A389117 Decimal expansion of the sum of the distinct entries of 1/A055204(n)^(1/2).

3, 7, 0, 9, 7, 5, 1, 2, 3, 3, 8, 9, 9, 2, 2, 2, 3, 6, 8, 8, 8, 7, 9, 6, 1, 4, 6, 6, 8, 4, 2, 0, 7, 8, 8, 8, 2, 1, 7, 4, 4, 3, 6, 4, 5, 0, 6, 4, 1, 6, 9, 7, 6, 6, 8, 0, 9, 7, 8, 9, 0, 9, 9, 3, 9, 3, 9, 4, 4, 8, 1, 4, 6, 5, 3, 8, 6, 4, 4, 8, 2, 2, 2, 6, 8, 1, 2, 3, 7, 0, 1, 8, 0, 6, 2, 7, 1, 7, 0, 5, 2, 3, 5, 7, 4

Offset

1,1

Comments

A result of Erdős and Selfridge implies that A055204(n) only repeats at squares, so one can simply sum 1/A055204(n)^(1/2) for all n other than squares greater than 1. The number of elements of A387186 up to x is asymptotic to this constant times x^(1/2). Conjecturally, the same is true for A388851 (see also Erdős problem #374). The series converges at an exponentially fast rate.

Links

Table of n, a(n) for n=1..105.

Thomas Bloom, Problem 374, Erdős Problems.

Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4 (1976), pp. 337-355.

Terence Tao, Products of consecutive integers with unusual anatomy, arXiv:2603.27990 [math.NT], 2026. See p. 7 (Section 1.3).

Example

3.70975123389922236888796146684207888217443645...

Maple

split := r -> ListTools:-Reverse(convert(floor(r), base, 10)):
core := n -> n / expand(numtheory:-nthpow(n, 2)):
A389117 := proc(upto) local total, cofact, n, term;
  total := 0.0; cofact := 1;
  for n from 2 to upto do
      cofact := core(cofact * n);
      if not issqr(n) then
          term := evalf(1 / sqrt(cofact));
          total := total + term;
      fi;
  od;
1 + total end:
Digits := 200: A389117(1000): split(% * 10^104);  # Peter Luschny, Sep 24 2025

Prog

(PARI) 1 + suminf(n=2, if(issquare(n), 0, 1/sqrt(core(n!)))) \\ Hugo Pfoertner, Sep 23 2025

Crossrefs

Cf. A055204, A387186, A388851.

Sequence in context: A103844 A383820 A199068 * A198490 A354797 A247956

Adjacent sequences: A389114 A389115 A389116 * A389118 A389119 A389120

Keyword

nonn,cons

Author

Terence Tao, Sep 23 2025

Status

approved