

A116864


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


4



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
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OFFSET

1,2


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 AbramowitzStegun (ASt), pp. 8312.
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.


LINKS

Table of n, a(n) for n=1..78.
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.


FORMULA

a(n,k)=product(part(i),i=1..m(n,k)) if the kth partition of n in the ASt 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.


EXAMPLE

[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 ASt 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.


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.
Sequence in context: A013372 A209640 A080300 * A255308 A079302 A138806
Adjacent sequences: A116861 A116862 A116863 * A116865 A116866 A116867


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang, Mar 24 2006


STATUS

approved



