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%I #21 Oct 28 2022 07:14:45
%S 1,1,3,0,5,3,7,0,3,5,11,0,13,7,15,0,17,3,19,0,21,11,23,0,5,13,3,0,29,
%T 15,31,0,33,17,35,0,37,19,39,0,41,21,43,0,15,23,47,0,7,5,51,0,53,3,55,
%U 0,57,29,59,0,61,31,21,0,65,33,67,0,69,35,71,0,73,37,15,0,77,39,79,0
%N Least positive k such that k^n is the sum of n consecutive integers, or 0 if no such k exists.
%C No k exists precisely when n == 0 (mod 4).
%H Amiram Eldar, <a href="/A076109/b076109.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = (n*A076107(n)+(n^2-n)/2)^(1/n) for n != 0 (mod 4).
%F a(n) = A076108^(1/n).
%F a(p) = p if p is a prime.
%F Multiplicative with a(2^1) = 1; a(2^e) = 0 if e >= 2; a(p^e) = p if p >= 3. - David W. Wilson, Jun 10 2005
%F a(n) = A007947(n) if n == 1 (mod 2); A007947(n/2) if n == 2 (mod 4); 0 if n == 0 (mod 4). - _David W. Wilson_, Jun 10 2005
%F a(4k) = 0; otherwise a(n) = p1*...*pm where p1, ..., pm are all distinct odd primes dividing n. - _Max Alekseyev_, Jun 10 2005
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) * Product_{p prime} (1 - 1/(p*(p+1))) = (3/8) * A065463 = 0.264165... . - _Amiram Eldar_, Oct 28 2022
%t f[p_, e_] := If[p == 2, Boole[e == 1], p]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 09 2020 *)
%o (PARI) for(n=1,100,t=n*(n-1)/2:f=0:for(r=1,10^4,if((r^n-t)%n==0,f=r:break)):print1(f","))
%o (PARI) { A076109(n) = if(n%4==0,return(0)); if(n%2==0,n\=2); vecprod(factorint(n)[,1]); } \\ _Max Alekseyev_, Jun 10 2005
%Y Cf. A007947, A065463, A076107, A076108.
%K nonn,easy,mult
%O 1,3
%A _Amarnath Murthy_, Oct 08 2002
%E Corrected and extended by _Ralf Stephan_, Mar 30 2003
%E More terms from _Max Alekseyev_, Jun 10 2005
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