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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. 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 (list; graph; refs; listen; history; text; internal format)
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 n-1 in factorial base representation ends in m-2 nonzero digits, so m = A339012(n-1) + 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*(k-1)). - 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. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

FORMULA

a(n) = A339012(n-1) + 2.

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]

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

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Last modified June 21 03:07 EDT 2021. Contains 345351 sequences. (Running on oeis4.)