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%I #31 Jan 24 2024 01:50:04
%S 0,1,1,2,1,1,1,2,2,3,1,2,1,1,1,2,2,2,1,1,1,2,1,1,1,2,2,3,1,2,2,3,3,4,
%T 2,3,1,2,2,3,2,2,1,2,2,3,1,2,1,1,1,2,2,2,1,2,2,3,2,2,2,2,2,2,3,3,1,1,
%U 1,2,2,2,1,1,1,2,1,1,1,2,2,3,1,2,1,1,1,2,2,2,2,2,2,2,2,2,1,1,1,2,1,1,1,2,2,3,1,2,1,1,1,2,2,2,1,1,1,2,1,1,1,2
%N a(n) = number of occurrences of a most frequent nonzero digit in factorial base representation (A007623) of n.
%H Antti Karttunen, <a href="/A264990/b264990.txt">Table of n, a(n) for n = 0..10080</a>
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%F a(0) = 0; for n >= 1, a(n) = max(A257511(n), a(A257684(n)).
%F Other identities. For all n >= 0:
%F From _Antti Karttunen_, Aug 15 2016: (Start)
%F a(n) = A275811(A225901(n)).
%F a(n) = A051903(A275735(n)).
%F (End)
%e n A007623(n) a(n) [highest number of times any nonzero digit occurs].
%e 0 = 0 0 (because no nonzero digits present)
%e 1 = 1 1
%e 2 = 10 1
%e 3 = 11 2
%e 4 = 20 1
%e 5 = 21 1
%e 6 = 100 1
%e 7 = 101 2
%e 8 = 110 2
%e 9 = 111 3
%e 10 = 120 1
%e 11 = 121 2
%e 12 = 200 1
%e 13 = 201 1
%e 14 = 210 1
%e 15 = 211 2
%e 16 = 220 2
%e 17 = 221 2
%e 18 = 300 1
%e and for n=63 we have:
%e 63 = 2211 2.
%t a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Max[Tally[Select[s, # > 0 &]][[;;,2]]]]; a[0] = 0; Array[a, 100, 0] (* _Amiram Eldar_, Jan 24 2024 *)
%o (Scheme with memoization-macro definec)
%o (definec (A264990 n) (if (zero? n) n (max (A257511 n) (A264990 (A257684 n)))))
%o (Python)
%o from sympy import prime, factorint
%o from operator import mul
%o import collections
%o def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
%o def a051903(n): return 0 if n==1 else max(factorint(n).values())
%o def a275735(n):
%o y=collections.Counter(map(int, list(str(a007623(n)).replace("0", "")))).most_common()
%o return 1 if n==0 else reduce(mul, [prime(y[i][0])**y[i][1] for i in range(len(y))])
%o def a(n): return 0 if n==0 else a051903(a275735(n))
%o print([a(n) for n in range(201)]) # _Indranil Ghosh_, Jun 20 2017
%Y Cf. A007623, A051903, A225901, A257511, A257684, A275735, A275811.
%Y Cf. A265349 (positions of terms <= 1), A265350 (positions of term > 1).
%Y Cf. also A266117, A266118.
%K nonn,base
%O 0,4
%A _Antti Karttunen_, Dec 22 2015
%E Name changed by _Antti Karttunen_, Aug 15 2016
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