A036691 -id:A036691 - cp's OEIS frontend

A111687 Comprimorial(n): the product of the first n primes and the first n composite numbers.

8, 144, 5760, 362880, 39916800, 6227020800, 1482030950400, 422378820864000, 155435406077952000, 81137281972690944000, 50305114823068385280000, 39087074217524135362560000, 35256540944206770097029120000

Offset: 1

Amarnath Murthy, Aug 17 2005

nonn

			a(1) = 2*4 = 8, a(2) = (2*3)*(4*6)=144.

Harvey P. Dale, Table of n, a(n) for n = 1..215

Cf. A002110 (primorials), A036691 (compositorials).

Module[{nn=50,prs,coms,len},prs=Select[Range[nn],PrimeQ];coms=Complement[ Range[4,nn],prs];len=Min[Length[prs],Length[coms]];Rest[FoldList[ Times, 1, Times@@@Thread[{Take[prs,len],Take[coms,len]}]]]] (* Harvey P. Dale, Jan 11 2014 *)

a(n) = A002110(n)*A036691(n) for n>=1. - Rick L. Shepherd, Aug 20 2005

Corrected and extended by Rick L. Shepherd, Aug 20 2005

A075070 a(n) = n-th compositorial number / (product of those primes which divide the n-th compositorial number).

Original entry on oeis.org

1, 2, 4, 32, 288, 576, 6912, 13824, 207360, 3317760, 59719680, 1194393600, 25082265600, 50164531200, 1203948748800, 30098718720000, 60197437440000, 1625330810880000, 45509262704640000, 1365277881139200000

Offset: 0

Amarnath Murthy, Sep 08 2002

nonn

Smallest integer of the form 'Product of first n composite number/ product of first k primes'.

Divide Compositorial(n) by Primorial(k) choosing k to give the smallest integer. (k+1)-th prime does not divide a(n).

			a(0) = 1, a(5) = (4*6*8*9*10)/(2*3*5) = 576, 10 is the fifth composite number.

Cf. A002808.

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ Product[ Composite[i], {i, 1, n}]/ Times @@ PrimeFactors[ Product[ Composite[i], {i, 1, n}]], {n, 0, 20}]

A036691/(prime factors of A036691)

Edited and extended by Robert G. Wilson v, Jul 15 2003

Further edited by N. J. A. Sloane, Sep 13 2008 at the suggestion of R. J. Mathar

A195529 Ultracompositorial: Compositorials raised to the power of themselves.

Original entry on oeis.org

1, 256, 1333735776850284124449081472843776

Offset: 0

Kausthub Gudipati, Sep 21 2011

nonn

Next term (192^192) has 439 digits.

Amiram Eldar, Table of n, a(n) for n = 0..3

Cf. A000312, A036691, A046882, A165812.

(* First run the program for A036691 *) Table[A036691[[n]]^A036691[[n]], {n, 4}] (* Alonso del Arte, Sep 21 2011 *)
(#^#) & /@ FoldList[Times, 1, Select[Range[6], CompositeQ]] (* Amiram Eldar, Jul 20 2025 *)

a(n) = A000312(A036691(n)). - Amiram Eldar, Jul 20 2025

a(2) = A114993(2). - R. J. Mathar, Aug 22 2025

a(2) corrected by Franklin T. Adams-Watters, Sep 21 2011

A132996 a(n) = gcd(Sum_{k=1..n} c(k), Product_{j=1..n} c(j)), where c(k) is the k-th composite.

Original entry on oeis.org

4, 2, 6, 27, 1, 1, 63, 6, 2, 112, 12, 9, 175, 1, 224, 250, 1, 5, 5, 1, 400, 14, 7, 5, 3, 6, 2, 8, 12, 3, 17, 847, 896, 22, 1, 1, 1, 6, 2, 1, 3, 3, 1, 2, 6, 31, 1, 1, 26, 4, 28, 2, 1, 1, 10, 2368, 2448, 9, 7, 2695, 20, 2, 1, 1, 31, 18, 2, 1, 9, 3596, 52, 10, 1, 1, 1, 5, 4300, 2, 74, 4624

Offset: 1

Leroy Quet, Nov 22 2007

nonn

			The first 8 composites are 4,6,8,9,10,12,14,15. 4+6+8+9+10+12+14+15 = 78 = 2*3*13. So a(8) = gcd(2*3*13, 4*6*8*9*10*12*14*15) = 6.

Cf. A053767, A036691.

lim=80;c[n_]:=n-PrimePi[n]-1;i=0;Do[Until[c[i]==m,i++];Cmp[m]=i,{m,lim}];Table[GCD[Sum[Cmp[k],{k,n}],Product[Cmp[j],{j,n}]],{n,lim}] (* James C. McMahon, Mar 09 2025 *)

More terms from R. J. Mathar, Jan 13 2008

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

A255383 Compositorial mod sum-of-composites.

