A110475 Number of symbols '*' and '^' to write the canonical prime factorization of n.

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

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: A084114 A294881 A355523 * A366739 A086971 A341677

Adjacent sequences: A110472 A110473 A110474 * A110476 A110477 A110478

Keyword

nonn

Author

Reinhard Zumkeller, Sep 08 2005

Status

approved