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%I #13 Jan 28 2023 12:09:13
%S 1,0,8,0,623,119,117646,0,2183,976558,25937424596,0,23298085122475,
%T 48444505197,29192926025390618,0,48661191875666868473,21523352,
%U 104127350297911241532832,0,278218429446951548637196391
%N First of n consecutive integers whose sum is a positive n-th power, or 0 if no such integers exist.
%C No sum exists precisely when n == 0 (mod 4). a(2) = 0 is a legitimate value.
%C The sum is given by A076108(n) = A076109(n)^n for n != 0 (mod 4).
%C a(p) = p^(p-1) - (p-1)/2 for prime p.
%F a(n) = A076108(n)/n - (n-1)/2 for n != 0 (mod 4).
%F a(4k)=0; otherwise a(n) = (2*A076108(n)/n - n + 1)/2 = (2*p1^n*...*pm^n/n - n + 1)/2 where p1, ..., pm are all distinct odd primes dividing n. - _Max Alekseyev_, Jun 10 2005
%e a(3) = 8 as 8+9+10 = 27 = 3^3. a(6) = 119 as 119+120+...+124 = 729 = 3^6.
%o (PARI) for(n=1,30,t=n*(n-1)/2:f=0:for(r=1,10^4,if((r^n-t)%n==0,f=(r^n-t)/n:break)):print1(f","))
%o (PARI) {A076107(n)=if(n%4==0,return(0));m=n;if(m%2==0,m\=2);f=factorint(m)[,1];p=1;(2*prod(i=1,length(f),f[i])^n/n-n+1)/2} (Alekseyev)
%Y Cf. A076108, A076109.
%K nonn
%O 1,3
%A _Amarnath Murthy_, Oct 08 2002
%E Corrected and extended by _Ralf Stephan_, Mar 30 2003
%E Revised by _Max Alekseyev_ and _David W. Wilson_, Jun 10 2005
%E More terms from _Max Alekseyev_, Jun 10 2005
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