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A076107
First of n consecutive integers whose sum is a positive n-th power, or 0 if no such integers exist.
3
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.
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
Sequence in context: A371990 A013456 A180738 * A076952 A209914 A375078
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