A365458 -id:A365458 - cp's OEIS frontend

A365459 a(n) = n - the largest power of 3 that is less than or equal to n.

0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 0, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Antti Karttunen, Sep 17 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..19683 (3^9 = 19683).

Cf. A000244, A053735 (gives the number of iterations needed to reach 0), A365458.

Array[# - 3^Floor@ Log[3, #] &, 88] (* Michael De Vlieger, Sep 17 2023 *)

A365459(n) = if(1==n,0,my(k=0); while((3^k) < n, k++); if((3^k) > n,k--); n-(3^k));

def A365459(n):
    kmin, kmax = 0, 1
    while 3**kmax <= n:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if 3**kmid > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return n-3**kmin # Chai Wah Wu, Sep 17 2023

a(n) = n - A365458(n).

a(n) = n - 3^floor((log n)/(log 3)). - Michael De Vlieger, Sep 17 2023

A374336 a(n) is the numerator of x(n) = (2*x(n-1) + c(n)) mod 1, where c(n) = 1/n if n is a power of 3 and 0 otherwise, with x(0) = 0.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4, 8, 16, 5, 10, 20, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4, 8, 16, 5, 10, 20, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4

Offset: 0

Paolo Xausa, Jul 06 2024

A constant alpha, defined as alpha = Sum_{n >= 1} p(n)/(q(n)*b^n), is b-normal if and only if the associated sequence, defined by x(0) = 0 and x(n) = (b*x(n-1) + p(n)/q(n)) mod 1, is equidistributed in the unit interval.
The present sequence gives the numerators of the associated sequence (where b = 2) for alpha_0 = Sum_{n >= 1} 1/((3^n)*2^(3^n)) = A192014. See Bailey and Borwein (2005), pp. 505-506 (third example of Theorem 3). They show that alpha_0, as well as any constant defined as Sum_{n >= 1} 1/((3^n)*2^(3^n+r_n)) (where r_n is the n-th binary digit of the real number r in the [0,1) interval), is 2-normal and transcendental.
Bailey and Borwein also note that terms follow a pattern of triply repeating segments, each of length 2*3^m and containing all integers relative prime to and less than 3^(m+1).
Denominators are given by A365458.

Paolo Xausa, Table of n, a(n) for n = 0..10000
David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the American Mathematical Society, May 2005, Vol. 52, No. 5, pp. 502-514.

Cf. A192014, A374332, A374334, A365458 (denominators).

Block[{n = 0}, Numerator[NestList[Mod[2*# + If[IntegerQ[Log[3, ++n]], 1/n, 0], 1] &, 0, 100]]]

cp's OEIS Frontend

A365459 a(n) = n - the largest power of 3 that is less than or equal to n.

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