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A297404
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A binary representation of the positive exponents that appear in the prime factorization of a number, shown in decimal.
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4
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0, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1, 8, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 4, 3, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 32, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 1, 9, 8, 1, 1, 3, 1, 1
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OFFSET
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1,4
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COMMENTS
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This sequence is similar to A087207; here we encode the exponents, there the prime numbers appearing in the prime factorization of a number.
The binary representation of a(n) shows which exponents appear in the prime factorization of n, but without multiplicities:
- for any prime number p and k > 0, if p^k divides n but p^(k+1) does not divide n, then a(n) AND 2^(k-1) = 2^(k-1) (where AND denotes the bitwise AND operator),
- conversely, if a(n) AND 2^(k-1) = 2^(k-1) for some k > 0, then there is prime number p such that p^k divides n but p^(k+1) does not divide n.
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LINKS
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FORMULA
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a(p^k) = 2^(k-1) for any prime number p and k > 0.
a(n^2) = A000695(2 * a(n)) / 2 for any n > 0.
a(n) <= 1 iff n is squarefree (A005117).
a(n) <= 3 iff n is cubefree (A004709).
a(n) is odd iff n belongs to A052485 (weak numbers).
a(n) is even iff n belongs to A001694 (powerful numbers).
a(n) AND 2 = 2 iff n belongs to A038109 (where AND denotes the bitwise AND operator).
A000120(a(n)) <= 1 iff n belongs to A072774 (powers of squarefree numbers).
If gcd(m, n) = 1, then a(m * n) = a(m) OR a(n) (where OR denotes the bitwise OR operator).
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EXAMPLE
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For n = 90:
- 90 = 5^1 * 3^2 * 2^1,
- the exponents appearing in the prime factorization of 90 are 1 and 2,
- hence a(90) = 2^(1-1) + 2^(2-1) = 3.
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MATHEMATICA
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Array[Total@ Map[2^(# - 1) &, Union[FactorInteger[#][[All, -1]] ]] - Boole[# == 1] &, 86] (* Michael De Vlieger, Dec 29 2017 *)
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PROG
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(PARI) a(n) = my (x=Set(factor(n)[, 2]~)); sum(i=1, #x, 2^(x[i]))/2
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CROSSREFS
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KEYWORD
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nonn,easy,base
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AUTHOR
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STATUS
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approved
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