A185975 - OEIS

%I #11 May 08 2018 15:11:56

%S 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,

%T 23,97,24,139,18,32,46,33,27,195,68,47,26,272,35,373,34,37,98,508,28,

%U 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

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

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

%C The prime number factorization

%C   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.

%C 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))].

%C   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].

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

%H %H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. pp. 831-2.

%F 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.

%e 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).

%Y Cf. A185974 (inverse map).

%K nonn

%O 2,2

%A _Wolfdieter Lang_, Feb 11 2011