A092670 -id:A092670 - cp's OEIS frontend

A020473 Egyptian fractions: number of partitions of 1 into reciprocals of positive integers <= n.

1, 2, 3, 5, 6, 13, 14, 24, 34, 60, 61, 168, 169, 252, 627, 1011, 1012, 2430, 2431, 7212, 15024, 16553, 16554, 50219, 60008, 64284, 92071, 260178, 260179, 844846, 844847, 1431187, 2608883, 2661217, 7946814, 22692855, 22692856, 22911815, 36004488, 120859171

Offset: 1

David W. Wilson

nonn

Number of ways to represent 1 = Sum_{k=1..n} b(k)/k, where the b(k) >= 0. - Franklin T. Adams-Watters, Aug 01 2006

Toshitaka Suzuki, Table of n, a(n) for n = 1..390
Index entries for sequences related to Egyptian fractions

Cf. A038034, A092666, A092670, A208480.

Table[Length[IntegerPartitions[1, All, 1/Range[n]]], {n, 1, 20}] (* Ben Branman, Apr 21 2012 *)

a(n) = Sum(A092666(i), i=1..n).

For prime p, a(p) = a(p-1) + 1. - Max Alekseyev, May 07 2012

A038034 Number of compositions (ordered partitions) of 1 into {1/1, 1/2, 1/3, ..., 1/n}.

Original entry on oeis.org

1, 2, 3, 7, 8, 52, 53, 288, 1209, 5247, 5248, 71395, 71396, 375779, 6957533, 52310862, 52310863, 1152622553, 1152622554, 45575902465, 1296407854551, 1580527987951, 1580527987952, 73245316681199, 584407520822198, 639887219617512, 11355804443049274, 516959218512416104, 516959218512416105, 29213061562205847736, 29213061562205847737, 886912328033731357358, 31286298736622399674197, 31349361777225437765677

Offset: 1

Christian G. Bower, Jun 15 1998

a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with max{x_i}<=n.

			a(4) = 7 since there are seven compositions into parts {1/1, 1/2, 1/3, 1/4}:
1 = 1/1, 1 = 1/2 + 1/2, 1 = 1/3 + 1/3 + 1/3, 1 = 1/2 + 1/4 + 1/4, 1 = 1/4 + 1/2 + 1/4, 1 = 1/4 + 1/4 + 1/2, and 1 = 1/4 + 1/4 + 1/4 + 1/4.

Index entries for sequences related to Egyptian fractions

Cf. A002967, A020473, A092667, A092670.

a(n) = Sum_{i=1..n} A092667(i).

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

More terms from Max Alekseyev, Mar 02 2004

A092669 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 5, 0, 11, 0, 0, 0, 19, 0, 0, 0, 73, 0, 86, 0, 0, 163, 0, 203, 286, 0, 0, 0, 803, 0, 1399, 0, 0, 2723, 0, 0, 4870, 0, 0, 0, 8789, 0, 13937, 14987, 42081, 0, 0, 0, 85577, 0, 0, 159982, 0, 117889, 437874, 0, 0, 0, 818640, 0

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

For a given n, the Mathematica program uses backtracking to count the solutions. The solutions can be printed by uncommenting the print statement. It is very time-consuming for large n. A092671 gives the n that yield a(n) > 0. - T. D. Noe, Mar 26 2004

			a(6) = 1 since there is the only fraction 1 = 1/2+1/3+1/6.

Toshitaka Suzuki, Table of n, a(n) for n = 1..610
Harry Ruderman and Paul Erdős, Problem E2427: Bounds of Egyptian fraction partitions of unity, Amer. Math. Monthly, Vol. 81, No. 7 (1974), 780-782.
Index entries for sequences related to Egyptian fractions

Cf. A092666, A092667, A092670, A092671, A092672, A006585.

n=20; try2[lev_, s_] := Module[{nmim, nmax, si, i}, AppendTo[soln, 0]; If[lev==1, nmin=2, nmin=1+soln[[ -2]]]; nmax=n-1; Do[If[iT. D. Noe, Mar 26 2004 *)

a(n) = A092670(n) - A092670(n-1).

