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 1-3 of 3 results.

A183037 a(n) = A001511(n)*2^A001511(n) where A001511(n) equals the 2-adic valuation of 2n.

Original entry on oeis.org

2, 8, 2, 24, 2, 8, 2, 64, 2, 8, 2, 24, 2, 8, 2, 160, 2, 8, 2, 24, 2, 8, 2, 64, 2, 8, 2, 24, 2, 8, 2, 384, 2, 8, 2, 24, 2, 8, 2, 64, 2, 8, 2, 24, 2, 8, 2, 160, 2, 8, 2, 24, 2, 8, 2, 64, 2, 8, 2, 24, 2, 8, 2, 896, 2, 8, 2, 24, 2, 8, 2, 64, 2, 8, 2, 24, 2, 8, 2, 160, 2, 8, 2, 24, 2, 8, 2, 64, 2, 8, 2, 24, 2
Offset: 1

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Comments

2n/2^A001511(n) is odd for n >= 1, so that A001511(n) is logarithmic in nature.

Examples

			L.g.f.: A(x) = 2*x + 8*x^2/2 + 2*x^3/3 + 24*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 64*x^8/8 + 2*x^9/9 + 8*x^10/10 + ...
The g.f. of A183036 begins:
exp(A(x)) = 1 + 2*x + 6*x^2 + 10*x^3 + 24*x^4 + 38*x^5 + 74*x^6 + ...

Links

Crossrefs

Cf. A183036.

Programs

  • Mathematica
    Array[# 2^# &[IntegerExponent[#, 2] + 1] &, 93] (* Michael De Vlieger, Nov 06 2018 *)
  • PARI
    {a(n)=valuation(2*n,2)*2^valuation(2*n,2)}
  • Python
    def A183037(n): return (m:=n&-n)*m.bit_length()<<1 # Chai Wah Wu, Jul 12 2022

Formula

Logarithmic derivative of A183036.

A183038 G.f.: exp( Sum_{n>=1} A051064(n)*3^A051064(n)*x^n/n ) where A051064(n) equals the 3-adic valuation of 3n.

Original entry on oeis.org

1, 3, 6, 15, 30, 51, 93, 156, 240, 387, 597, 870, 1311, 1920, 2697, 3873, 5448, 7422, 10278, 14016, 18636, 25098, 33402, 43548, 57333, 74757, 95820, 123780, 158637, 200391, 254778, 321798, 401451, 503490, 627915, 774726, 960156, 1184205, 1446873
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Comments

Compare to B(x), the g.f. of the number of partitions of 3n into powers of 3 (A005704):
B(x) = exp( Sum_{n>=1} 3^A051064(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(3^n)).

Examples

			G.f.: A(x) =  1 + 3*x + 6*x^2 + 15*x^3 + 30*x^4 + 51*x^5 + 93*x^6 +...
log(A(x)) = 3*x + 3*x^2/2 + 18*x^3/3 + 3*x^4/4 + 3*x^5/5 + 18*x^6/6 + 3*x^7/7 + 3*x^8/8 + 81*x^9/9 + 3*x^10/10 + 3*x^11/11 + 18*x^12/12 +...
G.f. satisfies A(x) = A(x^3)*G(x) where G(x) is the g.f. of A161809:
G(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 21*x^4 + 33*x^5 + 51*x^6 +...
TRISECTIONS of g.f. begin:
T_0(x) = 1 + 15*x + 93*x^2 + 387*x^3 + 1311*x^4 + 3873*x^5 +...
T_1(x) = 3 + 30*x + 156*x^2 + 597*x^3 + 1920*x^4 + 5448*x^5 +...
T_2(x) = 6 + 51*x + 240*x^2 + 870*x^3 + 2697*x^4 + 7422*x^5 +...
where the ratios involve Fibonacci numbers:
T_1(x)/T_0(x) = 3*(1 - 5*x + 34*x^2 - 233*x^3 +...+ (-1)^n*Fibonacci(4n+1)*x^n +...);
T_2(x)/T_0(x) = 3*(2 - 13*x + 89*x^2 - 610*x^3 +...+ (-1)^n*Fibonacci(4n+3)*x^n +...).

