A076109 Least positive k such that k^n is the sum of n consecutive integers, or 0 if no such k exists.

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, 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, 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

Offset

1,3

Comments

No k exists precisely when n == 0 (mod 4).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = (n*A076107(n)+(n^2-n)/2)^(1/n) for n != 0 (mod 4).

a(n) = A076108^(1/n).

a(p) = p if p is a prime.

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

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

a(4k) = 0; otherwise a(n) = p1*...*pm where p1, ..., pm are all distinct odd primes dividing n. - Max Alekseyev, Jun 10 2005

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

Mathematica

f[p_, e_] := If[p == 2, Boole[e == 1], p]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 09 2020 *)

Prog

(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", "))
(PARI) { A076109(n) = if(n%4==0, return(0)); if(n%2==0, n\=2); vecprod(factorint(n)[, 1]); } \\ Max Alekseyev, Jun 10 2005

Crossrefs

Cf. A007947, A065463, A076107, A076108.

Sequence in context: A326989 A326937 A336597 * A078788 A284599 A005069

Adjacent sequences: A076106 A076107 A076108 * A076110 A076111 A076112

Keyword

nonn,easy,mult

Author

Amarnath Murthy, Oct 08 2002

Extensions

Corrected and extended by Ralf Stephan, Mar 30 2003

More terms from Max Alekseyev, Jun 10 2005

Status

approved