%I #36 Oct 03 2023 13:14:24
%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,3,1,1,
%T 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,4,1,1,1,1,
%U 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1
%N Number of factorizations into distinct prime powers greater than 1.
%C 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 Antti Karttunen, <a href="/A050361/b050361.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from Reinhard Zumkeller)
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F Dirichlet g.f.: Product_{n is a prime power >1}(1 + 1/n^s).
%F Multiplicative with a(p^e) = A000009(e).
%F a(A002110(k))=1.
%F a(n) = A050362(A101296(n)). - _R. J. Mathar_, May 26 2017
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.26020571070524171076..., where f(x) = (1-x) * Product_{k>=1} (1 + x^k). - _Amiram Eldar_, Oct 03 2023
%e From _Gus Wiseman_, Jul 30 2022: (Start)
%e The A000688(216) = 9 factorizations of 216 into prime powers are:
%e (2*2*2*3*3*3)
%e (2*2*2*3*9)
%e (2*2*2*27)
%e (2*3*3*3*4)
%e (2*3*4*9)
%e (2*4*27)
%e (3*3*3*8)
%e (3*8*9)
%e (8*27)
%e Of these, the a(216) = 4 strict cases are:
%e (2*3*4*9)
%e (2*4*27)
%e (3*8*9)
%e (8*27)
%e (End)
%p A050361 := proc(n)
%p local a,f;
%p if n = 1 then
%p 1;
%p else
%p a := 1 ;
%p for f in ifactors(n)[2] do
%p a := a*A000009(op(2,f)) ;
%p end do:
%p end if;
%p end proc: # _R. J. Mathar_, May 25 2017
%t Table[Times @@ PartitionsQ[Last /@ FactorInteger[n]], {n, 99}] (* _Arkadiusz Wesolowski_, Feb 27 2017 *)
%o (Haskell)
%o a050361 = product . map a000009 . a124010_row
%o -- _Reinhard Zumkeller_, Aug 28 2014
%o (PARI)
%o A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
%o A050361(n) = factorback(apply(A000009,factor(n)[,2])); \\ _Antti Karttunen_, Nov 17 2019
%Y Cf. A000009, A050360, A050362, A050363, A050364, A101296.
%Y Cf. A124010.
%Y This is the strict case of A000688.
%Y Positions of 1's are A004709, complement A046099.
%Y The case of primes (instead of prime-powers) is A008966, non-strict A000012.
%Y The non-strict additive version allowing 1's A023893, ranked by A302492.
%Y The non-strict additive version is A023894, ranked by A355743.
%Y The additive version (partitions) is A054685, ranked by A356065.
%Y The additive version allowing 1's is A106244, ranked by A302496.
%Y A001222 counts prime-power divisors.
%Y A005117 lists all squarefree numbers.
%Y A034699 gives maximal prime-power divisor.
%Y A246655 lists all prime-powers (A000961 includes 1), towers A164336.
%Y A296131 counts twice-factorizations of type PQR, non-strict A295935.
%Y Cf. A001970, A002110, A025487, A055887, A063834, A076610, A085970, A279786, A302590, A302601, A354911, A355742.
%K nonn,easy,mult
%O 1,8
%A _Christian G. Bower_, Oct 15 1999