A185977 Minimal number of parts of multiset repetition class defining partitions of n.

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

Offset

1,2

Comments

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.

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 A369879 A106251

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. Adams-Watters, Apr 02 2011.

Status

approved