1,2

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.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 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, December 1972.

W. Lang: First 10 rows.

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.

[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.

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: A013371 A013372 A080300 this_sequence A079302 A138806 A104117

Adjacent sequences: A116861 A116862 A116863 this_sequence A116865 A116866 A116867

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 24 2006