cp's OEIS Frontend

A317145 Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.

%I A317145 #17 Oct 12 2018 04:42:51
%S A317145 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,5,1,1,1,2,1,3,1,4,1,1,
%T A317145 1,7,1,1,1,5,1,3,1,2,2,1,1,15,1,2,1,2,1,5,1,5,1,1,1,11,1,1,2,11,1,3,1,
%U A317145 2,1,3,1,26,1,1,2,2,1,3,1,15,2,1,1,11,1,1,1,5,1,11,1,2,1,1,1,52,1,2,2,7,1,3,1,5,3
%N A317145 Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.
%C A317145 If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.
%C A317145 a(n) depends only on prime signature of n (cf. A025487). - _Antti Karttunen_, Oct 08 2018
%H A317145 Antti Karttunen, <a href="/A317145/b317145.txt">Table of n, a(n) for n = 1..11520</a>
%H A317145 Antti Karttunen, <a href="/A317145/a317145.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H A317145 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A317145 a(prime^n) = A002846(n).
%F A317145 a(n) = A320105(A064988(n)). - _Antti Karttunen_, Oct 08 2018
%e A317145 The a(36) = 7 maximal chains:
%e A317145   (2*2*3*3) < (2*2*9) < (2*18) < (36)
%e A317145   (2*2*3*3) < (2*2*9) < (4*9)  < (36)
%e A317145   (2*2*3*3) < (2*3*6) < (2*18) < (36)
%e A317145   (2*2*3*3) < (2*3*6) < (3*12) < (36)
%e A317145   (2*2*3*3) < (2*3*6) < (6*6)  < (36)
%e A317145   (2*2*3*3) < (3*3*4) < (3*12) < (36)
%e A317145   (2*2*3*3) < (3*3*4) < (4*9)  < (36)
%o A317145 (PARI)
%o A317145 A064988(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); factorback(f); }; \\ From A064988
%o A317145 memoA320105 = Map();
%o A317145 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)));
%o A317145 A317145(n) = A320105(A064988(n)); \\ _Antti Karttunen_, Oct 08 2018
%Y A317145 Cf. A001055, A002846, A007716, A045778, A064988, A162247, A213427, A275024, A281113, A299202, A300385, A317144, A317146, A320105.
%K A317145 nonn
%O A317145 1,12
%A A317145 _Gus Wiseman_, Jul 22 2018
%E A317145 Data section extended to 105 terms by _Antti Karttunen_, Oct 08 2018