Crossrefs

Cf. A161809, A183039, A051064, A005704, A183036.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,valuation(3*m,3)*3^valuation(3*m,3)*x^m/m)+x*O(x^n)),n)}

Formula

G.f. satisfies: A(x) = (1-x^3)/(1-x)^3 * A(x^3)^2/A(x^9).
G.f. satisfies: A(x) = A(x^3)*G(x) where G(x) = G(x^3)*(1+x+x^2)/(1-x)^2 is the g.f. of A161809.
Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:
T_1(x)/T_0(x) = 3*(1 + 2*x)/(1 + 7*x + x^2) and
T_2(x)/T_0(x) = 3*(2 + x)/(1 + 7*x + x^2).

A195761 G.f.: exp( Sum_{n>=1} A055457(n) * 5^A055457(n) * x^n/n ) where 5^A055457(n) exactly divides 5*n.

Original entry on oeis.org

1, 5, 15, 35, 70, 135, 255, 465, 810, 1345, 2180, 3480, 5465, 8410, 12645, 18720, 27405, 39690, 56785, 80120, 111840, 154805, 212590, 289485, 390495, 522640, 694955, 918490, 1206310, 1573495, 2040260, 2631955, 3379065, 4317210, 5487145, 6941780, 8746180, 10977565, 13725310
Offset: 0

Views

Author

Paul D. Hanna, Sep 23 2011

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 135*x^5 + 255*x^6 +...
log(A(x)) = 5*x + 5*x^2/2 + 5*x^3/3 + 5*x^4/4 + 50*x^5/5 + 5*x^6/6 + 5*x^7/7 + 5*x^8/8 + 5*x^9/9 + 50*x^10/10 +...
The coefficients in the QUINTISECTIONS of g.f. A(x) begin:
Q0: [1, 135, 2180, 18720, 111840, 522640, 2040260, 6941780, ...];
Q1: [5, 255, 3480, 27405, 154805, 694955, 2631955, 8746180, ...];
Q2: [15, 465, 5465, 39690, 212590, 918490, 3379065, 10977565, ...];
Q3: [35, 810, 8410, 56785, 289485, 1206310, 4317210, 13725310, ...];
Q4: [70, 1345, 12645, 80120, 390495, 1573495, 5487145, 17090945, ...].

Links

Crossrefs

Cf. A195760, A183038, A183036, A055457.

Programs

  • PARI
    {a(n)=local(N=ceil(log(n+6)/log(5)));polcoeff(1/prod(k=0,N,(1-x^(5^k) +x*O(x^n))^(4*k+5)),n)}
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, valuation(5*m, 5)*5^valuation(5*m, 5)*x^m/m)+x*O(x^n)), n)}

Formula

G.f.: A(x) = Product_{n>=0} 1/(1 - x^(5^n))^(4*n+5).
G.f. satisfies: A(x) = (1-x^5)/(1-x)^5 * A(x^5)^2/A(x^25).
G.f. satisfies: A(x) = A(x^5)*G(x) where G(x) = G(x^5)*(1-x^5)/(1-x)^5 is the g.f. of A195760.
Let the QUINTISECTIONS of A(x) be defined by:
A(x) = Q0(x^5) + x*Q1(x^5) + x^2*Q3(x^5) + x^3*Q3(x^5) + x^4*Q4(x^5),
then
_ Q0(x) = (1 + 121*x + 381*x^2 + 121*x^3 + x^4)/R(x)
_ Q1(x) = 5*(1 + 37*x + 73*x^2 + 14*x^3)/R(x)
_ Q2(x) = 5*(3 + 51*x + 64*x^2 + 7*x^3)/R(x)
_ Q3(x) = 5*(7 + 64*x + 51*x^2 + 3*x^3)/R(x)
_ Q4(x) = 5*(14 + 73*x + 37*x^2 + 1*x^3)/R(x)
where R(x) = (1-x)^5*Product_{n>=0} (1 - x^(5^n))^(4*n+9).
Further, the quintisections are related by:
_ Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2.
Showing 1-3 of 3 results.

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