A333312 Positive integers m >= 2 such that there are at least two values of v such that the sequence {v, ..., v + m - 1} of m nonnegative integers can be partitioned into two subsets of consecutive integers with the same sum.

9, 15, 20, 21, 24, 25, 27, 28, 33, 35, 36, 39, 40, 44, 45, 48, 49, 51, 52, 55, 56, 57, 60, 63, 65, 68, 69, 72, 75, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 111, 112, 115, 116, 117, 119, 120, 121, 123, 124, 125, 129, 132, 133

Offset

1,1

Comments

Numbers m such that either m is odd and composite, or > 12, divisible by 4 and not a power of 2. - Robert Israel, Dec 08 2025

There are no such v for m == 2 (mod 4) and just one for odd prime numbers m.

References

Wilfried Haag, Die Wurzel. Problem 2020 - 14. (March/April 2020: www.wurzel.org)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

From Robert Israel, Dec 08 2025: (Start)

Let the subsets be {v, ..., v + k - 1} and {v + k, ..., v + m - 1}.

If m is odd and divisible by odd number p > 1, then

v = m^2/(4*p) - m/2 - p/4 + 1/2 and k = (m+p)/2 - 1 works.

If m is divisible by 4, then v = (m^2 - 4*m)/8 and k = m/2 works.

If m > 12 is divisible by 4 and by odd number p > 1, then v = m^2/(8*p) - (m + p - 1)/2 and k = p + m/2 - 1 works. (End)

For m = 6 * k + 3, you can always find two subsets of {v, ..., v+m-1} of length 3 * k + 2 with v = (3 * k + 1)^2 and length 3 * k + 3 with v = 3*k^2-1 elements.

Example

m = 9, v=16: 16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24.
m = 9, v=2: 2 + 3 + 4 + 5 + 6 + 7 = 8 + 9 + 10 = 27.

Maple

filter:= proc(m)
   if m::odd then not isprime(m)
   else (m mod 4 = 0) and (m > 12) and (m <> 2^padic:-ordp(m, 2))
   fi
end proc:
select(filter, [$2..1000]); # Robert Israel, Dec 07 2025

Prog

(Python)
for m in range(3, 1000):
    anz = 0
    for i in range(m // 2 + 1, m):
        l = (2 * i * i - 2 * i - m * (m - 1)) / (2 * (m - 2 * i))
        if l - int(l) == 0 and l >= 0:
            anz = anz + 1
    if anz > 1:
        print(m)

Crossrefs

Sequence in context: A330438 A321342 A338599 * A255763 A079364 A160666

Adjacent sequences: A333309 A333310 A333311 * A333313 A333314 A333315

Keyword

nonn

Author

Reiner Moewald, Mar 14 2020

Extensions

Edited and definition corrected by Robert Israel, Dec 07 2025

Status

approved