A020473 -id:A020473 - cp's OEIS frontend

A378271 Number of partitions of 1 into {1/1^3, 1/2^3, 1/3^3, ..., 1/n^3}.

1, 2, 3, 11, 12, 435, 436, 6748, 42360, 1252676, 1252677, 302302546, 302302547

Offset: 1

Ilya Gutkovskiy, Nov 21 2024

			a(4) = 11 because we have 64 * (1/64) = 56 * (1/64) + 1/8 = 48 * (1/64) + 2 * (1/8) = 40 * (1/64) + 3 * (1/8) = 32 * (1/64) + 4 * (1/8) = 24 * (1/64) + 5 * (1/8) = 16 * (1/64) + 6 * (1/8) = 8 * (1/64) + 7 * (1/8) = 27 * (1/27) = 8 * (1/8) = 1.

Index entries for sequences related to Egyptian fractions

Cf. A000578, A020473, A038034, A378270.

a(p) = a(p-1) + 1 for prime p. - Jinyuan Wang, Dec 11 2024

a(9)-a(13) from Jinyuan Wang, Dec 11 2024

A378842 Number of compositions (ordered partitions) of n into reciprocals of positive integers <= n.

Original entry on oeis.org

1, 1, 5, 154, 127459, 1218599617, 2319241469466200, 32824171395278825785183, 115384552858168166552304749413033, 22529589324775724210737089575811718669447945, 1255772217551224641521320538899160332818484462756697922572, 885355014578065534254256068634855343582928219947780981811219956595305584

Offset: 0

Ilya Gutkovskiy, Dec 09 2024

nonn

			a(2) = 5 because we have [1/2, 1/2, 1/2, 1/2], [1/2, 1/2, 1], [1/2, 1, 1/2], [1, 1/2, 1/2] and [1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..15

Cf. A020473, A038034, A208480.

b:= proc(n, r) option remember; `if`(r=0, 1,
      add(`if`(r*j<1, 0, b(n, r-1/j)), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..10);  # Alois P. Heinz, Dec 12 2024

from functools import lru_cache
from fractions import Fraction
def A378842(n):
    @lru_cache(maxsize=None)
    def f(r): return 1 if r==0 else sum(f(r-Fraction(1,j)) for j in range(int(Fraction(1,r))+(r.numerator!=1),n+1))
    return f(n) # Chai Wah Wu, Dec 14 2024

More terms from Alois P. Heinz, Dec 12 2024

A212658 Number of multisets {1^k1, 2^k2, ..., n^kn}, ki >= 0, with the sum of reciprocals <= 1.

Original entry on oeis.org

1, 2, 4, 8, 17, 37, 86, 199, 475, 1138, 2769, 6748, 16613, 40904, 101317, 251401, 624958, 1555940, 3882708, 9701790, 24276866, 60817940, 152508653, 382828565, 961859364, 2418662434, 6086480305, 15327208770, 38622901484, 97384378728, 245686368946, 620158662562

Offset: 0

Max Alekseyev, May 23 2012

nonn

The number of distinct sums of reciprocals is given by A212606.

Dexter Senft, Table of n, a(n) for n = 0..36

Cf. A212657, A212606, A212607, A020473, A092669, A092671, A208480, A003418.

a(24)-a(25) from Alois P. Heinz, Nov 20 2017

a(26)-a(31) from Dexter Senft, Feb 07 2019

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

A333437 Triangle read by rows: T(n,k) is the number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 3, 2, 1, 0, 0, 0, 0, 2, 2, 3, 2, 1, 0, 0, 0, 1, 3, 6, 7, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 8, 15, 21, 24, 20, 11, 4, 1

Offset: 1

Seiichi Manyama, Mar 24 2020

			1 = 1/2 + 1/6 + 1/6 + 1/6 = 1/3 + 1/3 + 1/6 + 1/6 = 1/3 + 1/4 + 1/4 + 1/6. So T(6,4) = 3.
Triangle begins:
n\k  | 1  2  3  4  5   6   7   8   9  10 11 12
-----+----------------------------------------
   1 | 1;
   2 | 0, 1;
   3 | 0, 0, 1;
   4 | 0, 0, 1, 1;
   5 | 0, 0, 0, 0, 1;
   6 | 0, 0, 1, 3, 2,  1;
   7 | 0, 0, 0, 0, 0,  0,  1;
   8 | 0, 0, 0, 1, 3,  3,  2,  1;
   9 | 0, 0, 0, 0, 2,  2,  3,  2,  1;
  10 | 0, 0, 0, 1, 3,  6,  7,  5,  3,  1;
  11 | 0, 0, 0, 0, 0,  0,  0,  0,  0,  0, 1;
  12 | 0, 0, 0, 3, 8, 15, 21, 24, 20, 11, 4, 1;

Index entries for sequences related to Egyptian fractions

Row sums give A092666.

