A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

18, 27, 30, 45, 63, 64, 72, 99, 105, 112, 117, 144, 153, 160, 162, 165, 171, 195, 207, 225, 243, 252, 255, 261, 279, 285, 288, 294, 320, 333, 336, 345, 352, 360, 369, 387, 396, 405, 416, 423, 435, 441, 465, 468, 477, 490, 504, 531, 544, 549, 555, 567, 576

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

A special case of a bound on d(n) by Erdős and Suranyi (1960) was used to get a limit: a = x/d(x) > 0.5*sqrt(x) and below 4194304 a computer test shows these values did not occur as x = a*d(x). For larger x this is impossible since if d(x) < sqrt(x), then x/d(x) > sqrt(4194304) = 2048 > the given terms.

A051521(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011

This sequence contains all numbers of the form k = 9p, p prime (i.e., k = 18, 27, 45, 63, 99, ...). - Jianing Song, Nov 25 2018

Examples

			No natural number x exists for which x = 18*d(x), so 18 is a term.
		

References

• P. Erdős and J. Suranyi, Selected Topics in Number Theory, Tankonyvkiado, Budapest, 1960 (in Hungarian).
• P. Erdős and J. Suranyi, Selected Topics in Number Theory, Springer, New York, 2003 (in English).

Links

• Donovan Johnson, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000005, A033950, A036761, A036762, A036764, A051278, A051279, A051280, A051521.

Programs

Haskell

a036763 n = a036763_list !! (n-1)
a036763_list = filter ((== 0) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

Maple

with(numtheory): A036763 := proc(n) local k,p: for k from 1 to 4*n^2 do p:=n*k: if(p=n*tau(p))then return NULL: fi: od: return n: end: seq(A036763(n),n=1..100); # Nathaniel Johnston, May 04 2011

Mathematica

noSolQ[n_] := !AnyTrue[Range[4*n^2], # == DivisorSigma[0, n*#]& ];
Reap[Do[If[noSolQ[n], Print[n]; Sow[n]], {n, 600}]][[2, 1]] (* Jean-François Alcover, Jan 30 2018 *)

Extensions

Definition corrected by N. J. A. Sloane, May 18 2022 at the suggestion of David James Sycamore.