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.)

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, 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, 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

Offset: 1

Author

Antti Karttunen, Oct 08 2018

Keywords

• nonn

Comments

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).

From Gus Wiseman, Oct 09 2018: (Start)

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

{{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}

{{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12960
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Crossrefs

Cf. A064988, A300385, A302242, A317145.

Programs

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));

PARI

memoA320105 = Map();
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.

Formula

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.

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