Cf. A020473, A333496.

T(n,n) = 1.

If n is prime, T(n,k) = 0 for 1 <= k < n.

A333496 Least k of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 7, 4, 5, 4, 11, 4, 13, 5, 4, 5, 17, 4, 19, 4, 5, 7, 23, 4, 8, 8, 6, 5, 29, 5, 31, 6, 5, 10, 5, 5, 37

Offset: 1

Seiichi Manyama, Mar 24 2020

			One of solutions
  n  |  [x_1, x_2, ... , x_a(n)]
-----+--------------------------------------
   4 | [2,  4,  4]
   6 | [2,  3,  6]
   8 | [2,  4,  8,  8]
   9 | [2,  6,  9,  9,  9]
  10 | [2,  5,  5, 10]
  12 | [2,  3, 12, 12]
  14 | [2,  7,  7,  7, 14]
  15 | [2,  3, 10, 15]
  16 | [2,  4,  8, 16, 16]
  18 | [2,  3,  9, 18]
  20 | [2,  4,  5, 20]
  21 | [2,  3, 14, 21, 21]
  22 | [2, 11, 11, 11, 11, 11, 22]
  24 | [2,  3,  8, 24]
  25 | [2,  4, 20, 25, 25, 25, 25, 25]
  26 | [2, 13, 13, 13, 13, 13, 13, 26]
  27 | [2,  3, 18, 27, 27, 27]
  28 | [2,  3, 12, 21, 28]
  30 | [2,  3, 10, 30, 30]
  32 | [2,  3, 16, 24, 32, 32]
  33 | [2,  3, 11, 22, 33]
  34 | [2, 17, 17, 17, 17, 17, 17, 17, 17, 34]
  35 | [2,  3, 14, 15, 35]
  36 | [2,  3,  9, 36, 36]

Index entries for sequences related to Egyptian fractions

Cf. A020473, A092666, A333437.

a(n) <= n.
a(m * n) <= a(n) + m - 1.
If p is prime, a(p) = p.
If m is odd, a(2 * m) <= (m - 1)/2 + 2 because 1 = 1/2 + (m - 1)/2 * 1/m + 1/(2 * m).

A378269 Number of partitions of 1 into {1/1, 1/3, 1/5, ..., 1/(2*n-1)}.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 28, 29, 30, 90, 91, 150, 294, 295, 296, 659, 2818, 2819, 4815, 4816, 4817, 26648, 26649, 38880, 55745, 55746, 247660, 322628, 322629, 322630, 1942493, 7597991, 7597992

Offset: 1

Ilya Gutkovskiy, Nov 21 2024

			a(5) = 7 because we have 9 * (1/9) = 6 * (1/9) + 1/3 = 3 * (1/9) + 2 * (1/3) = 7 * (1/7) = 5 * (1/5) = 3 * (1/3) = 1.

Index entries for sequences related to Egyptian fractions

Cf. A005408, A020473, A038034.

If 2*n-1 is prime, then a(n) = a(n-1) + 1. - Chai Wah Wu, Dec 26 2024

a(20)-a(34) from Jinyuan Wang, Dec 11 2024

A379528 Number of compositions (ordered partitions) of 1 into {1/1^2, 1/2^2, 1/3^2, ..., 1/n^2}.

Original entry on oeis.org

1, 2, 3, 97, 98, 40917543, 40917544, 2901109178066823, 81221415992592163051371926, 373220766236315864054296758124337507430, 373220766236315864054296758124337507431

Offset: 1

Ilya Gutkovskiy, Dec 24 2024

Index entries for sequences related to Egyptian fractions

Cf. A000290, A020473, A038034, A348625, A378270, A378842.

a(p) = a(p-1) + 1 for p prime. - Chai Wah Wu, Dec 27 2024

a(6)-a(11) from Alois P. Heinz, Dec 26 2024

A275666 Multisets of numbers such that the sum of reciprocals is 1 and each element e occurs at most lpf(e) - 1 times.

