A083854 -id:A083854 - cp's OEIS frontend

A083855 Multiplicands in the list of numbers which are squares, twice squares, three times squares, or six times squares (A083854).

1, 2, 3, 1, 6, 2, 1, 3, 1, 2, 6, 1, 3, 2, 1, 3, 1, 2, 6, 1, 2, 3, 1, 6, 2, 1, 3, 1, 2, 1, 3, 6, 2, 1, 3, 1, 2, 6, 1, 2, 3, 1, 2, 1, 6, 3, 1, 2, 1, 3, 6, 2, 1, 3, 1, 2, 1, 6, 3, 2, 1, 1, 2, 3, 6, 1, 2, 3, 1, 2, 6, 1, 3, 1, 2, 1, 6, 3, 2, 1, 1, 2, 3, 6, 1, 2, 3, 1, 2, 1, 6, 3, 1, 2, 1, 3, 6, 2, 1, 1, 3, 2, 1, 6, 2

Offset: 1

Henry Bottomley, May 06 2003

nonn

1, 2, 3 and 6 appear in the ratios sqrt(6):sqrt(3):sqrt(2):1, i.e. with proportions 0.3713..., 0.2626..., 0.2144... and 0.1516... respectively. A083855 is that which would appear using the d'Hondt method (A055440) on these.

			A083854 starts 1, 2, 3, 4, 6, 8, 9, 12, 16, etc.; i.e. 1*1^2, 2*1^2, 3*1^2, 1*2^2, 6*1^2, 2*2^2, 1*3^2, 3*2^2, 1*4^2, etc. giving 1, 2, 3, 1, 6, 2, 1, 3, 1, etc.

a(n) =A007913(A083854(n)).

A225837 Numbers of form 2^i3^j(6k+1), i, j, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 25, 26, 27, 28, 31, 32, 36, 37, 38, 39, 42, 43, 48, 49, 50, 52, 54, 55, 56, 57, 61, 62, 63, 64, 67, 72, 73, 74, 75, 76, 78, 79, 81, 84, 85, 86, 91, 93, 96, 97, 98, 100, 103, 104, 108, 109, 110, 111, 112

Offset: 1

Ralf Stephan, May 16 2013

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022
From Peter Munn, Nov 16 2023: (Start)
Contains all nonzero squares.
Dividing by 5 the terms that are multiples of 5 gives its complement, A225838.
(A352272, 2*A352272, 3*A352272, 6*A352272) is a partition of the terms.
The terms form a subgroup of the positive integers under the operation A059897(.,.) and are the positive integers in an index 2 multiplicative subgroup of rationals that is generated by 2, 3 and integers congruent to 1 modulo 6. See A225857 and A352272 for further information about such subgroups.
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A225838.
Subsequences: A003136\{0}, A083854\{0}, A260488\{0}, A352272.
Symmetric difference of A026225 and A036554; of A036668 and A189716.
Cf. A016921, A059897, A225857.

[n: n in [1..200] | IsOne(d mod 6) where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013

mx = 122; t = {}; Do[n = 2^i*3^j (6 k + 1); If[n <= mx, AppendTo[t, n]], {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, mx/6}]; Union[t] (* T. D. Noe, May 16 2013 *)

for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%6==1),print1(n,",")))

from sympy import integer_log
def A225837(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x//3**i>>j)+5)//6 for i in range(integer_log(x,3)[0]+1) for j in range((x//3**i).bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Feb 02 2025

A083583 a(n) = (83^n - 50^n)/3.

Original entry on oeis.org

1, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616, 2259436291848

Offset: 0

Paul Barry, May 01 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A083854.

[(8*3^n-5*(0)^n)/3: n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

Join[{1},NestList[3#&,8,30]] (* Harvey P. Dale, Apr 08 2020 *)

a(n) = (8*3^n - 5*0^n)/3.
G.f.: (1+5*x)/(1-3*x).
E.g.f.: (8*exp(3x) - 5*exp(0))/3.
a(n) = A005051(n-1), n > 0. - R. J. Mathar, Sep 17 2008

A083585 (85^n - 52^n)/3.

Original entry on oeis.org

1, 10, 60, 320, 1640, 8280, 41560, 208120, 1041240, 5207480, 26039960, 130204920, 651034840, 3255194680, 16276014360, 81380153720, 406900932440, 2034504989880, 10172525604760, 50862629334520, 254313149294040

Offset: 0

Paul Barry, May 01 2003

Binomial transform of A083854.

Index entries for linear recurrences with constant coefficients, signature (7,-10).

LinearRecurrence[{7, -10}, {1, 10}, 20] (* T. D. Noe, Aug 06 2013 *)

a(n) = (8*5^n-5*2^n)/3.
G.f. (1+3x)/((1-2x)(1-5x)).
E.g.f. (8*exp(5x) - 5*exp(2x))/3.

cp's OEIS Frontend

A083855 Multiplicands in the list of numbers which are squares, twice squares, three times squares, or six times squares (A083854).

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A225837 Numbers of form 2^i3^j(6k+1), i, j, k >= 0.

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A083583 a(n) = (83^n - 50^n)/3.

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A083585 (85^n - 52^n)/3.

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cp's OEIS Frontend

A083855 Multiplicands in the list of numbers which are squares, twice squares, three times squares, or six times squares (A083854).

Views

Author

Keywords

Comments

Examples

Formula

A225837 Numbers of form 2^i*3^j*(6k+1), i, j, k >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A083583 a(n) = (8*3^n - 5*0^n)/3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A083585 (8*5^n - 5*2^n)/3.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A225837 Numbers of form 2^i3^j(6k+1), i, j, k >= 0.

A083583 a(n) = (83^n - 50^n)/3.

A083585 (85^n - 52^n)/3.