A397799 a(n) is the number of terms in the lexicographically earliest sequence of distinct integers greater than 1 that are coprime to half the product of the first n primes and whose reciprocals sum to 1, or a(n) = -1 if no such finite sequence exists.

3, 4, 6, 8, 9, 6, 10, 8, 15, 24, 18, 8, 11, 10, 14, 16, 15, 9, 10, 12, 22, 16, 24

Offset

1,1

Comments

The sequence continues with ?, 17, 34, 27, 29, 22, 21, 15, 12, 22, ?, 12, 11, 19, ?, 16, 16, 21, 15, 25, 14, 15, ?, 16, 18, ... (where '?' denotes unknown terms).

Links

Table of n, a(n) for n=1..23.

Index entries for sequences related to Egyptian fractions.

Example

   n | a(n) | sequence
  ---+------+-----------------------------------------------------------------------------------------
   1 |   3  | 2, 3, 6
   2 |   4  | 2, 4, 5, 20
   3 |   6  | 2, 4, 7, 11, 62, 9548
   4 |   8  | 2, 4, 8, 11, 31, 547, 213176, 19881539876
   5 |   9  | 2, 4, 8, 13, 23, 218, 86914, 1618636672, 2292486589831694464
   6 |   6  | 2, 4, 8, 16, 17, 272
   7 |  10  | 2, 4, 8, 16, 19, 103, 6263, 65369021, 1831332323130311, 23476446544193030920580885566736
   8 |   8  | 2, 4, 8, 16, 23, 53, 6502, 63407504

Mathematica

next[sum_, prev_, np_] := Module[{pr = Product[Prime[i], {i, 2, np}], m = Max[prev + 1, Ceiling[1/(1 - sum)]]}, While[!CoprimeQ[m, pr], m++]; m];
a[n_] := Module[{sum = 0, prev = 1, i = 0, m}, While[sum < 1, m = next[sum, prev, n]; prev = m; sum += 1/m; i++]; i];
Array[a, 23]

Prog

(PARI) nxt(sm, prev, np) = {my(pr = prod(i = 2, np, prime(i)), m = max(prev + 1, ceil(1/(1 - sm)))); while(gcd(m, pr) > 1, m++); m; }
a(n) = {my(sm = 0, prev = 1, i = 0, m); while(sm < 1, m = nxt(sm, prev, n); prev = m; sm += 1/m; i++); i; }

Crossrefs

Cf. A070826, A130738, A397798.

Cf. A001651, A229829, A236206.

Sequence in context: A240675 A072152 A199015 * A196098 A099356 A397405

Adjacent sequences: A397796 A397797 A397798 * A397800 A397802 A397803

Keyword

nonn,hard,more,new

Author

Amiram Eldar, Jul 10 2026

Status

approved