A076107 First of n consecutive integers whose sum is a positive n-th power, or 0 if no such integers exist.

1, 0, 8, 0, 623, 119, 117646, 0, 2183, 976558, 25937424596, 0, 23298085122475, 48444505197, 29192926025390618, 0, 48661191875666868473, 21523352, 104127350297911241532832, 0, 278218429446951548637196391

Offset: 1

Author

Amarnath Murthy, Oct 08 2002

Keywords

• nonn

Comments

No sum exists precisely when n == 0 (mod 4). a(2) = 0 is a legitimate value.

The sum is given by A076108(n) = A076109(n)^n for n != 0 (mod 4).

a(p) = p^(p-1) - (p-1)/2 for prime p.

Examples

			a(3) = 8 as 8+9+10 = 27 = 3^3. a(6) = 119 as 119+120+...+124 = 729 = 3^6.
		

Crossrefs

Cf. A076108, A076109.

Programs

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","))

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)

Formula

a(n) = A076108(n)/n - (n-1)/2 for n != 0 (mod 4).

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

Extensions

Corrected and extended by Ralf Stephan, Mar 30 2003

Revised by Max Alekseyev and David W. Wilson, Jun 10 2005

More terms from Max Alekseyev, Jun 10 2005