[seqfan] Re: interesting relationship between A008507, A000217 and A033286

Don Reble djr at nk.ca
Wed Apr 13 21:23:37 CEST 2016


> A008507 = number of odd composites < n-th odd prime:
>      {0, 0, 0, 1, 1, 2, 2, 3, 5, 5, 7, 8...}, offset 1.
> A000217 = Triangular numbers:
>      {0, 1, 3, 6, 10, 15, 21, 28, 36...}, offset 0.
> A033286 = n*prime(n): {2, 6, 15, 28, 55, 78, 119...}, offset 1.
>
> It appears that for n>=4, A008507(n) = k+1, where A000217(j) is the
> smallest triangular number such that A000217(j) - A033286(n+1) also is a
> triangular number, i.e., A000217(k).
>
> Example n=29, A008507(29) = 27:
>
> A033286(30) = 3390, A000217(86) = 3741.
> 3741-3390 = 351 = A000217(26); k=26, 26+1 = 27.
>
> ...Is there a proof...?

    Let n = a*b, where b is odd.
    Let j = (b-1)/2 + a, k = (b-1)/2 - a;
    then n = (j^2+j)/2 - (k^2+k)/2, the difference of those triangular
        numbers.
    n may have many triangular-difference representations. We want the
    smallest j, and therefore the smallest (k^2+k)/2. k should be close
    to -1/2, so b should be close to 2a.

    If b is prime and b >= 2a, the above representation has the least j.
    This is the case, if b is the a'th prime and a >= 5. (The 4th prime
    is only 7.)

    A033286(a) = a*prime(a):
        j = (prime(a)-1)/2 + a, k = (prime(a)-1)/2 - a.
    A008507(n) = number of odd composites below n'th odd prime,
               = (number of odd numbers - number of odd primes - 1)
                 below n'th odd prime,
        (Minus one, because "1" isn't composite nor prime),
               = (prime(n+1)-1)/2 - (n-1) - 1,
               = (prime(n+1)-1)/2 - (n+1) + 1
    Let a=n+1: that last expression is (prime(a)-1)/2 - a + 1 = k+1,
    just as Bob conjectures.

-- 
Don Reble  djr at nk.ca



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