cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 30 results. Next

A049650 Compositorial numbers (A036691) + 1.

Original entry on oeis.org

2, 5, 25, 193, 1729, 17281, 207361, 2903041, 43545601, 696729601, 12541132801, 250822656001, 5267275776001, 115880067072001, 2781121609728001, 69528040243200001, 1807729046323200001, 48808684250726400001, 1366643159020339200001
Offset: 0

Views

Author

N. J. A. Sloane, May 05 2001

Keywords

Comments

This is to Euclid numbers (A006862): 1 + product of first n consecutive primes, as nonprimes (A018252) are to primes (A000040). These numbers - 1, times the appropriate Euclid numbers - 1, are the factorials. Primes in this sequence include a(1) = 2, a(2) = 5, a(4) = 193, a(8) = 2903041, a(12) = 250822656001, a(17) = 1807729046323200001. - Jonathan Vos Post, Jun 07 2008

Links

Crossrefs

Cf. A002808, A036691, A060880, A006862.

Programs

  • Mathematica
    Composite[n_] := FixedPoint[n + PrimePi[#] + 1 &, n + PrimePi[n] + 1]; Table[Product[Composite[i], {i, 1, n}] + 1, {n, 0, 30}] (* G. C. Greubel, Dec 05 2017 *)

Formula

a(n) = 1 + Product_{i=1..n} A002808(i). - Jonathan Vos Post, Jun 07 2008

A060880 Compositorial numbers (A036691) - 1.

Original entry on oeis.org

0, 3, 23, 191, 1727, 17279, 207359, 2903039, 43545599, 696729599, 12541132799, 250822655999, 5267275775999, 115880067071999, 2781121609727999, 69528040243199999, 1807729046323199999, 48808684250726399999, 1366643159020339199999
Offset: 0

Views

Author

N. J. A. Sloane, May 05 2001

Keywords

Links

Crossrefs

Cf. A036691, A049650.

A065899 a(n) is the index of the n-th compositorial number, A036691(n), in the sequence of composites (A002808).

Original entry on oeis.org

1, 14, 148, 1458, 15293, 188782, 2692726, 40909988, 660637057, 11976280879, 240871231369, 5080851687840, 112183659405198, 2700581280109040, 67686358108129808, 1763651979163805444, 47707175694652299653, 1337959106215345951164, 40196133912310028013721, 1287910861213828031657392
Offset: 1

Views

Author

Labos Elemer, Nov 28 2001

Keywords

Examples

			a(2) = 14 because 4*6 = 24, the 2nd compositorial number is the 14th composite number: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24.

Crossrefs

Cf. A002808, A000720, A036691, A065855.

Programs

  • Mathematica
    Table[A036691[n]-(PrimePi[A036691[n]])-1, {n, 1, 9}]
Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[c = Product[ Composite[i], {i, 1, n} ]; c - PrimePi[c] - 1, {n, 1, 10} ]
  • Python
    from sympy import factorial, primepi, composite, primorial, compositepi
def A065899(n):
    return compositepi(factorial(composite(n))//primorial(primepi(composite(n)))) # Chai Wah Wu, Sep 08 2020

Formula

a(n) = A036691(n) - primepi(A036691(n))-1.
a(n) = A065855(A036691(n)). - Chai Wah Wu, Sep 08 2020

Extensions

One more term from Robert G. Wilson v, Nov 29 2001
a(11)-a(19) from Chai Wah Wu, Sep 08 2020
a(20) from Chai Wah Wu, Sep 09 2020
Name rewritten by Felix Fröhlich, Jun 01 2021

A233437 Floor(Primorial(n) / compositorial(n)), that is, floor(A002110(n) / A036691(n)).

Original entry on oeis.org

1, 2, 4, 8, 18, 39, 85, 191, 425, 940, 2185, 5183, 12814, 32711, 84715, 218141, 555741, 1376723, 3457106, 9544621, 26048861, 72830491, 202468765, 591526393, 1717701641, 4994058475, 14800573301, 44137423952, 133960953399, 413431218250, 1247184175056, 3842131894125
Offset: 12

Views

Author

Alex Ratushnyak, Dec 09 2013

Keywords

Crossrefs

Cf. A002110, A036691.

Programs

  • PARI
    a(n)=my(c,p,N);N=n;c=4;p=1;while(n>0,if(!isprime(c),p=p*c;n=n-1);c=c+1);floor(prod(i=1,N,prime(i))/p) \\ Ralf Stephan, Dec 21 2013

Formula

a(n) = floor(A002110(n) / A036691(n)).

A233448 Compositorial(n) mod n!, that is, A036691(n) mod A000142(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25401600, 174182400, 1437004800, 46942156800, 301771008000, 0, 188305108992000, 5272543051776000, 30128817438720000, 964122158039040000, 24517325190758400000, 118315866081853440000, 16505063318418554880000, 283958078596448256000000
Offset: 1

Views

Author

Alex Ratushnyak, Dec 10 2013

Keywords

Comments

The sequence of numbers k such that a(k) > a(k+1) begins: 15, 81, 135, 337, 57517, ...

