cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

David W. Wilson

Keywords

Comments

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

Links

Crossrefs

Cf. A038034, A092666, A092670, A208480.

Programs

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

Formula

a(n) = Sum(A092666(i), i=1..n).
For prime p, a(p) = a(p-1) + 1. - Max Alekseyev, May 07 2012

A212606 Number of distinct sums <= 1 of reciprocals of positive integers <= n.

Original entry on oeis.org

1, 2, 3, 6, 10, 26, 34, 103, 175, 393, 599, 2015, 2551, 8681, 14254, 19620, 34700, 129557, 161272, 595304, 695175, 1094164, 1903859, 7654850, 9413484, 29625309
Offset: 0

Views

Author

Max Alekseyev, May 22 2012

Keywords

Examples

			a(3) = 6 counts numbers { 0, 1/3, 1/2, 2/3, 5/6, 1 }, each of which is can be represented as the sum of reciprocals 1/1, 1/2, and 1/3.

Crossrefs

For distinct sums of distinct reciprocals, see A212607.
Cf. A020473, A092669, A092671, A208480

Extensions

a(24)-a(25) from Dexter Senft, Feb 07 2019

A212607 Number of distinct sums <= 1 of distinct reciprocals of integers <= n.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 21, 38, 70, 129, 238, 440, 504, 949, 1790, 2301, 4363, 8272, 12408, 23604, 26675, 45724, 87781, 168549, 181989, 351076, 677339, 1306894, 1399054, 2709128, 2795144, 5423805, 10525050
Offset: 0

Views

Author

Max Alekseyev, May 22 2012

Keywords

Examples

			a(3) = 5 counts numbers { 0, 1/3, 1/2, 5/6, 1 }, each of which is can be represented as the sum of distinct reciprocals 1/1, 1/2, and 1/3.

Crossrefs

For possibly non-distinct reciprocals, see A212606.
Cf. A020473, A092669, A092671, A208480.

Programs

  • Maple
    s:= proc(n) option remember;
      `if`(n=0, {0}, map(x-> `if`(n-1 nops(s(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, May 23 2012
  • Mathematica
    s[_] := s[n] = If[n == 0, {0}, If[n-1 < n*#, #, {#, # + 1/n}]& /@ s[n-1] // Flatten];
a[n_] := Length[s[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 32}] (* Jean-François Alcover, May 13 2019, after Alois P. Heinz *)

Extensions

a(27)-a(32) from Alois P. Heinz, May 23 2012

A212657 Number of subsets of {1,2,...,n} with the sum of reciprocals <= 1.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 26, 46, 83, 151, 276, 503, 921, 1689, 3113, 5730, 10549, 19441, 35868, 66209, 122316, 226157, 418373, 774394, 1434130, 2657246, 4925837, 9135444, 16949660, 31460444, 58415377, 108502932, 201604081, 374708242, 696650259, 1295562800, 2410001851, 4484208954, 8345621293
Offset: 0

Views

Author

Max Alekseyev, May 23 2012

Keywords

Comments

The number of distinct sums of reciprocals is given by A212607.
a(n) grows as 2^(b*n) with b=0.911... (Tikhomirov et al. 2017).

Links

Crossrefs

Cf. A212658 (reciprocals can appear multiple times).
Cf. A212606, A212607, A020473, A092669, A092671, A208480.

Programs

  • PARI
    { A212657(n) = my(L=lcm(vector(n,i,i))); polcoeff( prod(i=1,n, 1+x^(L/i)+O(x^(L+1)) )/(1-x), L); }

Extensions

a(32)-a(35) from Alois P. Heinz, May 23 2012
a(36)-a(95) from Robert Gerbicz, May 23 2012

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

Views

Author

Ilya Gutkovskiy, Dec 09 2024

Keywords

Examples

			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].

Links

Crossrefs

Cf. A020473, A038034, A208480.

Programs

  • Maple
    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
  • Python
    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

Extensions

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

Views

Author

Max Alekseyev, May 23 2012

Keywords

Comments

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

Links

Crossrefs

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

Extensions

a(24)-a(25) from Alois P. Heinz, Nov 20 2017
a(26)-a(31) from Dexter Senft, Feb 07 2019
Showing 1-6 of 6 results.

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