A129595 Array A(i,j): A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), ... of elementwise sums of partitions encoded in the prime factorizations of i and j.

1, 2, 2, 3, 4, 3, 4, 9, 9, 4, 5, 8, 6, 8, 5, 6, 25, 27, 27, 25, 6, 7, 18, 15, 16, 15, 18, 7, 8, 49, 12, 125, 125, 12, 49, 8, 9, 16, 35, 54, 10, 54, 35, 16, 9, 10, 27, 81, 343, 45, 45, 343, 81, 27, 10, 11, 50, 18, 32, 21, 24, 21, 32, 18, 50, 11, 12, 121, 30, 81, 625, 175

Offset: 1

Author

Antti Karttunen, May 01 2007, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

Keywords

• nonn
• tabl

Comments

As described by Marc LeBrun, we can map integers 1-to-1 to partitions in a "crazy" order: factor n, take the (finite) tuple of exponents, add 1 to the first, use the rest as successive differences between parts and finally subtract 1 from the last part, thus we get the following partitions (elements in ascending order): 2 -> [1] -> 1, 3 -> [0,1] -> 1+1, 4 -> [2] -> 2, 5 -> [0,0,1] -> 1+1+1, 6 -> [1,1] -> 2+2, 7 -> [0,0,0,1] -> 1+1+1+1, 8 -> [3] -> 3, 9 -> [0,2] -> 1+2, 10 -> [1,0,1] -> 2+2+2, etc.

Inverse process: from a sorted (elements in ascending order) partition of n, subtract 1 from the first part, then take the first differences of parts and add 1 to the last (of differences or the first part if a singular partition) and use them as the exponents for A000040(1), A000040(2), etc. and multiply.

This array is obtained when we encode in such a way the partition obtained as an element-wise sum of two partitions encoded by i and j. The element-wise addition begins from the largest elements of the partitions, continuing towards the smaller elements and if the partitions do not contain the same number of elements, the shorter is prepended with as many zeros as needed to make them of equal length.

On what condition does A(i,j) = i*j ? E.g., A(3,5)=15, A(3,10)=30, A(5,11)=55. However A(3,7)=35 and A(5,7)=21.

Examples

			a(54) = A(9,2) = 27 because when we add element-wise partition 1+2 encoded by 9 to a singular partition 1 encoded by 2, we get partition 1+3, which maps to exponent tuple [0,3] and 27 = 2^0 * 3^3.
		

Links

• A. Karttunen, Table of n, a(n) for n = 1..1275

Crossrefs

Cf. A129593, A129596, A129597, A129598.

A122111 gives the involution of natural numbers induced when partition conjugation (see A129594) is applied to the same encoding.