A094756 -id:A094756

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A095366

Least k > 1 such that k divides 1^n + 2^n +...+ (k-1)^n.

+10
4

3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 17, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence is similar to A094756 but seems to have a simpler periodicity rules:

a(n)=3 when n=1 (mod 2), otherwise

a(n)=5 when n=2 (mod 4), otherwise

a(n)=7 when n=4*m (mod 12) for some m=1,2, otherwise

a(n)=11 when n=12*m (mod 60) for some m=1,2,3,4, otherwise

a(n)=17 when n=60*m (mod 240) for some m=1,2,3, otherwise

a(n)=19 when n=240*m (mod 720) for some m=1,2, otherwise

a(n)=23 when n=720*m (mod 7920) for some m=1,..,10, etc.

Note that only odd primes p given by A095365 seem to appear in this sequence. Given the definition of f(p) in that sequence, let q=A095365(i) and p=A095365(i-1), then the general rule for this sequence seems to be a(n)=q when n=f(p)*m (mod f(q)) for some m=1,...,f(q)/f(p)-1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

EXAMPLE

a(4) = 7 because k divides 1^4 + 2^4 +...+ k^4 for k=7 but no smaller k > 1.

MATHEMATICA

Table[k=2; s=0; While[s=s+(k-1)^n; Mod[s, k]>0, k++ ]; k, {n, 100}]

PROG

(PARI) A095366(n) = { my(k=1, s=0); while(1, k++; s += ((k-1)^n); if(!(s%k), return(k))); }; \\ Antti Karttunen, Dec 19 2018

CROSSREFS

Cf. A094756, A095365.

KEYWORD

nonn

AUTHOR

T. D. Noe, Jun 03 2004

STATUS

approved

A094755

Least integer value of (1 + 2^n + 3^n + ... + k^n)/(1 + 2 + 3 + ... + k), k > 1.

+10
2

1, 3, 3, 167, 11, 489, 43, 282407, 171, 110865, 683, 3710553451913, 2731, 27323481, 10923, 1293248801687, 43691, 6910715937, 174763, 2983746256027727, 699051, 1762357129833, 2796203, 734630194457006903941170593, 11184811, 450614156030769, 44739243

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..719

EXAMPLE

a(4) = (1^4 + 2^4 + 3^4 + 4^4 + 5^4 + 6^4 + 7^4)/(1+2+3+4+5+6+7) = 4676/28 = 167, k = 7.

a(5) = (1^5 + 2^5)/(1 + 2) = 11, k = 2.

MAPLE

a:= proc(n) option remember; local k, r, s, t; s, t:=1$2;

for k from 2 do s, t:= s+k, t+k^n;

if irem(t, s, 'r')=0 then return r fi

od:

end:

seq(a(n), n=1..28); # Alois P. Heinz, Mar 07 2024

MATHEMATICA

f[n_] := Block[{k = 2}, While[s = 2Sum[i^n, {i, k}]/(k(k + 1)); !IntegerQ[s], k++ ]; s]; Table[ f[n], {n, 25}] (* Robert G. Wilson v, Jun 02 2004 *)

CROSSREFS

Cf. A000217, A094756.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, May 29 2004

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jun 02 2004

STATUS

approved

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