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A217785 Smallest integer s>n such that 1+2*s+3*s^2+...+n*s^{n-1} is prime.

%I A217785 #32 May 13 2013 01:54:22
%S A217785 3,12,12,9,21,12,26,23,30,24,138,33,80,32,54,192,48,40,4500,48,50,192,
%T A217785 30,88,32,114,178,48,45,42,356,41,53,138,174,66,44,990,120,819,2898,
%U A217785 112,1052,122,164,132,108,77,540,198,106,135,237,98,234,162,83,720,3870,135,188,1014,94,489,180,110,204,180,107,468,1542,508,218,608,88,102,228,140,3890,93,361,1848,462,99,125,390,92,237,933,172,606,303,208,924,114,266,156,410,1330
%N A217785 Smallest integer s>n such that 1+2*s+3*s^2+...+n*s^{n-1} is prime.
%C A217785 Conjecture: For each n=2,3,... there are infinitely many primes of the form 1+2*s+...+n*s^{n-1}, where s is a positive integer; moreover, we have a(n)<12*n^2.
%C A217785 This is related to the following conjecture of the author: The polynomials s_n(x)=sum_{k=0}^n(k+1)x^k (n=1,2,3,...) are all irreducible over the field of rational numbers; moreover, s_n(x) is reducible modulo every prime if and only if n has the form 8k(k+1), where k is a positive integer.
%C A217785 Sum_{k=1..n} k*s^(k-1) = (1+n*s^(n+1)-s^n*(n+1))/(s-1)^2, see A059045. - _R. J. Mathar_, Mar 29 2013
%H A217785 Zhi-Wei Sun and Charles R Greathouse IV, <a href="/A217785/b217785.txt">Table of n, a(n) for n = 2..1000</a> (first 450 terms from Sun)
%e A217785 a(20)=4500<12*20^2=4800 since 4500 is the least integer s>20 with 1+2*s+3*s^2+...+20*s^{19} prime.
%t A217785 A[n_,x_]:=A[n,x]=Sum[(k+1)*x^k,{k,0,n-1}]
%t A217785 Do[Do[If[PrimeQ[A[n,s]]==True,Print[n," ",s];Goto[aa]],{s,n+1,12*n^2-1}];
%t A217785 Print[n," ",counterexample];Label[aa];Continue,{n,2,100}]
%o A217785 (PARI) f(n,s)=my(t);forstep(k=n,1,-1,t=s*t+k);t
%o A217785 a(n)=my(s=n);while(!ispseudoprime(f(n,s++)),);s \\ _Charles R Greathouse IV_, Mar 25 2013
%Y A217785 Cf. A000040.
%K A217785 nonn
%O A217785 2,1
%A A217785 _Zhi-Wei Sun_, Mar 24 2013