 and d_1 | n, d_2 | n and d_3 | n.
Then let m_i * d_i = n for i in (1, 2, 3).
Dividing d_1 + d_2 + d_3 = n by n gives d_1/n + d_2/n + d_3/n = n/n, i.e., 1/m_1 + 1/m_2 + 1/m_3 = 1. There are 3 solutions to that equation namely (m_1, m_2, m_3) in {(2, 4, 4), (2, 3, 6), (3, 3, 3)}. The lcm of these numbers is 12 so the sequence is periodic with a period of 12 and then calculating the first 12 terms defines the sequence.
Alternative name: number of divisors of n from {3, 4, 6}. (End)

			a(12) = 3 via 2 + 4 + 6 = 3 + 3 + 6 = 4 + 4 + 4. - _David A. Corneth_, Oct 08 2022

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A002966, A069905, A355200.

Block[{c}, c[n_] := 1 - Ceiling[n] + Floor[n]; Array[Sum[Sum[c[#/j]*c[#/i]*c[#/(# - i - j)], {i, j, Floor[(# - j)/2]} ], {j, Floor[#/3]} ] &, 105]] (* Michael De Vlieger, Oct 21 2021 *)
Table[Count[IntegerPartitions[n,{3}],?(Mod[n,#]=={0,0,0}&)],{n,100}] (* _Harvey P. Dale, Apr 07 2025 *)

a(n) = [0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3][(n-1)%12 + 1] \\ David A. Corneth, Oct 08 2022

a(n) = { my(d = divisors(n), res = 0); d = d[^#d]; forvec(x = vector(2, i, [1, #d]), s = d[x[1]] + d[x[2]]; if(n - s >= d[x[2]], if(n % (n - s) == 0, print([d[x[1]], d[x[2]], n-s]); res++ ) ) , 1 ); res } \\ David A. Corneth, Oct 08 2022

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(n/j) * c(n/i) * c(n/(n-i-j)), where c(n) = 1 - ceiling(n) + floor(n).

a(n + 12) = a(n) where n >= 1. - David A. Corneth, Oct 08 2022

A130691 Number of distinct unit fractions required to sum to n when using the "splitting algorithm".

Original entry on oeis.org

1, 4, 16, 172, 4331, 232388, 4865293065, 40149851165417480, 18146043304242768613568943751063, 5522398183372890742378015411585945396419106762128927

Offset: 1

Hugo van der Sanden, Jun 10 2010, with contributions from Franklin T. Adams-Watters and Robert Gerbicz

The splitting algorithm decomposes a rational p/q to distinct unit fractions by first creating the multiset with p copies of 1/q, then repeatedly replacing a duplicated element 1/q' with the pair 1/(q'+1), 1/q'(q'+1) until no duplicates remain.

			For n=2, the algorithm generates the multisets {1/1, 1/1}, {1/1, 1/2, 1/2}, {1/1, 1/2, 1/3, 1/6}. The final multiset has no duplicate elements, so the algorithm terminates, and has 4 elements, so a(2)=4.

Hugo van der Sanden and others, Table of n, a(n) for n = 1..14
L. Beeckmans, The Splitting Algorithm for Egyptian Fractions, J. Number Th. 43, 173-185, 1993.
Hugo van der Sanden and others, Table of n, a(n) for n = 1..17 [Included as an "a-file", since the last three terms exceed the limit for terms in b-files.]

Cf. A002966. - Robert G. Wilson v, Jun 10 2010

A185074 Number of representations of n in the form sum(i=1..n, c(i)/i ), where each of the c(i)'s is in {0,1,...,n}.

