

A185977


Minimal number of parts of multiset repetition class defining partitions of n.


1



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, 10, 11, 8, 9, 10, 10, 11, 12, 11, 12, 13, 9, 10, 11, 11, 12, 13, 12, 13, 14, 14, 10, 11, 12, 12, 13, 14, 13, 14, 15, 15, 14, 11, 12, 13, 13, 14
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OFFSET

1,2


COMMENTS

For the notion of mmultiset 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 AbramowitzStegun order.
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.


LINKS

Table of n, a(n) for n=1..70.


FORMULA

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.
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.


EXAMPLE

The multiset repetition class defining partitions with minimal number of parts a(n) are, for n=1,...,12:
1^1; 1^2; 1,2; 1^2,2; 1^3,2; 1,2,3; 1^2,2,3; 1^3,2,3;
1^2,2^2,3; 1,2,3,4; 1^2,2,3,4;
1^3,2,3,4, 1^2,2^2,3^2;...


CROSSREFS

Sequence in context: A256094 A063712 A340458 * A204006 A106251 A134478
Adjacent sequences: A185974 A185975 A185976 * A185978 A185979 A185980


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Mar 07 2011


EXTENSIONS

Changed by the author in response to comments from Franklin T. AdamsWatters, Apr 02 2011.


STATUS

approved



