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A339013 Class number m containing n in a partitioning of the natural numbers into classes B_m by William J. Keith.

%I A339013 #14 Feb 15 2021 02:32:06
%S A339013 2,3,2,4,2,4,2,3,2,5,2,5,2,3,2,5,2,5,2,3,2,5,2,5,2,3,2,4,2,4,2,3,2,6,
%T A339013 2,6,2,3,2,6,2,6,2,3,2,6,2,6,2,3,2,4,2,4,2,3,2,6,2,6,2,3,2,6,2,6,2,3,
%U A339013 2,6,2,6,2,3,2,4,2,4,2,3,2,6,2,6,2,3,2
%N A339013 Class number m containing n in a partitioning of the natural numbers into classes B_m by William J. Keith.
%C A339013 a(n)=m when n is in class B_m. Keith's residues formula in lemma 1 is equivalent to requiring that n-1 in factorial base representation ends in m-2 nonzero digits, so m = A339012(n-1) + 2.
%C A339013 a(n)=m iff n mod m! is among certain residue classes determined by m. The residues for A339012 are rows of A227157 and here add +1 to each residue (mod m!).  For example 3 or 5 (mod 24) in A339012 becomes here 4 or 6 (mod 24).
%C A339013 The frequency of appearance of the term k = 2, 3, ... in this sequence is 1/(k*(k-1)). - _Amiram Eldar_, Feb 15 2021
%H A339013 Kevin Ryde, <a href="/A339013/b339013.txt">Table of n, a(n) for n = 1..10080</a>
%H A339013 William J. Keith, <a href="https://www.emis.de/journals/INTEGERS/papers/k19/k19.Abstract.html">Sequences of Density zeta(K) - 1</a>, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also <a href="http://arxiv.org/abs/0905.3765">arXiv preprint</a>, arXiv:0905.3765 [math.NT], 2009 and <a href="http://www.math.drexel.edu/~keith/ZetaKMinusOne.pdf">author's copy</a>.
%F A339013 a(n) = A339012(n-1) + 2.
%F A339013 a(n) = m iff n == 1 + Sum_{j=1..m-2} d[j]*j! (mod m!) with d[j] in ranges 1 <= d[j] <= j. [Keith, section 2.1 lemma 1]
%F A339013 a(n)=2 iff n mod 2 = 1. [Keith section 4 residues].
%F A339013 a(n)=3 iff n mod 6 = 2.
%F A339013 a(n)=4 iff n mod 24 = 4 or 6.
%F A339013 a(n)=5 iff n mod 120 = any of 10, 12, 16, 18, 22, 24.
%t A339013 a[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; m]; Array[a, 30] (* _Amiram Eldar_, Feb 15 2021 after _Kevin Ryde_'s PARI code *)
%o A339013 (PARI) a(n) = n--; my(b=2,r); while([n,r]=divrem(n,b);r!=0, b++); b;
%Y A339013 Cf. A005408 (class B_2), A016933 (class B_3).
%Y A339013 Cf. A161189 (class number in partition A_k), A339012.
%K A339013 base,nonn
%O A339013 1,1
%A A339013 _Kevin Ryde_, Nov 19 2020