A140761 -id:A140761 - 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.

A140763 A051838 gives numbers m such that the sum of first m primes divides the product of the first m primes. This sequence gives corresponding values of the sum of first m primes.

Original entry on oeis.org

2, 10, 77, 238, 874, 2747, 2914, 3266, 3638, 4661, 5117, 5830, 6601, 6870, 7141, 9523, 10191, 10887, 11966, 13490, 16401, 19113, 21037, 23069, 40313, 41741, 46191, 50887, 53342, 54998, 58406, 60146, 61910, 65534, 68341, 72179, 75130, 76127, 80189, 82253

Offset: 1

Enoch Haga, May 28 2008

nonn

Sums (divisors) associated with A140761.

			a(2)=10 because when 30 is divided by 10, the quotient is 3 and integral.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A051838, A116536, A140761, A159578, A116536.

Module[{nn=200,s,p},s=Accumulate[Prime[Range[nn]]];p=FoldList[ Times,Prime[ Range[nn]]];Select[Thread[{p,s}],Divisible[#[[1]],#[[2]]]&]][[All,2]] (* Harvey P. Dale, Jun 07 2022 *)

a(n)=A116536(n)/A159578(n) = A007504(A051838(n)). - R. J. Mathar, Jun 09 2008

Sum_{i=1..A051838(n)} prime(i).

Corrected and edited by N. J. A. Sloane, Oct 01 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

A141091 Integral quotients of products of consecutive composites divided by their sums: sums (divisors).

Original entry on oeis.org

4, 27, 63, 112, 175, 224, 250, 400, 847, 896, 2368, 2448, 2695, 3596, 4300, 4624, 4961, 5076, 5546, 6032, 6156, 6664, 8750, 9048, 9200, 9976, 10295, 11620, 12312, 13572, 14697, 15872, 16275, 18139, 18352, 23572, 24304, 25544, 26814, 27072, 29986

Offset: 1

Enoch Haga, Jun 01 2008

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

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

Cf. A196415, A141089, A141090, A141092.

Compare with A140761, A159578, A140763, A116536.

comp = Select[Range[500], CompositeQ]; sum = Accumulate[comp]; sum[[Position[ Rest@FoldList[Times, 1, comp]/sum, ?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.

a(37) corrected by Amiram Eldar, Jan 12 2020

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.

cp's OEIS Frontend

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

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A140763 A051838 gives numbers m such that the sum of first m primes divides the product of the first m primes. This sequence gives corresponding values of the sum of first m primes.

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

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A141091 Integral quotients of products of consecutive composites divided by their sums: sums (divisors).

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

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