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Search: id:A115588
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| A115588 |
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Number of unique prime numbers necessary to represent a natural number n > 1. |
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+0
1
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| 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 3, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 1, 3, 1, 3, 3, 2
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OFFSET
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2,5
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COMMENT
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The sequence gives the number of unique prime numbers needed to represent a given natural number greater than or equal to 2. In order to do this, we must factor any subsequent composite number that may appear on the exponents of the next factorizations (i.e. 4 in 48=2^4*3), until only prime numbers are used. - Lucas Vieira Barbosa (dnukem(AT)gmail.com), Mar 15 2006
In this sequence, a(n)=1 if n is prime, or a power tower (tetration or iterated exponentiation) of a prime base (i.e. 2^2, 3^3^3^3, 7^7, etc). The sequence reaches a new boundary whenever n is a primorial number (factorial of primes). - Lucas Vieira Barbosa (dnukem(AT)gmail.com), Mar 15 2006
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EXAMPLE
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a(4)=1, since 4=2^2 and the only prime used was 2; a(30)=3 because 30=2*3*5, and three primes were necessary; a(65536)=1, since 65536=2^16=2^(2^4)=2^(2^(2^2)) and, again, only one prime was needed; a(1) would be undefined, so it is not included.
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CROSSREFS
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Cf. A000040, A001221, A002110.
Sequence in context: A055188 A084989 A128538 this_sequence A105220 A083654 A029428
Adjacent sequences: A115585 A115586 A115587 this_sequence A115589 A115590 A115591
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KEYWORD
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nonn
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AUTHOR
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Lucas Vieira Barbosa (dnukem(AT)gmail.com), Mar 09 2006
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