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A275964 Total number of nonzero digits with multiple occurrences in factorial base representation of n (counted with multiplicity): a(n) = A275812(A275735(n)).

%I A275964 #24 Feb 07 2024 01:18:10
%S A275964 0,0,0,2,0,0,0,2,2,3,0,2,0,0,0,2,2,2,0,0,0,2,0,0,0,2,2,3,0,2,2,3,3,4,
%T A275964 2,3,0,2,2,3,2,4,0,2,2,3,0,2,0,0,0,2,2,2,0,2,2,3,2,4,2,2,2,4,3,3,0,0,
%U A275964 0,2,2,2,0,0,0,2,0,0,0,2,2,3,0,2,0,0,0,2,2,2,2,2,2,4,2,2,0,0,0,2,0,0,0,2,2,3,0,2,0,0,0,2,2,2,0,0,0,2,0,0,0
%N A275964 Total number of nonzero digits with multiple occurrences in factorial base representation of n (counted with multiplicity): a(n) = A275812(A275735(n)).
%H A275964 Antti Karttunen, <a href="/A275964/b275964.txt">Table of n, a(n) for n = 0..40320</a>
%H A275964 Indranil Ghosh, <a href="/A275964/a275964.txt">Python program for computing this sequence</a>.
%H A275964 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%F A275964 a(n) = A275812(A275735(n)).
%F A275964 Other identities and observations. For all n >= 0.
%F A275964 a(n) = A275962(A225901(n)).
%F A275964 a(n) = A060130(n) - A275948(n).
%F A275964 a(n) >= A275949(n).
%e A275964 For n=0, with factorial base representation (A007623) also 0, there are no nonzero digits, thus a(0) = 0.
%e A275964 For n=2, with factorial base representation "10", there are no nonzero digits that are present multiple times, thus a(2) = 0.
%e A275964 For n=3 ("11") there is one nonzero digit which occurs more than once, and it occurs two times in total, thus a(3) = 2.
%e A275964 For n=41 ("1221") there are two distinct nonzero digits ("1" and "2"), and both occur more than once, namely twice each, thus a(41) = 2+2 = 4.
%e A275964 For n=44 ("1310") there are two distinct nonzero digits ("1" and "3"), but only the other (1) occurs more than once (two times), thus a(44) = 2.
%e A275964 For n=279 ("21211") there are two distinct nonzero digits present that occur more than once, digit 2 twice, and digit 1 for three times, thus a(279) = 2+3 = 5.
%t A275964 a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Total[Select[Tally[Select[s, # > 0 &]][[;;,2]], # > 1 &]]]; Array[a, 100, 0] (* _Amiram Eldar_, Feb 07 2024 *)
%o A275964 (Scheme) (define (A275964 n) (A275812 (A275735 n)))
%Y A275964 Cf. A275735, A275812.
%Y A275964 Cf. A265349 (indices of zeros), A265350 (of terms > 0).
%Y A275964 Cf. also A060130, A225901, A275948, A275949.
%K A275964 nonn,base
%O A275964 0,4
%A A275964 _Antti Karttunen_, Aug 15 2016