A344338 Smallest number that is the sum of two or more consecutive positive n-th powers in more than one way.

9, 365, 33075

Offset

1,1

Comments

a(4) > 10^24. - Bert Dobbelaere, May 16 2021

Conjecture: no terms exist for n >= 4. - Jon E. Schoenfield, May 16 2021

Links

Table of n, a(n) for n=1..3.

Example

9 = 2 + 3 + 4 = 4 + 5.
365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.
33075 = 11^3 + 12^3 + 13^3 + 14^3 + 15^3 + 16^3 + 17^3 + 18^3 + 19^3 = 15^3 + 16^3 + 17^3 + 18^3 + 19^3 + 20^3.

Prog

(Python)
N=3 # <== Adapt here
import heapq
sigma=1+2**N
h=[(sigma, 1, 2)]
nextcount=3
oldv, olds, oldl=0, 0, 0
while True:
    (v, s, l)=heapq.heappop(h)
    if v==oldv:
        break
    if v>=sigma:
        sigma += nextcount**N
        heapq.heappush(h, (sigma, 1, nextcount))
        nextcount+=1
    oldv, olds, oldl = v, s, l
    v-=s**N ; s+=1 ; l+=1 ;    v+=l**N
    heapq.heappush(h, (v, s, l))
print("a(%d) = %d = sum(i^%d, i=%d..%d) = sum(i^%d, i=%d..%d)"%
    (N, v, N, olds, oldl, N, s, l))
# Bert Dobbelaere, May 16 2021

Crossrefs

Cf. A062681, A062682, A091414, A105441, A230718.

Sequence in context: A162133 A197179 A373536 * A195888 A162084 A195518

Adjacent sequences: A344335 A344336 A344337 * A344339 A344340 A344341

Keyword

nonn,bref,more,hard

Author

Ilya Gutkovskiy, May 15 2021

Status

approved