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A110475
Number of symbols '*' and '^' to write the canonical prime factorization of n.
2
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
OFFSET
1,12
COMMENTS
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, 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, Sep 11 2005
LINKS
Eric Weisstein's World of Mathematics, Prime Factorization.
FORMULA
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.
From Amiram Eldar, Sep 27 2024: (Start)
a(n) = A238949(n) - 1 for n >= 2.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C - 1), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^2 (A085548). (End)
EXAMPLE
a(208029250) = a(2*5^3*11^2*13*23^2) = 4 '*' + 3 '^' = 7.
PROG
(Haskell)
a110475 1 = 0
a110475 n = length us - 1 + 2 * length vs where
(us, vs) = span (== 1) $ a118914_row n
-- Reinhard Zumkeller, Mar 23 2014
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Sep 08 2005
STATUS
approved