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

%I A344338 #14 May 17 2021 08:11:24
%S A344338 9,365,33075
%N A344338 Smallest number that is the sum of two or more consecutive positive n-th powers in more than one way.
%C A344338 a(4) > 10^24. - _Bert Dobbelaere_, May 16 2021
%C A344338 Conjecture: no terms exist for n >= 4. - _Jon E. Schoenfield_, May 16 2021
%e A344338 9 = 2 + 3 + 4 = 4 + 5.
%e A344338 365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.
%e A344338 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 A344338 (Python)
%o A344338 N=3 # <== Adapt here
%o A344338 import heapq
%o A344338 sigma=1+2**N
%o A344338 h=[(sigma,1,2)]
%o A344338 nextcount=3
%o A344338 oldv,olds,oldl=0,0,0
%o A344338 while True:
%o A344338     (v,s,l)=heapq.heappop(h)
%o A344338     if v==oldv:
%o A344338         break
%o A344338     if v>=sigma:
%o A344338         sigma += nextcount**N
%o A344338         heapq.heappush(h, (sigma,1,nextcount))
%o A344338         nextcount+=1
%o A344338     oldv,olds,oldl = v,s,l
%o A344338     v-=s**N ; s+=1 ; l+=1 ;    v+=l**N
%o A344338     heapq.heappush(h,(v,s,l))
%o A344338 print("a(%d) = %d = sum(i^%d, i=%d..%d) = sum(i^%d, i=%d..%d)"%
%o A344338     (N,v,N,olds,oldl,N,s,l))
%o A344338 # _Bert Dobbelaere_, May 16 2021
%Y A344338 Cf. A062681, A062682, A091414, A105441, A230718.
%K A344338 nonn,bref,more,hard
%O A344338 1,1
%A A344338 _Ilya Gutkovskiy_, May 15 2021