Original entry on oeis.org

1, 2, 4, 16, 36, 447, 1274, 9443, 54094, 995169, 3013040, 79403971, 244277081, 5853252222, 171545158710, 2586069434760, 8747524457442, 290539678831816, 1002826545775653, 37782799964911391, 1405277934671848125, 53429557586727235246, 189496067102901557686

Offset: 1

John W. Layman, Mar 02 2012

nonn

			For n=3, 1/1+2/2+3/3 = 2/1+0/2+3/3 = 2/1+2/2+0/3 = 3/1+0/2+0/3 = 3 and no other sums of the required type give 3, so a(3)=4.  For n=4, 0/1+4/2+3/3+4/4 and 15 other sums of the required type give 4, so a(4)=16.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..26

Cf. A002966, A020473.

b:= proc(r, i, n) option remember;
      `if`(r=0, 1, `if`(i>n, 0,
      add(b(r-j/i, i+1, n), j=0..min(n, r*i))))
    end:
a:= n-> b(n, 1, n):
seq(a(n), n=1..10);  # Alois P. Heinz, Mar 06 2012

b[r_, i_, n_] := b[r, i, n] = If[r == 0, 1, If[i>n, 0, Sum[b[r-j/i, i+1, n], {j, 0, Min[n, r*i]}]]]; a[n_] := b[n, 1, n]; Table[Print[a[n]]; a[n], {n, 1, 13}] (* Jean-François Alcover, Feb 27 2014, after Alois P. Heinz *)

A185074(n,i=1,m)={n || return(1); m || m=n; i>m & return; sum(j=0,min(m, n*i),A185074(n-j/i, i+1, m))} \\ - M. F. Hasler, Mar 07 2012

/* version with memoization - seems not faster */ R185074=Set("[0]"); A185074(n,i=1,m)={n || return(1); m || m=n; i>m & return; my(t=eval(R185074[setsearch(R185074,[n,i,m],1)-1])); t[1]==n & t[2]==i & t[3]==m & return(t[4]); t=sum(j=0,min(m, n*i),A185074(n-j/i, i+1, m)); R185074=setunion(R185074,Set([[n,i,m,t]])); t} \\ - M. F. Hasler, Mar 07 2012

a(7)-a(10) from R. J. Mathar, a(11)-a(13) from Alois P. Heinz, Mar 06 2012
a(14) from Alois P. Heinz, Sep 27 2014
a(15)-a(23) from Hiroaki Yamanouchi, Oct 03 2014

A316889 Heinz numbers of aperiodic integer partitions whose reciprocal sum is 1.

Original entry on oeis.org

2, 147, 195, 3185, 6475, 6591, 7581, 10101, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 107653, 123823, 142805, 164255, 164983, 171941, 218855, 228085, 267883, 304175, 312785, 333925, 333935, 335405, 343735, 355355, 390963, 414295, 442975, 444925, 455975

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.

			Sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (4,4,2), (6,3,2), (6,4,4,3), (12,4,3,3), (6,6,6,2), (8,8,4,2), (12,6,4,2), (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, A058360, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

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

A316891 Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 5, 2, 7, 4, 7, 6, 13, 7, 18, 12, 20, 17, 32, 20, 39, 31, 47, 45, 74, 56, 96, 83, 109, 105, 151, 130, 199, 183, 234, 232, 319, 286, 404, 386, 473, 488, 638, 599, 782, 767, 931, 960, 1197, 1165, 1465, 1477, 1747, 1814, 2212, 2196

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

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

A partition is aperiodic if its multiplicities are relatively prime.

			The a(17) = 13 partitions:
(6443),
(44441),
(3332222), (6322211),
(44222111),
(222222221), (333221111), (632111111),
(4421111111),
(22222211111), (33311111111),
(2222111111111),
(221111111111111).

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

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

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]==1,IntegerQ[Sum[1/m,{m,#}]]]&]],{n,50}]

a(51)-a(60) from Alois P. Heinz, Jul 18 2018

cp's OEIS Frontend

A369470 a(n) = number of integer solutions to 1 <= x1 <= x2 <= ... <= xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

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A181700 Smallest positive integer that can be represented as the sum of n of its distinct divisors in the maximum number of ways, or a(n)=0 if no such number exists.

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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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Extensions

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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Maple

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A348536 Number of partitions of n into 3 parts that divide n.

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PARI

PARI

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A130691 Number of distinct unit fractions required to sum to n when using the "splitting algorithm".

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A185074 Number of representations of n in the form sum(i=1..n, c(i)/i ), where each of the c(i)'s is in {0,1,...,n}.

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