Crossrefs

Cf. A000142, A036691.

Programs

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

A131206 a(n)=(A002866(n+1) - A036691(n))/192.

Original entry on oeis.org

0, 0, 0, 0, 1, 30, 600, 11760, 257040, 6048000, 147571200, 802982400, 105409382400, 3116065075200, 97103204352000, 3208697597952000, 111992720007168000
Offset: 0

Views

Author

Alexander R. Povolotsky, Oct 20 2007, Oct 28 2007

Keywords

Comments

The divisor in the formula is A002866(4) = 192

Crossrefs

Cf. A002866, A036691.

A181335 Partial products of A036691.

Original entry on oeis.org

1, 4, 96, 18432, 31850496, 550376570880, 114126085737676800, 331312591939905257472000, 14427205603578338379772723200000, 10051861189298894268003697526046720000000
Offset: 0

Views

Author

Jonathan Vos Post, Jan 28 2011

Keywords

Comments

Product of the first n compositorial numbers (which are themselves the product of the first n composite numbers). This is to compositorial numbers (A036691), as superfactorials (A000178) are to factorials (A000142).

Examples

			a(3) = 1 * 4 * 24 * 192 = (1) * (1*4) * (1*4*6) * (1*4*6*8), since the first 4 composite numbers are (4, 6, 8) and the 0th compositorial is 1.

Crossrefs

Cf. A000142, A000178, A002808, A036691.

Programs

  • Mathematica
    nn=20;cnos=Complement[Range[nn],Prime[Range[PrimePi[nn]]]];Rest[ FoldList[ Times,1,Rest[FoldList[Times,1,cnos]]]] (* Harvey P. Dale, Jun 28 2011 *)

Formula

a(n) = Product_{i=0..n} A036691(i) = Product_{i=0..n} Product_{j=1..i} A002808(j).

A233438 Primorial(n) mod compositorial(n), that is, A002110(n) mod A036691(n).

Original entry on oeis.org

0, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 2153462358810, 72490129383210, 1958274892758030, 58665460642891410, 50035643372444730, 19221664375883039070, 1123712842678138983270, 27456249893723439879090, 350421246400567367415390
Offset: 0

Views

Author

Alex Ratushnyak, Dec 09 2013

Keywords

Comments

The sequence of numbers k such that a(k) > a(k+1) begins: 15, 42, 645, 805, 1566, 5430, 53698, ...

Crossrefs

Cf. A002110, A036691, A233437, A233448.

Programs

  • PARI
    a(n)=my(c,p,N);N=n;c=4;p=1;while(n>0,if(!isprime(c),p=p*c;n=n-1);c=c+1);prod(i=1,N,prime(i))%p \\ Ralf Stephan, Dec 21 2013

Formula

a(n) = A002110(n) mod A036691(n).

A233447 Floor(compositorial(n) / n!), that is, floor(A036691(n) / A000142(n)).

Original entry on oeis.org

4, 12, 32, 72, 144, 288, 576, 1080, 1920, 3456, 6283, 10996, 18609, 31901, 53169, 86400, 137223, 213458, 337040, 539264, 847415, 1309642, 1992933, 2989400, 4543888, 6815833, 10097530, 15146295, 22980586, 34470879, 51150337, 76725506, 113925752, 167537870, 244126611
Offset: 1

Views

Author

Alex Ratushnyak, Dec 10 2013

Keywords

Crossrefs

Cf. A000142, A036691.

Programs

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

Formula

a(n) = floor(A036691(n) / A000142(n)).

A337366 Number of representations of A036691(n) as a sum of 3 nonnegative cubes.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 1, 2, 1, 4, 6, 3, 8, 8, 14, 7
Offset: 0

Views

Author

Altug Alkan, Aug 25 2020

Keywords

Comments

Conjecture I: a(n) = 0 only for n = 1. That is, any product of first n > 1 composite numbers is a sum of at most 3 positive cubes. For example,
A036691(100) = 2563573191821442299652988946477367093137353211904000000000^3 + 21431289850849406740917647451954098598503667204096000000000^3 + 26409890400237152457638095665189553529771293409280000000000^3.
Conjecture II: For any term t >= 1, there are only finitely many values of n such that a(n) = t.

Examples

			a(4) = 2 because A036691(4) = 1728 = 12^3 = 6^3 + 8^3 + 10^3.

Crossrefs

Cf. A025447, A036691, A226955, A267414.

Programs

  • Mathematica
    A036691 = Join[{1}, FoldList[Times, Select[Range[20], CompositeQ]]];
Table[Length@ PowersRepresentations[A036691[[n]], 3, 3], {n, 10}] (* Robert Price, Sep 08 2020 *)

Formula

a(n) = A025447(A036691(n)).
Showing 1-10 of 30 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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