cp's OEIS Frontend

A108462 Number of factorizations of (n,n) into pairs (i,j) with i,j >= 1, not both 1.

%I A108462 #24 Jan 18 2022 03:32:33
%S A108462 1,2,2,9,2,15,2,31,9,15,2,92,2,15,15,109,2,92,2,92,15,15,2,444,9,15,
%T A108462 31,92,2,203,2,339,15,15,15,712,2,15,15,444,2,203,2,92,92,15,2,1903,9,
%U A108462 92,15,92,2,444,15,444,15,15,2,1663,2,15,92,1043,15,203,2,92,15,203,2
%N A108462 Number of factorizations of (n,n) into pairs (i,j) with i,j >= 1, not both 1.
%C A108462 The rule of building products is (a,b)*(x,y) = (a*x,b*y).
%C A108462 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
%H A108462 R. J. Mathar, <a href="/A108462/b108462.txt">Table of n, a(n) for n = 1..999</a>
%H A108462 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A108462 a(A025487(n)) = A108463(n).
%F A108462 a(p^k) = A002774(k).
%F A108462 a(A002110(n)) = A020557(n).
%F A108462 a(n) = A108461(n,n).
%e A108462 From _Alois P. Heinz_ and _Antti Karttunen_, Nov 24 2017: (Start)
%e A108462 a(4) = 9 because for pair (4,4) there are nine factorizations:
%e A108462   (4,4)
%e A108462   (1,4)*(4,1)
%e A108462   (1,2)*(4,2)
%e A108462   (2,1)*(2,4)
%e A108462   (2,2)*(2,2)
%e A108462   (1,2)*(2,1)*(2,2)
%e A108462   (1,4)*(2,1)*(2,1)
%e A108462   (4,1)*(1,2)*(1,2)
%e A108462   (1,2)*(1,2)*(2,1)*(2,1)
%e A108462 (End)
%e A108462 a(pq) = 15 for primes p<>q: (pq,pq); (p,1)(q,pq); (p,1)(q,1)(1,pq); (p,1)(q,1)(1,p)(1,q); (p,1)(q,q)(1,p); (p,1)(q,p)(1,q); (p,q)(q,p); (p,q)(q,1)(1,p); (p,p)(q,q) ; (p,p)(q,1)(1,q); (p,pq)(q,1); (pq,1)(1,pq); (pq,1)(1,p)(1,q); (pq,q)(1,p); (pq,p)(1,q). - _R. J. Mathar_, Nov 30 2017
%o A108462 (PARI) a(n) = if(n==1, return(1)); my(b, c, r, x, y, v=List([]), w=List([[n]])); while(#w>r, c++; for(k=r+1, r=#w, y=w[k]; if(!isprime(x=y[c]), fordiv(x, d, if(d!=1&&d!=x, listput(w, concat([y[1..c-1], d, x/d]))))))); for(i=1, #w, x=w[i]; r=#x; for(j=1, #w, y=w[j]; for(k=0, 2^r-1, b=concat(b=binary(k), vector(r-#b)); if(#y>=t=vecsum(b), c=0; listput(v, vecsort(vector(r+#y-t, m, if(m>r, [1, y[m-r+t]], if(b[m], [x[m], y[c++]], [x[m], 1]))))))))); #Set(v); \\ _Jinyuan Wang_, Jan 17 2022
%Y A108462 Main diagonal of A108461.
%Y A108462 Cf. A002774, A020557, A108463.
%K A108462 nonn
%O A108462 1,2
%A A108462 _Christian G. Bower_, Jun 03 2005