A157248 -id:A157248 - cp's OEIS frontend

A258252 Sequence of distinct positive integers having lowest possible denominators of sums of 1/a(n).

1, 2, 6, 3, 4, 12, 15, 10, 14, 35, 5, 30, 42, 7, 8, 24, 18, 9, 33, 88, 40, 60, 84, 63, 99, 22, 26, 143, 11, 154, 238, 51, 21, 28, 20, 55, 66, 78, 91, 56, 72, 90, 110, 132, 156, 13

Offset: 1

Ivan Neretin, May 24 2015

nonn

a(n) is chosen among the unused positive integers so that the denominator of Sum_{i=1..n} 1/a(i), is as low as possible.
Presumably a permutation of positive integers.
Primes do not always occur in natural order.
The numbers which retain their natural positions (that is, a(n)=n) are 1, 2, 48, 80601...
Inverse (A258253): 1, 2, 4, 5, 11, 3, 14, 15, 18, 8, 29, 6, 46, 9, 7, 47, 73, 17, 134, 35, 33, 26, 153, 16, 96, ..., . - Robert G. Wilson v, Jun 18 2015
Records: 1, 2, 6, 12, 15, 35, 42, 88, 99, 143, 154, 238, 260, 460, 544, 840, 1645, 1666, 2109, 2622, 3876, 4599, 5644, 6565, 6734, 8701, 9492, 10272, ..., . - Robert G. Wilson v, Jun 18 2015

			After 2 terms, the partial sum of 1/a(i) reaches 3/2. Adding 1 or 1/2 is impossible, since 1 and 2 are already taken. The rest of positive integers lead to the following sums: +1/3 -> 11/6, +1/4 -> 7/4, +1/5 -> 17/10, +1/6 -> 5/3 with denominator 3 which is the lowest we can get. Hence a(3)=6.
For this specific term, the fractions that are encountered are 3/2 + 1/k with k>2. The resulting sequence of denominators are: 6, 4, 10, 3, 14, 8, 18, 5, 22, 12, ... (see A145979) within which the smallest term is indeed 3. - _Michel Marcus_, Jun 04 2015

Ivan Neretin, Table of n, a(n) for n = 1..25000
Ivan Neretin, C++ program generating this sequence and A258254

Cf. A157248 (another reordering of the harmonic series), A258253 (putative inverse), A258254 (denominators of partial sums of 1/a(n)), A258255 (positions where partial sums reach integers).

f[lst_] := Block[{c = 0, d, dk, k, mk, mn = Infinity, t = Total[1/lst]}, d = Denominator@ t; k = d; While[c < 101, If[ !MemberQ[lst, k], c++; dk = Denominator[t + 1/k]; If[dk < mn, mn = dk; mk = k]]; k += d]; Append[lst, mk]]; Nest[f, {}, 60] (* Robert G. Wilson v, Jun 18 2015 *)

A375802 Lexicographically earliest sequence of positive integers such that the sum of the inverses of the indices where the sequence has the same value is at most 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 3, 3, 3, 3, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6

Offset: 1

Rémy Sigrist, Aug 29 2024

In other words, if a(n_1) = ... = a(n_k) with n_1 < ... < n_k then 1/n_1 + ... + 1/n_k <= 1.

Each positive integer appears in the sequence, a finite number of times. This is a consequence of the fact that the greedy algorithm for Egyptian fractions terminates in a finite number of steps for any rational starting value.

			The first terms, alongside the sums of the inverses of the indices so far where the sequence has the same value, are:
  n   a(n)  Sums
  --  ----  ---------
   1     1  1
   2     2  1/2
   3     2  5/6
   4     3  1/4
   5     3  9/20
   6     2  1
   7     3  83/140
   8     3  201/280
   9     3  2089/2520
  10     3  2341/2520

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Wikipedia, Greedy algorithm for Egyptian fractions
Index entries for sequences related to Egyptian fractons

Cf. A157248, A323725.

{ b = vector(6); for (n = 1, 87, for (v = 1, oo, if (b[v] + 1/n <= 1, b[v] += 1/n; print1 (v", "); break;););); }

from fractions import Fraction
from itertools import count, islice
from collections import defaultdict
def agen(): # generator of terms
    invsum, mink = defaultdict(int), 1
    for n in count(1):
        an = next(k for k in count(mink) if invsum[k] + Fraction(1, n) <= 1)
        yield an
        invsum[an] += Fraction(1, n)
        while invsum[mink] == 1: mink += 1
print(list(islice(agen(), 87))) # Michael S. Branicky, Aug 31 2024

a(A157248(n)) <= a(A157248(n+1)).

cp's OEIS Frontend

A258252 Sequence of distinct positive integers having lowest possible denominators of sums of 1/a(n).

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