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

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, 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, 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

Offset

0,4

Links

Antti Karttunen, Table of n, a(n) for n = 0..40320

Indranil Ghosh, Python program for computing this sequence.

Index entries for sequences related to factorial base representation.

Formula

a(n) = A275812(A275735(n)).

Other identities and observations. For all n >= 0.

a(n) = A275962(A225901(n)).

a(n) = A060130(n) - A275948(n).

a(n) >= A275949(n).

Example

For n=0, with factorial base representation (A007623) also 0, there are no nonzero digits, thus a(0) = 0.
For n=2, with factorial base representation "10", there are no nonzero digits that are present multiple times, thus a(2) = 0.
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.
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.
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.
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.

Mathematica

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 *)

Prog

(Scheme) (define (A275964 n) (A275812 (A275735 n)))

Crossrefs

Cf. A275735, A275812.

Cf. A265349 (indices of zeros), A265350 (of terms > 0).

Cf. also A060130, A225901, A275948, A275949.

Sequence in context: A143063 A210703 A340455 * A284272 A175070 A343873

Adjacent sequences: A275961 A275962 A275963 * A275965 A275966 A275967

Keyword

nonn,base

Author

Antti Karttunen, Aug 15 2016

Status

approved