A344338 - OEIS

%I #14 May 17 2021 08:11:24

%S 9,365,33075

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

%C a(4) > 10^24. - _Bert Dobbelaere_, May 16 2021

%C Conjecture: no terms exist for n >= 4. - _Jon E. Schoenfield_, May 16 2021

%e 9 = 2 + 3 + 4 = 4 + 5.

%e 365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.

%e 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.

%o (Python)

%o N=3 # <== Adapt here

%o import heapq

%o sigma=1+2**N

%o h=[(sigma,1,2)]

%o nextcount=3

%o oldv,olds,oldl=0,0,0

%o while True:

%o     (v,s,l)=heapq.heappop(h)

%o     if v==oldv:

%o         break

%o     if v>=sigma:

%o         sigma += nextcount**N

%o         heapq.heappush(h, (sigma,1,nextcount))

%o         nextcount+=1

%o     oldv,olds,oldl = v,s,l

%o     v-=s**N ; s+=1 ; l+=1 ;    v+=l**N

%o     heapq.heappush(h,(v,s,l))

%o print("a(%d) = %d = sum(i^%d, i=%d..%d) = sum(i^%d, i=%d..%d)"%

%o     (N,v,N,olds,oldl,N,s,l))

%o # _Bert Dobbelaere_, May 16 2021

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

%K nonn,bref,more,hard

%O 1,1

%A _Ilya Gutkovskiy_, May 15 2021