A016017 Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways.

1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192, 1048576, 2097152, 60, 8388608, 216, 768, 67108864, 288, 1536, 536870912, 1073741824, 120, 576, 8589934592, 6144, 34359738368, 68719476736, 180, 864

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn

Comments

From Jianing Song, Aug 30 2021: (Start)

a(n) is the smallest number whose square has exactly 2n-1 divisors.

a(n) is the earliest occurrence of 2n-1 in A048691. (End)

Examples

			a(1)=1 and a(2)=2 because 1/2 = 1/3 + 1/6 = 1/4 + 1/4.
a(3)=4 because 1/4 = 1/5 + 1/20 = 1/6 + 1/12 = 1/8 + 1/8.
a(4)=8 because 1/8 = 1/9 + 1/72 = 1/10 + 1/40 = 1/12 + 1/24 = 1/16 + 1/16.
a(5)=6 because 1/6 = 1/7 + 1/42 = 1/8 + 1/24 = 1/9 + 1/18 = 1/10 + 1/15 = 1/12 + 1/12.
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..1000 (terms 122, 365 and 608 corrected by _Amiram Eldar_, Nov 08 2024)

Crossrefs

Identical to A071571 shifted right.

Cf. A000005, A000290, A005408, A005179, A003680, A037992, A055079, A048691.

Programs

Mathematica

f[j_, n_] := (Times @@ (j(Last /@ FactorInteger[n]) + 1) + j - 1)/j; t = Table[0, {50}]; Do[a = f[2, n]; If[a < 51 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 2^30}] (* Robert G. Wilson v, Aug 03 2005 *)

PARI

a(n) = {k = 1; while (numdiv(k^2) != (2*n-1), k++); return (k); }; \\ Amiram Eldar, Jan 07 2019 after Michel Marcus at A071571

Formula

a(n+1) <= 2^n.

From Labos Elemer, May 22 2001: (Start)

a(n) = sqrt(A061283(n)).

a(n) = sqrt(Min{k| A000005(k)=2n-1}).

a((p+1)/2) = 2^((p-1)/2) = 2^A005097(i) if p is the i-th odd prime. [Corrected by Jianing Song, Aug 30 2021] (End)

a(n) is the least k such that (tau(k^2) + 1)/2 = n. - Vladeta Jovovic, Aug 01 2001

Extensions

Entry revised by N. J. A. Sloane, Aug 14 2005

Offset corrected by David W. Wilson, Dec 27 2018