cp's OEIS Frontend

A050361 Number of factorizations into distinct prime powers greater than 1.

%I A050361 #40 Jun 12 2025 10:24:31
%S A050361 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 A050361 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 A050361 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 A050361 Number of factorizations into distinct prime powers greater than 1.
%C A050361 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).
%C A050361 The number of unordered factorizations of n into 1 and exponentially odd prime powers, i.e., p^e where p is a prime and e is odd (A246551). - _Amiram Eldar_, Jun 12 2025
%H A050361 Antti Karttunen, <a href="/A050361/b050361.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from Reinhard Zumkeller)
%H A050361 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A050361 Dirichlet g.f.: Product_{n is a prime power >1}(1 + 1/n^s).
%F A050361 Multiplicative with a(p^e) = A000009(e).
%F A050361 a(A002110(k))=1.
%F A050361 a(n) = A050362(A101296(n)). - _R. J. Mathar_, May 26 2017
%F A050361 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 A050361 From _Gus Wiseman_, Jul 30 2022: (Start)
%e A050361 The A000688(216) = 9 factorizations of 216 into prime powers are:
%e A050361   (2*2*2*3*3*3)
%e A050361   (2*2*2*3*9)
%e A050361   (2*2*2*27)
%e A050361   (2*3*3*3*4)
%e A050361   (2*3*4*9)
%e A050361   (2*4*27)
%e A050361   (3*3*3*8)
%e A050361   (3*8*9)
%e A050361   (8*27)
%e A050361 Of these, the a(216) = 4 strict cases are:
%e A050361   (2*3*4*9)
%e A050361   (2*4*27)
%e A050361   (3*8*9)
%e A050361   (8*27)
%e A050361 (End)
%p A050361 A050361 := proc(n)
%p A050361     local a,f;
%p A050361     if n = 1 then
%p A050361         1;
%p A050361     else
%p A050361         a := 1 ;
%p A050361         for f in ifactors(n)[2] do
%p A050361             a := a*A000009(op(2,f)) ;
%p A050361         end do:
%p A050361     end if;
%p A050361 end proc: # _R. J. Mathar_, May 25 2017
%t A050361 Table[Times @@ PartitionsQ[Last /@ FactorInteger[n]], {n, 99}] (* _Arkadiusz Wesolowski_, Feb 27 2017 *)
%o A050361 (Haskell)
%o A050361 a050361 = product . map a000009 . a124010_row
%o A050361 -- _Reinhard Zumkeller_, Aug 28 2014
%o A050361 (PARI)
%o A050361 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 A050361(n) = factorback(apply(A000009,factor(n)[,2])); \\ _Antti Karttunen_, Nov 17 2019
%Y A050361 Cf. A000009, A050360, A050362, A050363, A050364, A101296, A246551.
%Y A050361 Cf. A124010.
%Y A050361 This is the strict case of A000688.
%Y A050361 Positions of 1's are A004709, complement A046099.
%Y A050361 The case of primes (instead of prime-powers) is A008966, non-strict A000012.
%Y A050361 The non-strict additive version allowing 1's A023893, ranked by A302492.
%Y A050361 The non-strict additive version is A023894, ranked by A355743.
%Y A050361 The additive version (partitions) is A054685, ranked by A356065.
%Y A050361 The additive version allowing 1's is A106244, ranked by A302496.
%Y A050361 A001222 counts prime-power divisors.
%Y A050361 A005117 lists all squarefree numbers.
%Y A050361 A034699 gives maximal prime-power divisor.
%Y A050361 A246655 lists all prime-powers (A000961 includes 1), towers A164336.
%Y A050361 A296131 counts twice-factorizations of type PQR, non-strict A295935.
%Y A050361 Cf. A001970, A002110, A025487, A055887, A063834, A076610, A085970, A279786, A302590, A302601, A354911, A355742.
%K A050361 nonn,easy,mult
%O A050361 1,8
%A A050361 _Christian G. Bower_, Oct 15 1999