A116864 Array of product of parts of the partitions of n with only prime parts.

0, 2, 0, 3, 0, 0, 0, 0, 4, 0, 0, 5, 0, 6, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 8, 0, 0, 0, 0, 7, 0, 10, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 20, 0, 27

Offset: 1

Author

Wolfdieter Lang, Mar 24 2006

Keywords

• nonn
• easy
• tabf

Comments

The inverse of sequence A001414 (sopfr(n)=sum of prime factors of n). See the examples and the W. Lang link.

The row length sequence of this array is p(n)=A000041(n) (number of partitions).

The partitions of n are ordered according to Abramowitz-Stegun (A-St), pp. 831-2.

Row n gives the values k for which A001414(k)=n>=2. E.g. n=10 appears 5 times in A001414, namely for the k values 21, 25, 30, 36 and 32.

Examples

			[0];
[2, 0];
[3, 0, 0];
[0, 0, 4, 0, 0];
[5, 0, 6, 0, 0, 0, 0];
...
a(4,3)=4 because the third partition of 4 is, in A-St order, (2,2)
with product 4. There is only this partition of 4 with only prime parts.
Row n=5 shows: n=5 appears twice in A001414(k), namely for k= 5 and
6. This is related to the two partitions (5) and (3,2) with only prime parts.
		

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
• W. Lang: First 10 rows.

Crossrefs

Row sums give A002098(n), n>=1.

Row sums (with nonzero numbers replaced by 1) give A000607(n), n>=1. See the array A116865.

Formula

a(n,k)=product(part(i),i=1..m(n,k)) if the k-th partition of n in the A-St order has only prime parts. Here m(n,k) is the number of parts of this partition. Otherwise a(n,k)=0. See A000040 for the prime numbers.