A110475 - OEIS

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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

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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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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.