

A076107


First of n consecutive integers whose sum is a positive nth 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
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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^(p1)(p1)/2 for prime p.


LINKS

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


FORMULA

a(n) = A076108(n)/n(n1)/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*(n1)/2:f=0:for(r=1, 10^4, if((r^nt)%n==0, f=(r^nt)/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/nn+1)/2} (Alekseyev)


CROSSREFS

Cf. A076108, A076109.
Sequence in context: A221421 A013456 A180738 * A076952 A209914 A342980
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



