[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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