

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
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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 n1 in the following mixedradix expansion,
.. m m ... m m m1 ... 3 2 radix
.. !=m1 m1 ... m1 0 !=0 ... !=0 !=0 digit of n1
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 m1 digits immediately above the 0. That part of n1 is floor((n1)/m!) and is equal to floor(n/m!) since any carry when incrementing n1 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^(k1)*m! for each m >= 2. The modulus covers digits up to and including !=m1 for the given k. There are (m1)! combinations of permitted digit values within the modulus, so density (m1)!/(m^(k1)*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^(k1)1. This is m=2 and n1 = binary 011..110 where the number of 1's is j=k2.
(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. 233241. 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] != m1. [Keith, section 3]
a(n) = 2 + A286563(1+floor(n/m!), m), where m=A339013(n) and A286563(q,m) is the madic 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 n1 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 n1
 2 digits m1, 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



