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

Author

Ilya Gutkovskiy, May 15 2021

Keywords

• nonn
• bref
• more
• hard

Comments

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

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

Examples

			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.
		

Crossrefs

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

Programs

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