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 A330773 Number of perfect compositions of n. 2
 1, 1, 1, 3, 1, 5, 1, 11, 3, 5, 1, 27, 1, 5, 5, 49, 1, 27, 1, 27, 5, 5, 1, 163, 3, 5, 11, 27, 1, 49, 1, 261, 5, 5, 5, 231, 1, 5, 5, 163, 1, 49, 1, 27, 27, 5, 1, 1109, 3, 27, 5, 27, 1, 163, 5, 163, 5, 5, 1, 435, 1, 5, 27, 1631, 5, 49, 1, 27, 5, 49, 1, 2055, 1, 5, 27, 27, 5, 49, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A perfect composition of n is one whose sequence of parts contains one composition of every positive integer less than n. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..20000 A. O. Munagi, Perfect Compositions of Numbers, J. Integer Seq. 23 (2020), art. 20.5.1. FORMULA a(1)=1, a(n) = Sum_{k=1..Omega(n+1)} k! * A251683(n+1,k), n>1. EXAMPLE a(7) = 11 because the perfect compositions are 1111111, 1222, 2221, 1114, 4111, 124, 142, 214, 241, 412, 421. For example, 241 generates the compositions of 1,...,6: 1,2,21,4,41,24. MAPLE b:= proc(n) option remember; expand(x*(1+add(b(n/d), d=numtheory[divisors](n) minus {1, n}))) end: a:= n-> (p-> add(coeff(p, x, i)*i!, i=1..degree(p)))(b(n+1)): seq(a(n), n=0..100); # Alois P. Heinz, Jan 15 2020 MATHEMATICA b[n_] := b[n] = x(1+Sum[b[n/d], {d, Divisors[n]~Complement~{1, n}}]); a[n_] := With[{p = b[n+1]}, Sum[Coefficient[p, x, i] i!, {i, Exponent[p, x]}]]; a /@ Range[0, 100] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *) CROSSREFS Cf. A001222, A002033, A074206, A251683, A330774. Sequence in context: A146935 A360756 A133730 * A112031 A146285 A146059 Adjacent sequences: A330770 A330771 A330772 * A330774 A330775 A330776 KEYWORD nonn,easy AUTHOR Augustine O. Munagi, Dec 30 2019 STATUS approved

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Last modified September 29 14:55 EDT 2023. Contains 365772 sequences. (Running on oeis4.)