This late at night I can’t remember which primes belong to whom, but think that you have found a nice algorithm here. I checked https://oeis.org/A055527 up to ten in my head, and a modified version of the factoring algorithm looks to work also for Pythagorean triples, though with every prime n a fail case. Example: n=10, 10^2=50*2, (50-2)/2=24, (50+2)/2=26. 10^2 + 24^2 -26^2 = 100 + (400 + 160 + 16) - (400 + 240 + 36) = 676 - 676 = 0. I don’t know how well this technique is known, but OEIS seems to be using brute force. Why shouldn’t A082183 cross-reference A055527? Wouldn’t it be easier to understand that way? —Brad
On Feb 19, 2020, at 11:03 PM, rcs@xmission.com wrote:
Quoting Neil Sloane <njasloane@gmail.com>:
Let T(i) = i(i+1)/2. Given n, let k be smallest number such that T(n) + T(k) = T(m) for some m. The k and m values are in A082183 and A082184. It must be classical that k and m always exist. - can someone supply a reference or a proof?
The graph of the k values is quite irregular. Is there an upper bound? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
2T(n) = 2T(m) - 2T(k) = m^2 + m - k^2 - k = (m-k) (m + k + 1)
(m-k) and (m+k+1) are of opposite parity because their sum is 2m+1, which is odd.
So, factor 2T(n) into the product of an odd number times an even number. One of these is m-k, and the other is m+k+1. 2T(n) = n^2 + n gives two obvious solutions, n * (n+1) and 1 * (n^2+n). Equating these to (m-k) * (m+k+1) gives the two "trivial" solutions k=0, m=n and k=T(n)-1, m=T(n).
Unless n is a Mersenne prime, or n+1 is a Fermat prime, there will be a non-trivial odd divisor of n(n+1) other than 1, n, or n+1. Select the odd divisor d logarithmicly closest to n+.5 that isn't n or n+1. Let q be the quotient n(n+1)/q. Then m-k = min(d,q) and m+k+1 = max(d,q). Solve for k, which is the required minimum.
Example: n=5, T(n)=15, 2T(n)=30 = 3*10, d=3, q=10, k=3, m=6, 15+6 = 21.
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun