cp's OEIS Frontend

Let n = p^a*q^b*...r^c be the prime power factorization of b(n). Then lambda(n) is

(-1)^s, where s is the sum of exponents a + b + ... + c.

In case B=A000027 (the natural numbers), G. Polya (1919) conjectured that L_B(n)<=0, for n>=2. But this was disproved in 1958 by B. Haselgrove, and in 1980 M. Tanaka found the smallest counterexample, 906150257.

However, for this sequence we conjecture that a(n)>=0 for all n other than 3. A reason for our conjecture is the later appearance of primes in A098550 than in A000027. By our conjecture, among the first N terms of A098550, the terms with odd s are never in the majority, if N is other than 3. Peter J. C. Moses verified the conjecture up to 2.5*10^5 and, moreover, in this range a(n)>0 for n>6.

To create the data, the author studied the b-file of Donovan Johnson in A189229.

For k>=1,

in the interval [a(2k-1), a(2k)], L(n)<=0,

in the interval [a(2k), a(2k+1)], L(n)>=0.

In particular, for k=1, in the interval [2, 906150256], L(n)<=0.

G. Polya (1919) conjectured that L(n)<=0, for n>=2. But this was disproved in 1958 by B. Haselgrove, and in 1980 M. Tanaka found the smallest counterexample, a(2)+1 = 906150257.