A076108 - OEIS

%I #11 Jan 28 2023 11:58:04

%S 1,1,27,0,3125,729,823543,0,19683,9765625,285311670611,0,

%T 302875106592253,678223072849,437893890380859375,0,

%U 827240261886336764177,387420489,1978419655660313589123979,0

%N Least positive n-th power that is the sum of n consecutive integers, or 0 if no such n-th power exists.

%C No n-th power exists precisely when n == 0 (mod 4).

%C The first term of the sum is A076107(n) for n != 0 (mod 4).

%C a(p) = p^p for prime p.

%F a(n) = n*A076107(n)+(n^2-n)/2 for n != 0 (mod 4).

%F a(n) = A076109(n)^n.

%F a(4k)=0; otherwise a(n)=p1^n*...*pm^n where p1, ..., pm are all distinct odd primes dividing n. - _Max Alekseyev_, Jun 10 2005

%e 27 = 3^3 = 8+9+10 is least positive cube that is sum of 3 consecutive integers, hence a(3) = 27.

%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:break)):print1(f","))

%o (PARI) {A076108(n)=if(n%4==0,return(0));m=n;if(m%2==0,m\=2);f=factorint(m)[,1];prod(i=1,length(f),f[i])^n} (Alekseyev)

%Y Cf. A076107, 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