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A355742
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Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.
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18
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1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2
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OFFSET
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1,7
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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LINKS
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FORMULA
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Totally multiplicative with a(prime(k)) = A001222(k).
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EXAMPLE
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The prime indices of 49 are {4,4}, and the a(49) = 4 choices are: (2,2), (2,4), (4,2), (4,4).
The prime indices of 777 are {2,4,12}, and the a(777) = 6 choices are: (2,2,2), (2,2,3), (2,2,4), (2,4,2), (2,4,3), (2,4,4).
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@PrimeOmega/@primeMS[n], {n, 100}]
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CROSSREFS
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Allowing all divisors (not just primes) gives A355731, firsts A355732.
Choosing only prime factors (not prime-powers) gives A355741.
Counting multisets of primes gives A355744.
A000688 counts factorizations into prime powers.
A003963 multiplies together the prime indices of n.
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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