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A161189 Set a(n) = k if n is in the set zeta(k) - 1 in the notation defined by William J. Keith in 2010. 11
2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 3, 5, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 6, 3, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 4, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 3, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 3, 5, 2, 2, 2, 3, 2, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 6, 2, 2, 2, 3, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The numbers 2, 3, and 4 occur with density 0.929... since (zeta(2) - 1) + (zeta(3) - 1) + (zeta(4) - 1) = (Pi^2/6 - 1) + 0.20205... + 0.0823... = 0.929...

From Kevin Ryde, Dec 05 2020: (Start)

a(n) can be calculated by writing n-1 in the following mixed-radix expansion,

  ..   m     m  ...  m   m  m-1 ...  3   2    radix

  .. !=m-1  m-1 ... m-1  0  !=0 ... !=0 !=0   digit of n-1

           |-----j-----|                      a(n) = j+2

The least significant digit is radix 2, the next is radix 3, etc, until a 0 digit is found at radix m.  Further higher digits are radix m.  j is the number of consecutive m-1 digits immediately above the 0.  That part of n-1 is floor((n-1)/m!) and is equal to floor(n/m!) since any carry when incrementing n-1 to n will not go past the 0 digit.

Those n in class k, i.e., a(n)=k, can be characterized by certain sets of remainders n mod m^(k-1)*m! for each m >= 2.  The modulus covers digits up to and including !=m-1 for the given k.  There are (m-1)! combinations of permitted digit values within the modulus, so density (m-1)!/(m^(k-1)*m!) = 1/m^k (and total Sum_{m>=2} 1/m^k = zeta(k)-1).

The smallest n with a(n)=k is n = 2^(k-1)-1.  This is m=2 and n-1 = binary 011..110 where the number of 1's is j=k-2.

(End)

LINKS

Kevin Ryde, Table of n, a(n) for n = 1..10080

William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

FORMULA

Given [Keith's array, section 4]; and A143028 through A143034, which partitions the set of natural numbers according to asymptotic density of zeta(k) - 1: A2 = [1, 2, 4, 5, 6, 10, 12,...] = A143028, density zeta(2) - 1 = .6449... A3 = [3, 11, 14, 19, 27, 32,...] = A143029, density zeta(3) - 1 = .2020... A4 = [7, 23, 39, 50, 55, 71,...] = A143030, density zeta(4) - 1 = .0823... A5 = [15, 47, 79, 111, 143,....] = A143031, density zeta(5) - 1 = ........ ...etc, where Sum_{k=2..inf} (zeta(k) - 1) = 1.0 or 100%; such that "2" will occur with a frequency zeta(2) - 1 = .644...; "3" will occur with the frequency zeta(3) - 1 = .20205...; and "k" will occur with the frequency zeta(k) - 1. Thus a(n) = the zeta(k) - 1 subset to which n belongs, according to the system discovered by Keith.

From Kevin Ryde, Dec 05 2020: (Start)

a(n) = j+2 where n = L + m!*(b[0]*m^0 + b[1]*m^1 + b[2]*m^2 + ...) where m=A339013(n), L in the range 0 < L < m!, each digit b[i] in the range 0 <= b[i] < m, and smallest j where b[j] != m-1. [Keith, section 3]

a(n) = 2 + A286563(1+floor(n/m!), m), where m=A339013(n) and A286563(q,m) is the m-adic valuation of q (including A286563(q,m)=0 when m>q).

(End)

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k*(zeta(k)-1) = Pi^2/6 + 1. - Amiram Eldar, Feb 15 2021

EXAMPLE

Examples: A143028 gives a subset of terms within the natural number system that tend to density zeta(2) - 1 = (Pi^2/6 - 1) = .644...: where A143028 = [1, 2, 4, 5, 6, 8, 9, 10, 12,...]. Terms a(1), a(2), a(4),... = 2.

Similarly, zeta(3) - 1 = .20205..., denoted by A143029: [3, 11, 14, 19, 27, 32,...]; so terms a(3), a(11), a(14),...= 3.

From Kevin Ryde, Dec 05 2020: (Start)

For n = 880644, the mixed radix expansion of n-1 is

            m           lowest 0 digit gives m

  6 6  6 6  6 5 4 3 2   radix

  5 3  5 5  0 3 1 2 1   digit of n-1

      |---|             2 digits m-1, a(n)=2+2=4

(End)

MATHEMATICA

a[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Array[a, 30] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde's PARI code *)

PROG

(PARI) a(n) = n--; my(m=2, r); while([n, r]=divrem(n, m); r!=0, m++); 2+valuation(n+1, m); \\ Kevin Ryde, Dec 05 2020

CROSSREFS

Cf. A143028, A143029, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A286563, A339013 (B class).

Sequence in context: A072814 A196229 A191302 * A328162 A067132 A336500

Adjacent sequences:  A161186 A161187 A161188 * A161190 A161191 A161192

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Jun 06 2009

STATUS

approved

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Last modified June 21 03:51 EDT 2021. Contains 345354 sequences. (Running on oeis4.)