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

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

Offset: 1

Author

Kevin Ryde, Nov 19 2020

Keywords

• base
• nonn

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.

Crossrefs

Cf. A005408 (class B_2), A016933 (class B_3).

Cf. A161189 (class number in partition A_k), A339012.

Programs

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 *)

PARI

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

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.