A141091 -id:A141091 - cp's OEIS frontend

A141092 Product of first k composite numbers divided by their sum, when the result is an integer.

1, 64, 46080, 111974400, 662171811840, 310393036800000, 7230916185292800, 108238138194410864640000, 23835710455777670400935290994688000000000, 1104077556971139123493322971152384000000000

Offset: 1

Enoch Haga, Jun 01 2008

Find the products and sums of first k composites, k = 1, 2, 3, .... When the products divided by the sums produce integral quotients, add terms to sequence.

			a(3)=46080 because 4*6*8*9*10*12*14=2903040 and 4+6+8+9+10+12+14=63; 2903040/63=46080, which is an integer, so 46080 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..90

Cf. A196415, A141089, A141090, A141091.
Compare with A140761, A159578, A140763, A116536.
Cf. A116536.

import Data.Maybe (catMaybes)
a141092 n = a141092_list !! (n-1)
a141092_list = catMaybes $ zipWith div' a036691_list a053767_list where
   div' x y | m == 0    = Just x'
            | otherwise = Nothing where (x',m) = divMod x y
-- Reinhard Zumkeller, Oct 03 2011

With[{cnos=Select[Range[50],CompositeQ]},Select[Table[Fold[ Times,1,Take[ cnos,n]]/ Total[Take[cnos,n]],{n,Length[cnos]}],IntegerQ]] (* Harvey P. Dale, Jan 14 2015 *)

s=0;p=1;forcomposite(n=4,100,p*=n;s+=n;if(p%s==0,print1(p/s", "))) \\ Charles R Greathouse IV, Apr 04 2013

a(n) = A036691(A196415(n)) / A053767(A196415(n)). [Reinhard Zumkeller, Oct 03 2011]

Checked by N. J. A. Sloane, Oct 02 2011.

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

Original entry on oeis.org

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

N. J. A. Sloane, Oct 02 2011

nonn

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

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

Cf. A051838, A141090, A141091, A141092.

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

# 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

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 *)

More terms from Arkadiusz Wesolowski, Oct 03 2011

A141090 Integral quotients of products of first k consecutive composites divided by their sums: products (dividends).

Original entry on oeis.org

4, 1728, 2903040, 12541132800, 115880067072000, 69528040243200000, 1807729046323200000, 43295255277764345856000000, 20188846756043686829592191472500736000000000, 989253491046140654650017382152536064000000000

Offset: 1

Enoch Haga, Jun 01 2008

Based on A141092.
Compare with A140761 A159578 A140763 A116536.
Take the first k composite numbers. If their product divided by their sum results in an integer, their product is a term of the sequence. - Harvey P. Dale, Apr 29 2018

			a(3) = 2903040 because 4*6*8*9*10*12*14 = 2903040 and 4+6+8+9+10+12+14 = 63; 2903040/63 = 46080, integral -- 2903040 is added to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..93

Cf. A196415, A141089, A141091, A141092.

With[{c=Select[Range[100],CompositeQ]},Table[If[IntegerQ[ Times@@Take[ c,n]/Total[ Take[ c,n]]], Times@@ Take[ c,n],0],{n,Length[c]}]]/.(0-> Nothing) (* Harvey P. Dale, Apr 29 2018 *)

Find the products and sums of first k consecutive composites. When the product divided by the sum produces an integral quotient, add product to sequence.

Checked by N. J. A. Sloane, Oct 02 2011

Edited by N. J. A. Sloane, Apr 29 2018

A141089 Integral quotients of products of consecutive composites divided by their sums: Last consecutive composite.

Original entry on oeis.org

4, 9, 14, 18, 22, 25, 26, 33, 48, 49, 78, 80, 84, 95, 105, 110, 114, 115, 119, 123, 124, 129, 147, 150, 152, 158, 160, 170, 175, 184, 190, 200, 202, 212, 213, 242, 245, 250, 256, 258, 272, 284, 287, 288, 291, 306, 309, 314, 319, 327, 332, 333, 336, 342, 343

Offset: 1

Enoch Haga, Jun 01 2008

			a(3) = 14 because 4*6*8*9*10*12*14 = 2903040 and 4+6+8+9+10+12+14 = 63; 2903040/63 = 46080, integral -- 14 is added to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A141090, A141091, A141092.

Compare with A140761, A159578, A140763, A116536.

comp = Select[Range[500], CompositeQ]; comp[[Position[Rest @ FoldList[Times, 1, comp]/Accumulate[comp], ?IntegerQ] // Flatten]] (* _Amiram Eldar, Jan 12 2020 *)

Find the products and sums of consecutive composites. When the products divided by the sums produce integral quotients, add terms to sequence.

A141278 Clusters of consecutive composites in A141089.

Original entry on oeis.org

25, 26, 48, 49, 114, 115, 123, 124, 212, 213, 287, 288, 332, 333, 342, 343, 398, 399, 415, 416, 440, 441, 446, 447, 470, 471, 488, 489, 510, 511, 512, 548, 549, 553, 554, 603, 604, 638, 639, 640, 648, 649, 675, 676, 771, 772, 785, 786, 818, 819, 836, 837

Offset: 1

Enoch Haga, Jun 21 2008

A141089 contains composites A002808(k) such that the partial sum A053767(k) divides the partial product A036691(k). The sequence contains the subsequences of A141089 that contain two or more consecutive integers.

			The first pair of consecutive integers is (25,26) in A141089(6,7), the second (48,49) in A141089(9,10).
Triples of consecutive integers in A141089 are (510,511,512), (638,639,640), (889,890,891), (912,913,914), quadruples are (987,988,989,990), etc, all members included here.

Cf. A141089 A141090 A141091 A141092 A141274 A141275.

Numbers A141089(i) such that either 1+A141089(i) = A141089(i+1) or A141089(i)-1 = A141089(i-1) or both.

Edited by R. J. Mathar, Jul 08 2008

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A141092 Product of first k composite numbers divided by their sum, when the result is an integer.

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

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A141090 Integral quotients of products of first k consecutive composites divided by their sums: products (dividends).

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A141089 Integral quotients of products of consecutive composites divided by their sums: Last consecutive composite.

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A141278 Clusters of consecutive composites in A141089.

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