|
|
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.
|
|
19
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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.
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
|
|
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
|
|
|
EXAMPLE
|
No natural number x exists for which x = 18*d(x), so 18 is a term.
|
|
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*#]& ];
|
|
PROG
|
(Haskell)
a036763 n = a036763_list !! (n-1)
a036763_list = filter ((== 0) . a051521) [1..]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|