cp's OEIS Frontend

A320105 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.)

%I A320105 #20 Oct 09 2018 15:11:40
%S A320105 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 A320105 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 A320105 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 A320105 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 A320105 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 A320105 From _Gus Wiseman_, Oct 09 2018: (Start)
%C A320105 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 A320105   {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}
%C A320105   {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}.
%C A320105 (End)
%H A320105 Antti Karttunen, <a href="/A320105/b320105.txt">Table of n, a(n) for n = 1..12960</a>
%H A320105 Antti Karttunen, <a href="/A320105/a320105.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%F A320105 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 A320105 a(A064988(n)) = A317145(n).
%o A320105 (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 A320105 (PARI)
%o A320105 memoA320105 = Map();
%o A320105 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 A320105 Cf. A064988, A300385, A302242, A317145.
%K A320105 nonn
%O A320105 1,12
%A A320105 _Antti Karttunen_, Oct 08 2018