

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
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OFFSET

1,12


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, 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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization


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


CROSSREFS

Cf. A050252, A001358, A025475, A000040.
Cf. A007304, A046386.
Cf. A118914.
Sequence in context: A063962 A084114 A294881 * A086971 A211159 A088434
Adjacent sequences: A110472 A110473 A110474 * A110476 A110477 A110478


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Sep 08 2005


STATUS

approved



