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A319711 Sum of A034968(d) over proper divisors d of n, where A034968 gives the sum of digits in factorial base.

%I A319711 #13 Mar 06 2024 00:59:39
%S A319711 0,1,1,2,1,4,1,4,3,5,1,7,1,4,6,6,1,8,1,10,5,6,1,11,4,5,6,9,1,15,1,10,
%T A319711 7,7,6,15,1,6,6,16,1,15,1,13,13,8,1,16,3,10,8,9,1,14,8,14,7,6,1,25,1,
%U A319711 5,13,13,7,18,1,13,9,18,1,21,1,6,12,12,7,15,1,25,9,8,1,26,9,7,7,20,1,29,6,16,6,9,8,21,1,10,14,19,1,18,1,15,22
%N A319711 Sum of A034968(d) over proper divisors d of n, where A034968 gives the sum of digits in factorial base.
%H A319711 Antti Karttunen, <a href="/A319711/b319711.txt">Table of n, a(n) for n = 1..65537</a>
%H A319711 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%F A319711 a(n) = Sum_{d|n, d<n} A034968(d).
%F A319711 a(n) = A319712(n) - A034968(n).
%t A319711 d[n_] := Module[{k = n, m = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := DivisorSum[n, d[#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Mar 05 2024 *)
%o A319711 (PARI)
%o A319711 A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
%o A319711 A319711(n) = sumdiv(n,d,(d<n)*A034968(d));
%Y A319711 Cf. A034968, A319712, A319713.
%K A319711 nonn,base
%O A319711 1,4
%A A319711 _Antti Karttunen_, Oct 02 2018