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Product of digits+1 of n in factorial base.
8

%I #27 Feb 14 2024 01:04:51

%S 1,2,2,4,3,6,2,4,4,8,6,12,3,6,6,12,9,18,4,8,8,16,12,24,2,4,4,8,6,12,4,

%T 8,8,16,12,24,6,12,12,24,18,36,8,16,16,32,24,48,3,6,6,12,9,18,6,12,12,

%U 24,18,36,9,18,18,36,27,54,12,24,24,48,36,72,4,8,8

%N Product of digits+1 of n in factorial base.

%H Antti Karttunen, <a href="/A227154/b227154.txt">Table of n, a(n) for n = 0..5040</a> (corrected by Ray Chandler, Jan 19 2019)

%H Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Edgar/edgar3.html">Some Combinatorics of Factorial Base Representations</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.

%e 5 has factorial base representation A007623(5) = "21" (as 2*2 + 1*1 = 5). Adding one to these digits and multiplying, we get 3*2 = 6, thus a(5)=6.

%t a[n_] := Module[{k = n, m = 2, r, p = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, p *= (r+1); m++]; p]; Array[a, 100, 0] (* _Amiram Eldar_, Feb 13 2024 *)

%o (MIT/GNU Scheme):

%o (define (A227154 n) (apply * (map 1+ (n->factbase n))))

%o (define (n->factbase n) (let loop ((n n) (fex (if (zero? n) (list 0) (list))) (i 2)) (cond ((zero? n) fex) (else (loop (floor->exact (/ n i)) (cons (modulo n i) fex) (1+ i))))))

%o (PARI) a(n)=my(b=2,t=1); while(n, t *= n%b + 1; n \= b; b++); t \\ _Charles R Greathouse IV_, Jun 06 2017

%Y Cf. A007623, A208575, A227153.

%K nonn,base

%O 0,2

%A _Antti Karttunen_, Jul 04 2013

%E a(0)=1 added by _Tom Edgar_, Jun 05 2017