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

1,3

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.

Links

Table of n, a(n) for n=1..21.

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

Example

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

Prog

(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)

Crossrefs

Cf. A076108, A076109.

Sequence in context: A371990 A013456 A180738 * A076952 A209914 A365237

Adjacent sequences: A076104 A076105 A076106 * A076108 A076109 A076110

Keyword

nonn

Author

Amarnath Murthy, Oct 08 2002

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

Status

approved