More terms from T. D. Noe, Mar 26 2004

More terms from T. Suzuki (suzuki(AT)scio.co.jp), Nov 24 2006

A379335 Number of subsets of {-n..n} whose sum of reciprocals is 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 48, 96, 192, 384, 768, 1536, 5632, 11264, 22528, 77312, 154624, 309248, 922624, 1845248, 6848512, 17096576, 34193152, 68386304, 272849152

Offset: 1

Ilya Gutkovskiy, Dec 21 2024

Number of ways of writing 1 as Sum_{k=-n..n, k<>0} e(k)/k, where e(k) is 0 or 1.

			a(3) = 4 subsets: {1}, {-3, 1, 3}, {-2, 1, 2}, {-3, -2, 1, 2, 3}.

Cf. A092670.

from functools import cache
from fractions import Fraction
@cache
def b(i, s):
    if i == 0: return 1 if s == 1 else 0
    return b(i-1, s) + b(i-1, s+Fraction(1, (-1)**(i&1)*((i+1)>>1)))
a = lambda n: b(2*n, 0)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Dec 21 2024

a(n) <= 2*a(n-1) since we count s and s union {-1/n, 1/n} for each subset s counted in a(n-1); equality holds for n prime (and other cases). - Michael S. Branicky, Dec 21 2024

a(12)-a(24) from Michael S. Branicky, Dec 21 2024

A305442 Number of subsets of {1, 2, ..., n} such that the sum of the reciprocals is strictly less than 1.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 501, 918, 1686, 3110, 5724, 10543, 19435, 35857, 66198, 122294, 226135, 418351, 774372, 1434089, 2657205, 4925796, 9135403, 16949546, 31460330, 58415177, 108502732, 201603881, 374707879, 696649896, 1295562234, 2410000999

Offset: 0

Peter Kagey, Jun 01 2018

nonn

			For n = 4 the a(4) = 7 subsets are:
{}     because 0 < 1,
{2}    because 1/2 < 1,
{2, 3} because 1/2 + 1/3 = 5/6 < 1,
{2, 4} because 1/2 + 1/4 = 3/4 < 1,
{3}    because 1/3 < 1,
{3, 4} because 1/3 + 1/4 = 7/12 < 1, and
{4}    because 1/4 < 1.

Cf. A092669, A092670, A212657, A272083.

a[n_] := 1 + Length@ Select[Subsets[Range[2,n], {1, n-1}], Total[1/#] < 1  &]; Array[a, 15] (* Giovanni Resta, Jun 01 2018 *)

a(n) = A212657(n) - A092670(n).

a(26)-a(36) from Giovanni Resta, Jun 01 2018

A339569 Number of subsets of {1..n} whose cardinality is equal to the root-mean-square of the elements.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 10, 16, 32, 56, 90, 134, 186, 304, 476, 746, 1308, 2522, 4845, 9129, 17260, 32684, 59908, 106181, 191779, 337793, 596689, 1061991, 1907311, 3518903, 6426672, 12093858, 22777645, 42886411, 81002076, 151575988, 285280108, 529313088

Offset: 1

Ilya Gutkovskiy, Dec 09 2020

nonn

			a(12) = 10 subsets: {1}, {1, 2, 4, 5, 7, 11}, {1, 3, 5, 6, 8, 9}, {3, 4, 5, 6, 7, 9}, {1, 2, 3, 6, 7, 10, 12}, {2, 3, 4, 5, 8, 9, 12}, {2, 3, 6, 7, 8, 9, 10}, {3, 4, 5, 6, 7, 8, 12}, {1, 2, 5, 6, 9, 10, 11, 12} and {1, 4, 6, 7, 8, 9, 11, 12}.

Eric Weisstein's World of Mathematics, Root-Mean-Square

Cf. A092670, A339454, A339484.

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, sos, c):
  if n == 0:
    if c>0:
      if sos==c*c*c: return 1
    return 0
  return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 44)]) # Michael S. Branicky, Dec 10 2020

a(24)-a(43) from Michael S. Branicky, Dec 09 2020

cp's OEIS Frontend

A020473 Egyptian fractions: number of partitions of 1 into reciprocals of positive integers <= n.

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A038034 Number of compositions (ordered partitions) of 1 into {1/1, 1/2, 1/3, ..., 1/n}.

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A092669 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

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A379335 Number of subsets of {-n..n} whose sum of reciprocals is 1.

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A305442 Number of subsets of {1, 2, ..., n} such that the sum of the reciprocals is strictly less than 1.

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Programs

Mathematica

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Extensions

A339569 Number of subsets of {1..n} whose cardinality is equal to the root-mean-square of the elements.

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Author

Keywords

Examples

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Programs

Python

Extensions