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%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
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