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A110475
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Number of symbols '*' and '^' to write the canonical prime factorization of n.
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1
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0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 0, 3, 0, 1, 2, 2, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 2, 0, 2, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,12
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COMMENTS
| a(n) = A001221(n) - 1 + A056170(n) for n>1;
a(n) = 0 iff n=1 or n is prime: a(A008578(n)) = 0;
a(n) = 1 iff n is a semiprime or a prime power p^e with e>1.
It is conjectured that 1,2,3,4,5,6,7,9,11 are the only positive integers which cannot be represented as the sum of two elements of indices n such that a(n) = 1. - Jonathan Vos Post (jvospost3(AT)gmail.com), Sep 11 2005
a(n) = 2 iff n is a sphenic number (A007304) or n is a prime p times a prime power q^e with e>1 and q not equal to p. a(n) = 3 iff n has exactly four distinct prime factors (A046386); or n is the product of two prime powers (p^e)*(q^f) with e>1, f>1 and p not equal to q; or n is a semiprime s times a prime power r^g with g>1 and r relatively prime to s. For a(n) > 3, Reinhard Zumkeller's description is a simpler description than the above compound descriptions. - Jonathan Vos Post (jvospost3(AT)gmail.com), Sep 11 2005
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LINKS
| Eric Weisstein's World of Mathematics, Prime Factorization
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EXAMPLE
| a(208029250) = a(2*5^3*11^2*13*23^2) = 4 '*' + 3 '^' = 7.
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CROSSREFS
| Cf. A050252, A001358, A025475, A000040.
Cf. A007304, A046386.
Sequence in context: A171182 A063962 A084114 * A086971 A088434 A205745
Adjacent sequences: A110472 A110473 A110474 * A110476 A110477 A110478
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KEYWORD
| nonn
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 08 2005
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