A131685 - OEIS

A131685

a(n) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^n + n) / n! takes integral values for all i>=0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 1, 1, 1, 1, 1, 11, 11, 11, 55, 143, 13, 91, 91, 91, 91, 91, 1001, 17017, 595595, 595595, 17017, 46189, 600457, 3002285, 3002285, 3002285, 3002285, 6605027, 3002285, 726869, 726869, 726869

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

OFFSET

1,14

COMMENTS

It appears that none of the terms are divisible by 3. - Alexander R. Povolotsky, Oct 18 2007

LINKS

Cyril Banderier, Table of n, a(n) for n = 1..100

Index to divisibility sequences

MAPLE

# Maple program from Cyril Banderier, Sep 18 2007:

List:=NULL: for n from 1 to 1000 do m:=1: #running till n=50 will last 2 min.

for i from 1 to numtheory[pi](n) do div:=ithprime(i): d:=1: e:=0: oldmini:=-1:mini:=0:

while oldmini<>mini do e:=e+1: #the last time consuming loop could be skipped by proving e<=floor(ln(n)/ln(div)):

d:=d*div; for x from 0 to d-1 do [seq((x &^k mod d)+k mod d, k=1..n)]:contrib[d, x]:=nops(select(has, %, 0)): od:

L:=seq(add(contrib[div^j, x mod div^j], j=1..e), x=0..div^e-1); oldmini:=mini: mini:=min(L): od:

if mini<padic[ordp](n!, div) then m:=m*div^(padic[ordp](n!, div)-mini) fi; od: print(n, m); List:=List, m: od:

[List];

CROSSREFS

Cf. A000027 (for n=1), A064808 (n=2), A131509 (n=3), A129995 (n=4), A131675 (n=5), ..., A131680 (n=10).