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

[Bob Selcoe]

Hi Don and Seqfans,

Don's proof looks good to me.

>    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.

The critical idea is that for b > 2a, j is minimum iff b is the a-th prime, 
as Don shows below.

Thanks for your help!

Cheers,
Bob
PS - Don, I tried sending you two emails off-list, the second one - which 
essentially is this post - was returned as undeliverable.  Hope you at least 
received the first one.

--------------------------------------------------
From: "Don Reble" <djr at nk.ca>
Sent: Wednesday, April 13, 2016 2:23 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: interesting relationship between A008507, A000217 and 
A033286

>> 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
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>