A320105 - OEIS

%I #20 Oct 09 2018 15:11:40

%S 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,4,1,1,1,2,1,3,1,1,1,1,

%T 1,6,1,1,1,4,1,3,1,2,2,1,1,8,1,2,1,2,1,4,1,4,1,1,1,11,1,1,2,1,1,3,1,2,

%U 1,3,1,16,1,1,2,2,1,3,1,8,2,1,1,11,1,1,1,4,1,10,1,2,1,1,1,16,1,2,2,6,1,3,1,4,3

%N If A001222(n) <= 2, a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)*primepi(q)) * (n/(p*q))), where p and q range over distinct primes dividing n. (See formula section for exact details.)

%C This is an auxiliary function for computing A317145 with help of A064988. Note the similarity of the formula to that of A300385, with only difference being in the value of a(1) and that here we have multiplication (*) instead of addition (+) between primepi(p) and primepi(q).

%C From _Gus Wiseman_, Oct 09 2018: (Start)

%C Combinatorial interpretation is: In the poset of multiset partitions ordered by refinement, number of maximal chains from the n-th multiset multisystem (A302242) to the maximal multiset partition of the same multiset, assuming n is odd. For example, the a(315) = 10 maximal chains are

%C   {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}

%C   {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}.

%C (End)

%H Antti Karttunen, <a href="/A320105/b320105.txt">Table of n, a(n) for n = 1..12960</a>

%H Antti Karttunen, <a href="/A320105/a320105.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%F If A001222(n) <= 2 [when n is one, a prime or semiprime], a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)*primepi(q)) * (n/(p*q))), where p ranges over all distinct primes dividing n, and q also ranges over primes dividing n, but with condition that q > p if p is a unitary prime factor of n, otherwise q >= p. Here primepi = A000720.

%F a(A064988(n)) = A317145(n).

%o (PARI) A320105(n) = if(bigomega(n)<=2,1,my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A320105(prime(primepi(f[i,1])*primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); (s));

%o (PARI)

%o memoA320105 = Map();

%o A320105(n) = if(bigomega(n)<=2,1,if(mapisdefined(memoA320105,n), mapget(memoA320105,n), my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A320105(prime(primepi(f[i,1])*primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA320105,n,s); (s))); \\ Memoized version.

%Y Cf. A064988, A300385, A302242, A317145.

%K nonn

%O 1,12

%A _Antti Karttunen_, Oct 08 2018