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A050361
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Number of factorizations into distinct prime powers greater than 1.
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21
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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, 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, 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
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OFFSET
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1,8
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COMMENTS
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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).
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LINKS
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FORMULA
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Dirichlet g.f.: Product_{n is a prime power >1}(1 + 1/n^s).
Multiplicative with a(p^e) = A000009(e).
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
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EXAMPLE
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The A000688(216) = 9 factorizations of 216 into prime powers are:
(2*2*2*3*3*3)
(2*2*2*3*9)
(2*2*2*27)
(2*3*3*3*4)
(2*3*4*9)
(2*4*27)
(3*3*3*8)
(3*8*9)
(8*27)
Of these, the a(216) = 4 strict cases are:
(2*3*4*9)
(2*4*27)
(3*8*9)
(8*27)
(End)
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MAPLE
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local a, f;
if n = 1 then
1;
else
a := 1 ;
for f in ifactors(n)[2] do
end do:
end if;
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MATHEMATICA
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Table[Times @@ PartitionsQ[Last /@ FactorInteger[n]], {n, 99}] (* Arkadiusz Wesolowski, Feb 27 2017 *)
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PROG
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(Haskell)
a050361 = product . map a000009 . a124010_row
(PARI)
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));
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CROSSREFS
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This is the strict case of A000688.
The case of primes (instead of prime-powers) is A008966, non-strict A000012.
The non-strict additive version allowing 1's A023893, ranked by A302492.
A001222 counts prime-power divisors.
A005117 lists all squarefree numbers.
A034699 gives maximal prime-power divisor.
Cf. A001970, A002110, A025487, A055887, A063834, A076610, A085970, A279786, A302590, A302601, A354911, A355742.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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