

A339013


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


12



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, 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, 2, 6, 2, 6, 2, 3, 2, 4, 2, 4, 2, 3, 2, 6, 2, 6, 2, 3, 2
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OFFSET

1,1


COMMENTS

a(n)=m when n is in class B_m. Keith's residues formula in lemma 1 is equivalent to requiring that n1 in factorial base representation ends in m2 nonzero digits, so m = A339012(n1) + 2.
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).
The frequency of appearance of the term k = 2, 3, ... in this sequence is 1/(k*(k1)).  Amiram Eldar, Feb 15 2021


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

a(n) = A339012(n1) + 2.
a(n) = m iff n == 1 + Sum_{j=1..m2} d[j]*j! (mod m!) with d[j] in ranges 1 <= d[j] <= j. [Keith, section 2.1 lemma 1]
a(n)=2 iff n mod 2 = 1. [Keith section 4 residues].
a(n)=3 iff n mod 6 = 2.
a(n)=4 iff n mod 24 = 4 or 6.
a(n)=5 iff n mod 120 = any of 10, 12, 16, 18, 22, 24.


MATHEMATICA

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 *)


PROG

(PARI) a(n) = n; my(b=2, r); while([n, r]=divrem(n, b); r!=0, b++); b;


CROSSREFS

Cf. A005408 (class B_2), A016933 (class B_3).
Cf. A161189 (class number in partition A_k), A339012.
Sequence in context: A076399 A286602 A286600 * A136529 A113982 A256542
Adjacent sequences: A339010 A339011 A339012 * A339014 A339015 A339016


KEYWORD

base,nonn


AUTHOR

Kevin Ryde, Nov 19 2020


STATUS

approved



