A339012 - OEIS

%I #18 Feb 15 2021 08:09:45

%S 0,1,0,2,0,2,0,1,0,3,0,3,0,1,0,3,0,3,0,1,0,3,0,3,0,1,0,2,0,2,0,1,0,4,

%T 0,4,0,1,0,4,0,4,0,1,0,4,0,4,0,1,0,2,0,2,0,1,0,4,0,4,0,1,0,4,0,4,0,1,

%U 0,4,0,4,0,1,0,2,0,2,0,1,0,4,0,4,0,1,0

%N Written in factorial base, n ends in a(n) consecutive non-0 digits.

%C Also, a(n) is the least p for which n mod (p+2)! < (p+1)!.  A small remainder like this means a 0 digit at position p in the factorial base representation of n, where the least significant digit is position p=0.  The least such p means only nonzero digits below.

%C Those n with a(n)=p are characterized by remainders n mod (p+2)!, per the formula below.  These remainders are terms of A227157 which is factorial base digits all nonzero.  A227157 can be taken by rows where row p lists the terms having p digits in factorial base.  Each digit ranges from 1 up to 1,2,3,... respectively so there are p! values in a row, and so the asymptotic density of terms p here is p!/(p+2)! = 1/((p+2)*(p+1)) = 1/A002378(p+1) = 1/2, 1/6, etc.

%C The smallest n with a(n)=p is the factorial base repunit n = 11..11 with p 1's = A007489(p).

%H Kevin Ryde, <a href="/A339012/b339012.txt">Table of n, a(n) for n = 0..10080</a>

%F a(n)=p iff n mod (p+2)! is a term in row p of A227157 (row p terms having p digits), including p=0 by reckoning an initial A227157(0) = 0 as no digits.

%F a(n)=0 iff n mod 2 = 0.

%F a(n)=1 iff n mod 6 = 1, which is A016921.

%F a(n)=2 iff n mod 24 = 3 or 5.

%e n = 10571 written in factorial base is 2,0,4,0,1,2,1.  It ends in 3 consecutive nonzero digits (1,2,1) so a(10571) = 3.  Remainder 10571 mod (3+2)! = 11 is in A227157 row 3.

%t a[n_] := Module[{k = n, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; m - 2]; Array[a, 30, 0] (* _Amiram Eldar_, Feb 15 2021 after _Kevin Ryde_'s PARI code *)

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

%Y Cf. A108731 (factorial base digits), A016921 (where a(n)=1), A339013 (+2), A230403 (ending 0's).

%Y In other bases: A215887, A328570.

%K base,nonn

%O 0,4

%A _Kevin Ryde_, Nov 19 2020