A161189 Set a(n) = k if n is in the set zeta(k) - 1 in the notation defined by William J. Keith in 2010.

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

Offset: 1

Author

Gary W. Adamson, Jun 06 2009

Keywords

• nonn

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)

Examples

			Examples: A143028 gives a subset of terms within the natural number system that tend to density zeta(2) - 1 = Pi^2/6 - 1 = 0.644...: where A143028 = [1, 2, 4, 5, 6, 8, 9, 10, 12, ...]. Terms a(1), a(2), a(4), ... = 2.
Similarly, zeta(3) - 1 = 0.20205..., 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)
		

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.

Crossrefs

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

Programs

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 *)

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

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 = 0.6449...

A3 = [3, 11, 14, 19, 27, 32, ...] = A143029, density zeta(3) - 1 = 0.2020...

A4 = [7, 23, 39, 50, 55, 71, ...] = A143030, density zeta(4) - 1 = 0.0823...

A5 = [15, 47, 79, 111, 143, ...] = A143031, density zeta(5) - 1 = ........ etc., where Sum_{k>=2} (zeta(k) - 1) = 1.0 or 100%; such that "2" will occur with a frequency zeta(2) - 1 = 0.644...; "3" will occur with the frequency zeta(3) - 1 = 0.20205...; 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