Original entry on oeis.org

0, 4, 12, 0, 1, 41, 0, 72, 2, 0, 48, 126, 0, 20, 0, 0, 90, 95, 115, 4, 0, 140, 161, 90, 261, 138, 208, 512, 72, 420, 51, 0, 0, 924, 899, 29, 893, 72, 840, 727, 129, 1185, 194, 732, 1080, 1612, 566, 175, 1352, 1192, 1204, 1360, 428, 957, 2170, 0, 0, 513, 2240

Offset: 1

Walter Carlini, May 14 2015

			For n = 5, a(5) = (4*6*8*9*10) mod (4+6+8+9+10) = 17280 mod 37 = 1.

Cf. A002808, A036691, A053767, A255217.

comp=Select[Range[2,83],!PrimeQ[#]&];Mod[Rest[FoldList[Times,1,comp]],Accumulate[comp]] (* Ivan N. Ianakiev, May 22 2015 *)

a(n) = A036691(n) mod A053767(n).

More terms from Alois P. Heinz, May 21 2015

A277005 Least prime greater than n-th compositorial.

Original entry on oeis.org

2, 5, 29, 193, 1733, 17291, 207367, 2903041, 43545611, 696729629, 12541132817, 250822656001, 5267275776047, 115880067072017, 2781121609728037, 69528040243200079, 1807729046323200001, 48808684250726400031, 1366643159020339200397

Offset: 0

Walter Carlini, Sep 25 2016

			a(0) = A151800(A036691(0)) = A151800(1) = 2; where the zeroth compositorial, A036691(0), is the empty product = 1.
a(3) = 193, which is the least prime number greater than the third compositorial number, 192 = 4 * 6 * 8.

Cf. A036691, A038710, A151800.

findComp[n_] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1] ; Table[NextPrime@ Product[findComp@ k, {k, n}], {n, 0, 18}] (* Michael De Vlieger, Sep 25 2016, after Robert G. Wilson v at A036691 *)

a(n) = A151800(A036691(n)). - Michel Marcus, Sep 25 2016

a(18) corrected by Sean A. Irvine, Sep 26 2023

A306848 Product of first n odd nonprimes, a(n) = Product_{k=1..n} A071904(k).

Original entry on oeis.org

1, 9, 135, 2835, 70875, 1913625, 63149625, 2210236875, 86199238125, 3878965715625, 190069320065625, 9693535323346875, 533144442784078125, 30389233238692453125, 1914521694037624546875, 124443910112445595546875, 8586629797758746092734375

Offset: 0

Zhandos Mambetaliyev, Mar 13 2019

nonn

Cf. A071904, A036691, A002866, A001147.

With[{nn = 70}, FoldList[Times, Complement[Range[1, nn, 2], Prime@ Range[2, PrimePi@ nn]]]] (* Michael De Vlieger, Apr 21 2019 *)

lista(nn) = {my(p=1); print1(p, ", "); forcomposite (n=1, nn, if (n%2, p *= n; print1(p, ", ")); ); } \\ Michel Marcus, Mar 13 2019

A371030 n written in compositorial base.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 40, 41, 42, 43, 50, 51, 52, 53, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 123, 130, 131, 132, 133, 140, 141, 142, 143, 150, 151, 152, 153, 200, 201, 202, 203, 210, 211, 212, 213

Offset: 0

James C. McMahon, Mar 08 2024

Compositorial base is a mixed-radix representation using the composite numbers (A002808) from least to most significant.
Places reading from right have values (1, 4, 24, 192, ...) = compositorial numbers (A036691).
a(n) = concatenation of decimal digits of n in compositorial base. This concatenated representation is unsatisfactory for large n (above 172799), when coefficients of 10 or greater start to appear.

			a(35)=123; 35 = 1*24 + 2*4 + 3*1.

Cf. A002808, A036691, A049345.

Table[FromDigits@ IntegerDigits[n,MixedRadix[Reverse@ ResourceFunction["Composite"]@ Range@ 8]], {n, 0,55}]

A380318 Product of the first n perfect powers (A001597).

Original entry on oeis.org

1, 1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 3583180800, 175575859200, 11236854988800, 910185254092800, 91018525409280000, 11013241574522880000, 1376655196815360000000, 176211865192366080000000, 25374508587700715520000000, 4288291951321420922880000000, 840505222458998500884480000000, 181549128051143676191047680000000

Offset: 0

Ilya Gutkovskiy, Jan 20 2025

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A000442, A001044, A001597, A002110, A024923, A036691, A076408, A111059.

Join[{1},FoldList[Times,Join[{1},Select[Range[250],GCD@@FactorInteger[#][[All,2]]>1&]]]] (* Harvey P. Dale, May 03 2025 *)

cp's OEIS Frontend

A111687 Comprimorial(n): the product of the first n primes and the first n composite numbers.

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A075070 a(n) = n-th compositorial number / (product of those primes which divide the n-th compositorial number).

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A195529 Ultracompositorial: Compositorials raised to the power of themselves.

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A132996 a(n) = gcd(Sum_{k=1..n} c(k), Product_{j=1..n} c(j)), where c(k) is the k-th composite.

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A141278 Clusters of consecutive composites in A141089.

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A255383 Compositorial mod sum-of-composites.

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A277005 Least prime greater than n-th compositorial.

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A306848 Product of first n odd nonprimes, a(n) = Product_{k=1..n} A071904(k).

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