A196415 Values of n such that (product of first n composite numbers) / (sum of first n composite numbers) is an integer.

1, 4, 7, 10, 13, 15, 16, 21, 32, 33, 56, 57, 60, 70, 77, 80, 83, 84, 88, 92, 93, 97, 112, 114, 115, 120, 122, 130, 134, 141, 147, 153, 155, 164, 165, 188, 191, 196, 201, 202, 213, 222, 225, 226, 229, 243, 245, 248, 252, 260, 264, 265, 268, 273, 274, 281

Offset: 1

Author

N. J. A. Sloane, Oct 02 2011

Keywords

• nonn

Comments

A036691(a(n)) mod A053767(a(n)) = 0, A141092(n) = A036691(a(n)) / A053767(a(n)). [Reinhard Zumkeller, Oct 03 2011]

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A051838, A141090, A141091, A141092.

Programs

Haskell

import Data.List (elemIndices)
a196415 n = a196415_list !! (n-1)
a196415_list =
   map (+ 1) $ elemIndices 0 $ zipWith mod a036691_list a053767_list
-- Reinhard Zumkeller, Oct 03 2011

Maple

# First define list of composite numbers:
tc:=[4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,
28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,
50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,
70,72,74,75,76,77,78,80,81,82,84,85,86,87,88];
a1:=n->mul(tc[i],i=1..n);
a2:=n->add(tc[i],i=1..n);
sn:=[];
s0:=[];
s1:=[];
s2:=[];
for n from 1 to 40 do
  t1:=a1(n)/a2(n);
  if whattype(t1) = integer then
   sn:= [op(sn),n];
   s0:= [op(s0),t1];
   s1:= [op(s1),a1(n)];
   s2:= [op(s2),a2(n)];
fi;
od:
sn; s0; s1; s2;
# alternatively
for n from 1 to 1000 do
        if type(A036691(n)/A053767(n),'integer') then
                printf("%d,",n);
        end if;
end do: # R. J. Mathar, Oct 03 2011

Mathematica

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 *)

Extensions

More terms from Arkadiusz Wesolowski, Oct 03 2011