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
KEYWORD
base,nonn
AUTHOR
Kevin Ryde, Nov 19 2020
STATUS
approved