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A347421 Numbers k such that the product of the first k semiprimes is divisible by the sum of the first k semiprimes.

%I A347421 #17 Sep 01 2021 02:05:57
%S A347421 1,9,19,29,30,31,32,33,35,36,40,44,45,46,47,51,55,57,64,67,70,71,72,
%T A347421 74,81,83,84,92,94,95,96,97,103,104,105,107,108,109,113,116,118,124,
%U A347421 125,127,130,131,132,133,136,138,140,142,144,158,159,160,167,177,182,184,188,191,196,202,203,206
%N A347421 Numbers k such that the product of the first k semiprimes is divisible by the sum of the first k semiprimes.
%C A347421 What are the asymptotics of a(n)/n as n -> infinity?
%H A347421 Robert Israel, <a href="/A347421/b347421.txt">Table of n, a(n) for n = 1..10000</a>
%e A347421 a(2) = 9 is a term because the first 9 semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, and 4*6*9*10*14*15*21*22*25 = 5239080000 is divisible by 4+6+9+10+14+15+21+22+25 = 126.
%p A347421 R:= NULL:
%p A347421 s:= 0: p:= 1: zcount:= 0: scount:= 0:
%p A347421 for n from 4 while zcount < 100 do
%p A347421   if numtheory:-bigomega(n) = 2 then
%p A347421     s:= s+n; p:= p*n;
%p A347421     scount:= scount+1;
%p A347421     if p mod s = 0 then zcount:= zcount+1; R:= R, scount fi
%p A347421   fi
%p A347421 od:
%p A347421 R;
%t A347421 sp = Select[Range[700], PrimeOmega[#] == 2 &]; Position[Divisible[Rest @ FoldList[Times, 1, sp], Accumulate @ sp], True] // Flatten (* _Amiram Eldar_, Aug 31 2021 *)
%o A347421 (Python)
%o A347421 from sympy import factorint
%o A347421 def aupto(limit):
%o A347421     alst, i, k, s, p = [], 1, 0, 0, 1
%o A347421     while k < limit:
%o A347421         if sum(factorint(i).values()) == 2:
%o A347421             k += 1; s += i; p *= i
%o A347421             if p%s == 0: alst.append(k)
%o A347421         i += 1
%o A347421     return alst
%o A347421 print(aupto(206)) # _Michael S. Branicky_, Aug 31 2021
%o A347421 (Julia)
%o A347421 using Nemo
%o A347421 function A347421List(upto)
%o A347421     c, s, p = 0, ZZ(0), ZZ(1)
%o A347421     list = Int32[]
%o A347421     for n in 4:typemax(Int32)
%o A347421         if 2 == sum([e for (p, e) in factor(n)])
%o A347421             s += n; p *= n; c += 1
%o A347421             if divisible(p, s)
%o A347421                 c > upto && return list
%o A347421                 push!(list, c)
%o A347421             end
%o A347421         end
%o A347421     end
%o A347421 end
%o A347421 A347421List(206) |> println # _Peter Luschny_, Aug 31 2021
%Y A347421 Cf. A001358, A062198, A112141, A347413.
%K A347421 nonn
%O A347421 1,2
%A A347421 _J. M. Bergot_ and _Robert Israel_, Aug 31 2021