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 1-4 of 4 results.

A050336 Number of ways of factoring n with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 9, 1, 3, 3, 14, 1, 9, 1, 9, 3, 3, 1, 23, 3, 3, 6, 9, 1, 12, 1, 27, 3, 3, 3, 31, 1, 3, 3, 23, 1, 12, 1, 9, 9, 3, 1, 57, 3, 9, 3, 9, 1, 23, 3, 23, 3, 3, 1, 41, 1, 3, 9, 58, 3, 12, 1, 9, 3, 12, 1, 83, 1, 3, 9, 9, 3, 12, 1, 57, 14, 3, 1, 41, 3, 3, 3, 23, 1, 41, 3, 9
Offset: 1

Views

Author

Christian G. Bower, Oct 15 1999

Keywords

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Examples

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2*2) = (3*2)*(2) = (3)*(2*2) = (3)*(2)*(2).

Links

Crossrefs

Cf. A001055, A050337, A050338, A050339, A050340, A050341.
a(p^k)=A001970. a(A002110)=A000258.

Formula

Dirichlet g.f.: Product_{n>=2}(1/(1-1/n^s)^A001055(n)).
a(n) = A050337(A101296(n)). - R. J. Mathar, May 26 2017

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Original entry on oeis.org

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472
Offset: 0

Views

Author

Labos Elemer, Apr 28 2003

Keywords

Comments

a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.

Examples

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...

Links

Crossrefs

Cf. A000010, A007283, A009195, A050339, A134059.
Essentially the same as A003945 (and perhaps also A058764).

Programs

  • Magma
    [0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009
  • Maple
    0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021
  • Mathematica
    {0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* Michael De Vlieger, Jul 03 2016, after Harvey P. Dale at A134059 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)
  • PARI
    {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */
  • PARI
    a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012
  • Sage
    [0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

Formula

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012
Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008
a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016
a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

Extensions

More terms from Klaus Brockhaus, Dec 02 2009

A050338 Number of ways of factoring n with 2 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 30, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 75, 4, 4, 4, 74, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 176, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 206, 4, 22, 1, 16, 4, 22, 1, 267, 1, 4, 16, 16, 4, 22, 1, 176, 30, 4, 1, 102
Offset: 1

Views

Author

Christian G. Bower, Oct 15 1999

Keywords

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Examples

			4 = ((4)) = ((2*2)) = ((2)*(2)) = ((2))*((2)).

Links

Crossrefs

Cf. A001055, A050336-A050341. a(p^k)=A007713. a(A002110)=A000307.

Formula

Dirichlet g.f.: Product_{n>=2}(1/(1-1/n^s)^A050336(n)).
a(n) = A050339(A101296(n)). - R. J. Mathar, May 26 2017

A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 8, 1, 8, 1, 8, 1, 4, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 2, 1, 16, 1, 2, 1, 8, 1, 4, 1, 8, 1, 2, 1, 8, 1, 8, 1, 32, 1, 2, 1, 4, 1, 4, 1, 32, 1, 2, 1, 4, 1, 32, 1, 32, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 16, 1, 64, 1, 8, 1
Offset: 1

Views

Author

Labos Elemer, Apr 28 2003

Keywords

Comments

a(n)=1 if and only if n is odd or n = 2. - Robert Israel, May 31 2018

Examples

			Different from A069177, analogous sequence with totient, instead of cototient.

Links

Crossrefs

Cf. A000010, A051953, A009195, A083250, A007283, A050339, A053576, A069177.

Programs

  • Maple
    f:= n -> padic:-ordp(n - numtheory:-phi(n), 2):
map(f, [$1..100]); # Robert Israel, May 31 2018
Showing 1-4 of 4 results.

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