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Values of n such that (product of first n composite numbers) / (sum of first n composite numbers) is an integer.
7

%I #34 May 04 2019 21:51:11

%S 1,4,7,10,13,15,16,21,32,33,56,57,60,70,77,80,83,84,88,92,93,97,112,

%T 114,115,120,122,130,134,141,147,153,155,164,165,188,191,196,201,202,

%U 213,222,225,226,229,243,245,248,252,260,264,265,268,273,274,281

%N Values of n such that (product of first n composite numbers) / (sum of first n composite numbers) is an integer.

%C A036691(a(n)) mod A053767(a(n)) = 0, A141092(n) = A036691(a(n)) / A053767(a(n)). [_Reinhard Zumkeller_, Oct 03 2011]

%H Arkadiusz Wesolowski, <a href="/A196415/b196415.txt">Table of n, a(n) for n = 1..10000</a>

%p # First define list of composite numbers:

%p tc:=[4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,

%p 28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,

%p 50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,

%p 70,72,74,75,76,77,78,80,81,82,84,85,86,87,88];

%p a1:=n->mul(tc[i],i=1..n);

%p a2:=n->add(tc[i],i=1..n);

%p sn:=[];

%p s0:=[];

%p s1:=[];

%p s2:=[];

%p for n from 1 to 40 do

%p t1:=a1(n)/a2(n);

%p if whattype(t1) = integer then

%p sn:= [op(sn),n];

%p s0:= [op(s0),t1];

%p s1:= [op(s1),a1(n)];

%p s2:= [op(s2),a2(n)];

%p fi;

%p od:

%p sn; s0; s1; s2;

%p # alternatively

%p for n from 1 to 1000 do

%p if type(A036691(n)/A053767(n),'integer') then

%p printf("%d,",n);

%p end if;

%p end do: # _R. J. Mathar_, Oct 03 2011

%t c = Select[Range[2,355], ! PrimeQ@# &]; p = 1; s = 0; Select[Range@ Length@c, Mod[p *= c[[#]], s += c[[#]]] == 0 &] (* _Giovanni Resta_, Apr 03 2013 *)

%o (Haskell)

%o import Data.List (elemIndices)

%o a196415 n = a196415_list !! (n-1)

%o a196415_list =

%o map (+ 1) $ elemIndices 0 $ zipWith mod a036691_list a053767_list

%o -- _Reinhard Zumkeller_, Oct 03 2011

%Y Cf. A051838, A141090, A141091, A141092.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Oct 02 2011

%E More terms from _Arkadiusz Wesolowski_, Oct 03 2011