A196415 -id:A196415 - cp's OEIS frontend

A036691 Compositorial numbers: product of first n composite numbers.

1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800, 250822656000, 5267275776000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000, 48808684250726400000, 1366643159020339200000

Offset: 0

Felice Russo

a(A196415(n)) = A141092(n) * A053767(A196415(n)). - Reinhard Zumkeller, Oct 03 2011
For n>11, A000142(n) < a(n) < A002110(n). - Chayim Lowen, Aug 18 2015
For n = {2,3,4}, a(n) is testably a Zumkeller number (A083207). For n > 4, a(n) is of the form 2^e_1 * p_2^e_2 * … * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) < e_1. Therefore, 2^e * p_m^e_m is primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number. Therefore, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m is a Zumkeller number. Therefore, for n > 1, a(n) is a Zumkeller number (see my proof at A002182 for details). - Ivan N. Ianakiev, May 04 2020

			a(3) = c(1)*c(2)*c(3) = 4*6*8 = 192.

T. D. Noe, Table of n, a(n) for n = 0..100
Googology Wiki, Compositorial

Cf. primorial numbers A002110. Distinct members of A049614. See also A049650, A060880.

Cf. A092435 (subsequence: A092435(n) = a(prime(n)-n-1)). - Chayim Lowen, Jul 23 2015

a036691_list = scanl1 (*) a002808_list -- Reinhard Zumkeller, Oct 03 2011

A036691 := proc(n)
        mul(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Product[ Composite[i], {i, 1, n}], {n, 0, 18}] (* Robert G. Wilson v, Sep 13 2003 *)
nn=50;cnos=Complement[Range[nn],Prime[Range[PrimePi[nn]]]];Rest[FoldList[ Times,1,cnos]] (* Harvey P. Dale, May 19 2011 *)
A036691 = Union[Table[n!/(Times@@Prime[Range[PrimePi[n]]]), {n, 29}]] (* Alonso del Arte, Sep 21 2011 *)
Join[{1},FoldList[Times,Select[Range[30],CompositeQ]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 14 2019 *)

a(n)=my(c,p);c=4;p=1;while(n>0,if(!isprime(c),p=p*c;n=n-1);c=c+1);p \\ Ralf Stephan, Dec 21 2013

from sympy import factorial, primepi, primorial, composite
def A036691(n):
    return factorial(composite(n))//primorial(primepi(composite(n))) if n > 0 else 1 # Chai Wah Wu, Sep 08 2020

From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A049614(A002808(n)) = A000142(A002808(n))/A034386(A002808(n)).
a(n) = Product_{k=1..A002808(n)-n-1} prime(k)^(A085604(A002808(n),k)-1).
Sum_{k >= 1} 1/a(k) = 1.2975167655550616507663335821769... is to this sequence as e is to the factorials. (End)

Corrected and extended by Niklas Eriksen (f95-ner(AT)nada.kth.se) and N. J. A. Sloane

A053767 Sum of first n composite numbers.

Original entry on oeis.org

0, 4, 10, 18, 27, 37, 49, 63, 78, 94, 112, 132, 153, 175, 199, 224, 250, 277, 305, 335, 367, 400, 434, 469, 505, 543, 582, 622, 664, 708, 753, 799, 847, 896, 946, 997, 1049, 1103, 1158, 1214, 1271, 1329, 1389, 1451, 1514, 1578, 1643, 1709, 1777, 1846, 1916, 1988

Offset: 0

G. L. Honaker, Jr., Mar 29 2000

nonn

a(A196415(n)) = A036691(A196415(n)) / A141092(n). - Reinhard Zumkeller, Oct 03 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
C. Rivera, See also

Cf. A002808, A051349.

First differences of A023539.

a053767 n = a053767_list !! (n-1)
a053767_list = scanl1 (+) a002808_list -- Reinhard Zumkeller, Oct 03 2011

A053767 := proc(n)
        add(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011
ListTools[PartialSums](remove(isprime,[$2..1000])); # Robert Israel, Jan 09 2015

lst={};s=0;Do[If[ !PrimeQ[n], s=s+n;AppendTo[lst, s]], {n, 2, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Complement[Range[2,100],Prime[Range[PrimePi[100]]]]] (* Harvey P. Dale, Dec 28 2010 *)
Accumulate[Select[Range[2, 100], ! PrimeQ[#] &]]

lista(nn) = {my(s=0); forcomposite(n=0, nn, print1(s, ", "); s += n;);} \\ Michel Marcus, Jan 09 2015

a(n) = A000217(A002808(n)) - A034387(A002808(n)) - 1 . - Robert Israel, Jan 09 2015

a(n) = A051349(n+1) - 1. - Michel Marcus, Feb 16 2018

a(0)=0 prepended by Max Alekseyev, Feb 10 2018

A051838 Numbers k such that sum of first k primes divides product of first k primes.

Original entry on oeis.org

1, 3, 8, 13, 23, 38, 39, 41, 43, 48, 50, 53, 56, 57, 58, 66, 68, 70, 73, 77, 84, 90, 94, 98, 126, 128, 134, 140, 143, 145, 149, 151, 153, 157, 160, 164, 167, 168, 172, 174, 176, 182, 191, 194, 196, 200, 210, 212, 215, 217, 218, 219, 222, 225, 228, 229

Offset: 1

G. L. Honaker, Jr., Dec 12 1999

nonn

			Sum of first 8 primes is 77 and product of first 8 primes is 9699690. 77 divides 9699690 therefore a(3)=8.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A007504, A002110, A159639, A196415.

A116536 gives the quotients, A140763 the divisors and A159578 the dividends.

P:=Filtered([1..2000],IsPrime);;
Filtered([1..Length(P)],n->Product([1..n],i->P[i]) mod Sum([1..n],i->P[i])=0); # Muniru A Asiru, Dec 20 2018

import Data.List (elemIndices)
a051838 n = a051838_list !! (n-1)
a051838_list =
   map (+ 1) $ elemIndices 0 $ zipWith mod a002110_list a007504_list
-- Reinhard Zumkeller, Oct 03 2011

p = Prime@ Range@ 250; Flatten@ Position[ Mod[ First@#, Last@#] & /@ Partition[ Riffle[ Rest[ FoldList[ Times, 1, p]], Accumulate@ p], 2], 0] (* Harvey P. Dale, Dec 19 2010 *)

for(n=1,100,P=prod(i=1,n,prime(i));S=sum(i=1,n,prime(i));if(!(P%S),print1(n,", "))) \\ Derek Orr, Jul 19 2015

isok(n) = my(p = primes(n)); (vecprod(p) % vecsum(p)) == 0; \\ Michel Marcus, Dec 20 2018

From Reinhard Zumkeller, Oct 03 2011: (Start)
A002110(a(n)) mod A007504(a(n)) = 0.
A116536(n) = A002110(a(n)) / A007504(a(n)). (End)

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

Original entry on oeis.org

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.

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

cp's OEIS Frontend

A036691 Compositorial numbers: product of first n composite numbers.

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A053767 Sum of first n composite numbers.

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A051838 Numbers k such that sum of first k primes divides product of first k primes.

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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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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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