A036763 - OEIS

%I #39 May 18 2022 16:59:25

%S 18,27,30,45,63,64,72,99,105,112,117,144,153,160,162,165,171,195,207,

%T 225,243,252,255,261,279,285,288,294,320,333,336,345,352,360,369,387,

%U 396,405,416,423,435,441,465,468,477,490,504,531,544,549,555,567,576

%N 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.

%C 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.

%C A051521(a(n)) = 0. - _Reinhard Zumkeller_, Dec 28 2011

%C 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

%D P. Erdős and J. Suranyi, Selected Topics in Number Theory, Tankonyvkiado, Budapest, 1960 (in Hungarian).

%D P. Erdős and J. Suranyi, Selected Topics in Number Theory, Springer, New York, 2003 (in English).

%H Donovan Johnson, <a href="/A036763/b036763.txt">Table of n, a(n) for n = 1..5000</a>

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

%p 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

%t noSolQ[n_] := !AnyTrue[Range[4*n^2], # == DivisorSigma[0, n*#]& ];

%t Reap[Do[If[noSolQ[n], Print[n]; Sow[n]], {n, 600}]][[2, 1]] (* _Jean-François Alcover_, Jan 30 2018 *)

%o (Haskell)

%o a036763 n = a036763_list !! (n-1)

%o a036763_list = filter ((== 0) . a051521) [1..]

%o -- _Reinhard Zumkeller_, Dec 28 2011

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

%K nonn

%O 1,1

%A _Labos Elemer_

%E Definition corrected by _N. J. A. Sloane_, May 18 2022 at the suggestion of _David James Sycamore_.