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Primevens and primeodds (Posted on 2016-01-10) Difficulty: 4 of 5
Let a primeven be a positive integer that is the product of an even number of primes. Let a primeodd be a positive integer that is the product of an odd number of primes. Then, 1 is a primeven because it is the product of 0 primes. 2 is a primeodd because it is the product of 1 prime. 3 is a primeodd because it is the product of 1 prime. 4 is a primeven because it is the product of 2 primes. Here are the first 10 positive integers.
Number:   Factorization:      Number of primes:   Type:
1                             0                   primeven
2         2                   1                   primeodd
3         3                   1                   primeodd
4         2*2                 2                   primeven
5         5                   1                   primeodd
6         2*3                 2                   primeven
7         7                   1                   primeodd
8         2*2*2               3                   primeodd
9         3*3                 2                   primeven
10        2*5                 2                   primeven
Suppose the primevens and primeodds had a race. First, the primevens would be ahead because 1 is a primeven. Then, there would be a tie because 2 is a primeodd. Then, the primeodds would be ahead because 3 is a primeodd. Then, there would be a tie because 4 is a primeven. Here are the winners from 1 to 10.
Number:   Type:          Primevens:     Primeodds:     Winner:
1         primeven       1              0              primevens
2         primeodd       1              1              tie
3         primeodd       1              2              primeodds
4         primeven       2              2              tie
5         primeodd       2              3              primeodds
6         primeven       3              3              tie
7         primeodd       3              4              primeodds
8         primeodd       3              5              primeodds
9         primeven       4              5              primeodds
10        primeven       5              5              tie
The primevens were ahead at the start, but have not been ahead since then. Do the primevens ever become the winner again?

No Solution Yet Submitted by Math Man    
Rating: 4.0000 (1 votes)

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Some Thoughts OEIS sequences Comment 8 of 8 |

OEIS A008836: 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, ...
"Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity)."

Basically, 1 = n is primeeven and -1 = n is primeodd.

OEIS A02819: 0, 1, 0, -1, 0, -1, 0, -1, -2, -1, 0, ...
"Liouville's function L(n) = partial sums of A008836."

This is positive when primeevens are in the lead, negative when primeodds are in the lead, and 0 when tied.

In the comments:
"George Polya conjectured in 1919 that L(n) <= 0 for all n >= 2. .... In 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257."

Charlie successfully verified this result.  Congratulations.


  Posted by Brian Smith on 2016-01-12 11:23:22
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