A185975 Prime number factorization of n mapped to a(n)-th partition in A-St order.

1, 2, 3, 4, 5, 7, 6, 9, 8, 12, 10, 19, 13, 14, 11, 30, 16, 45, 15, 21, 20, 67, 17, 22, 31, 25, 23, 97, 24, 139, 18, 32, 46, 33, 27, 195, 68, 47, 26, 272, 35, 373, 34, 37, 98, 508, 28, 49, 36, 69, 50, 684, 40, 48, 38, 99, 140, 915, 39, 1212, 196, 53, 29, 70, 51, 1597, 72, 141, 52, 2087, 42, 2714, 273, 54, 103, 71, 73, 3506, 41, 59, 374, 4508, 56, 100, 509, 197, 55, 5763, 58, 101, 145, 274, 685, 142, 43, 7338, 75, 76, 57

Offset: 2

Author

Wolfdieter Lang, Feb 11 2011

Keywords

• nonn

Comments

This is the inverse of the map n->A185974(n), n>=1.

The prime number factorization

n = p(1)^e(1)*p(2)^e(2)*...*p(M)^e(M), with e(M)>=1, and e(j)>=0, j=1,..,M-1, with the prime numbers p(j):=A000040(j), is mapped to the partition 1^e(1),2^e(2),...,M^e(M), with M=M(n) = A061395(n). Note that j^0 means that j does not show up in the partition, it is not 1. a(n) is the position of this partition of N=N(n):=sum(j*e(j), j=1..M(n)) in the A-St (Abramowitz-Stegun) list of all partitions. See A036036 and the reference for this order.

In order to obtain an exponent list of length N=N(n), appropriate for a partition of N(n), one has to append N(n)-M(n)>=0 zeros to the list [e(1),e(2),...,e(M(n))].

E.g., n=10, M(10)=3, N(10)=4, from the partition 1^2,3^1; N(10)-M(10)= 1, hence the complete exponent list for this partition is [2,0,1,0].

In order to have offset 1 one could add a(1):=0.

Examples

			a(20)=15 because 20=p(1)^2 p(3)^1 which maps to the partition 1^2,3^1 (of N(20)=5) which appears at position 15 in the A-St list. M(20)=3, hence the exponent list for this partition is [2,0,1,0,0] (2=5-3 zeros added to the list [2,0,1] from the prime number factorization exponent list).
		

Links

• %H 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]. pp. 831-2.

Crossrefs

Cf. A185974 (inverse map).

Formula

a(n) gives the a(n)-th position of the partition in A-St order (see A036036 and the reference) obtained from the prime number factorization of n read as partition. This mapping is explained in the comment above.