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.
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
a(n) ~ 11.3*n - Chai Wah Wu, Nov 28 2025
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
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Donovan Johnson)
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*#]& ];
Reap[Do[If[noSolQ[n], Print[n]; Sow[n]], {n, 600}]][[2, 1]] (* Jean-François Alcover, Jan 30 2018 *)
PROG
(Haskell)
a036763 n = a036763_list !! (n-1)
a036763_list = filter ((== 0) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011
(Python)
from itertools import count, islice
from math import prod
from collections import Counter
from sympy import factorint
def A036763_gen(startvalue=1): # generator of terms >= startvalue
for n in count(max(startvalue, 1)):
f = Counter(factorint(n))
if all(prod(a+1 for a in (Counter(factorint(k))+f).values())!=k for k in range(1, (n<<2)|1)):
yield n
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Definition corrected by N. J. A. Sloane, May 18 2022 at the suggestion of David James Sycamore.
STATUS
approved