Original entry on oeis.org

2, 3, 6, 2, 5, 5, 10, 3, 5, 5, 6, 10, 2, 4, 6, 12, 3, 3, 4, 12, 4, 5, 5, 6, 10, 12, 2, 7, 7, 7, 14, 3, 6, 7, 7, 7, 14, 4, 6, 7, 7, 7, 12, 14, 5, 5, 7, 7, 7, 10, 14, 2, 3, 10, 15, 2, 4, 10, 12, 15, 2, 5, 6, 15, 15, 3, 3, 5, 15, 15, 3, 3, 6, 10, 15, 3, 5, 5, 5, 15, 3, 4

Offset: 1

David A. Corneth, Aug 23 2016

lpf(n) gives the least prime factor of n (A020639(n)).
The multisets are ordered primarily by largest element, then secondly by length, then thirdly by smallest distinct element.
Sets are placed in nondecreasing order. The separation between two sets is where an element a(n) > a(n + 1).
Let M be the largest element of a multiset. No primes >= (M + 1)/2 are in that multiset.
This sequence may be used to find multisets that have the sum of reciprocals an integer, along with the following operations.
In a set we can replace a number by twice its double. For example, in (2, 3, 6), we can replace the 6 by two twelves, giving {2, 3, 12, 12}. The sum of reciprocals is still 1. However 12 occurs twice in the set, more than lpf(12) - 1 = 1.
{3, 4, 4, 6} isn't included as we could replace two fours by one 2 to get {2, 3, 6}. To exclude such possibilities, each element e is in a multiset at most lpf(e) - 1.
We can also take the union of two sets, for example the sets {2, 3, 6} and {2, 5, 5, 10} giving {2, 2, 3, 5, 5, 6, 10} which has sum of reciprocals 1 + 1 = 2.
We can put 1 in a set, so from {2, 3, 6} we can find {1, 2, 3, 6}.
Using any such operations, we get out of the sequence but more terms of A034708 can be found and A020473 may be investigated further.

			{2, 3, 6}
{2, 5, 5, 10}
{3, 5, 5, 6, 10}
{2, 4, 6, 12}
{3, 3, 4, 12}
{4, 5, 5, 6, 10, 12}
{2, 7, 7, 7, 14}
{3, 6, 7, 7, 7, 14}
{4, 6, 7, 7, 7, 12, 14}
{5, 5, 7, 7, 7, 10, 14}
{2, 3, 10, 15}
{2, 4, 10, 12, 15}
{2, 5, 6, 15, 15}
{3, 3, 5, 15, 15}
{3, 3, 6, 10, 15}
{3, 5, 5, 5, 15}
...
{3, 3, 4, 12} comes before {2, 7, 7, 7, 14} because the largest element of the first is less than the one from the second.
{2, 5, 5, 10} comes before {3, 5, 5, 6, 10} because they both have the largest element 10 but the latter has more elements.
{2, 4, 10, 12, 15} comes before {2, 5, 6, 15, 15} because they both have the largest element 15 and the same number of elements but the first smallest different element, 4 resp. 5, is less for the first.

Cf. A020473, A020639, A034708, A091784.

A377284 Number of partitions of 1 into {1/1^n, 1/2^n, 1/3^n, ..., 1/n^n}.

Original entry on oeis.org

1, 2, 3, 19, 36, 522332, 6117036, 1183731130981

Offset: 1

Ilya Gutkovskiy, Dec 12 2024

			a(3) = 3 because we have 27 * (1/27) = 8 * (1/8) = 1.

Index entries for sequences related to Egyptian fractions

Cf. A020473, A378270, A378271.

a(6)-a(8) from Jinyuan Wang, Dec 13 2024

cp's OEIS Frontend

A378271 Number of partitions of 1 into {1/1^3, 1/2^3, 1/3^3, ..., 1/n^3}.

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A378842 Number of compositions (ordered partitions) of n into reciprocals of positive integers <= n.

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A212658 Number of multisets {1^k1, 2^k2, ..., n^kn}, ki >= 0, with the sum of reciprocals <= 1.

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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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A333437 Triangle read by rows: T(n,k) is the number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.

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A333496 Least k of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.

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A378269 Number of partitions of 1 into {1/1, 1/3, 1/5, ..., 1/(2*n-1)}.

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A379528 Number of compositions (ordered partitions) of 1 into {1/1^2, 1/2^2, 1/3^2, ..., 1/n^2}.

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A275666 Multisets of numbers such that the sum of reciprocals is 1 and each element e occurs at most lpf(e) - 1 times.

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