login
a(n) is the smallest odd number that is greater than n^2 and is the product of two distinct primes.
3

%I #30 Dec 06 2021 16:25:43

%S 15,15,15,21,33,39,51,65,85,111,123,145,177,201,235,259,291,327,365,

%T 403,445,485,533,579,629,679,731,785,843,901,965,1027,1099,1157,1227,

%U 1299,1371,1457,1527,1603,1685,1765,1851,1937,2031,2117,2215,2305,2407,2501,2603,2705,2811,2921,3027,3139,3261,3365,3487,3601,3737,3845,3973,4097,4227,4359,4497,4627

%N a(n) is the smallest odd number that is greater than n^2 and is the product of two distinct primes.

%C A099611(n) < A000290(n) < a(n); subsequence of A046388.

%C This is an "arithmetic" sequence (like sigma(n)), so it has offset 1. - _N. J. A. Sloane_, Dec 06 2021

%H N. J. A. Sloane, <a href="/A099610/b099610.txt">Table of n, a(n) for n = 1..10000</a>

%p with(numtheory);

%p A099610 := proc(n) local M,i,t1,tt;

%p M:=100; t1:=n^2;

%p for i from 1 to M do

%p tt:=t1+i;

%p if (tt mod 2) = 1 and tau(tt) = 4 and nops(factorset(tt)) = 2 then return(tt); fi;

%p od:

%p lprint("error: the internal parameter M needs to be increased");

%p end proc; # _N. J. A. Sloane_, Dec 05 2021

%t Module[{nn=70,p2p},p2p=Union[Times@@@Subsets[Prime[Range[ 2,PrimePi[ Ceiling[ nn^2/3]]]],{2}]];Table[SelectFirst[p2p,#>n^2&],{n,nn}]] (* _Harvey P. Dale_, Dec 06 2021 *)

%o (Python)

%o from itertools import count

%o from sympy import factorint

%o def A099610(n):

%o for i in count(n**2+(n%2)+1,2):

%o fs = factorint(i)

%o if len(fs) == 2 == sum(fs.values()):

%o return i # _Chai Wah Wu_, Dec 05 2021

%Y Cf. A000290, A046388, A099611, A349806 (a(n)-n^2).

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Oct 25 2004

%E Edited and corrected by _Harvey P. Dale_ and _N. J. A. Sloane_, Dec 05 2021