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%I #12 Mar 30 2012 18:49:33
%S 1,2,2,3,4,3,4,5,5,4,5,6,6,7,5,6,7,7,8,8,6,7,8,8,9,10,9,7,8,9,9,10,11,
%T 10,11,8,9,10,10,11,12,11,12,13,9,10,11,11,12,13,12,13,14,14,10,11,12,
%U 12,13,14,13,14,15,15,14,11,12,13,13,14
%N Minimal number of parts of multiset repetition class defining partitions of n.
%C For the notion of m-multiset repetition class defining partitions of n see a comment in A185976 (with N replaced by n), and the characteristic array A176723 of such partitions in Abramowitz-Stegun order.
%C Note that there may be more than one multiset repetition class defining partition of n with minimal number of parts a(n). E.g., n=12, a(12)= 6, with two such partitions 1^2,2^2,3^2 and 1^3,2,3,4.
%F a(n)= min(sum(e[j],j=1..M)) with sum(j*e[j],j=1..M)=n, e[1]>=e[2]>=...>=e[M]>=1, and largest part M.
%F M takes all values from 1,...,Mmax(n), where Mmax(n) is the index of the largest triangular number from A000217 smaller or equal to n. E.g., Mmax(7) = 3.
%e The multiset repetition class defining partitions with minimal number of parts a(n) are, for n=1,...,12:
%e 1^1; 1^2; 1,2; 1^2,2; 1^3,2; 1,2,3; 1^2,2,3; 1^3,2,3;
%e 1^2,2^2,3; 1,2,3,4; 1^2,2,3,4;
%e 1^3,2,3,4, 1^2,2^2,3^2;...
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Mar 07 2011
%E Changed by the author in response to comments from Franklin T. Adams-Watters, Apr 02 2011.
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