A247592 - OEIS

%I #12 Sep 21 2014 16:23:06

%S 2,8,10,25,42,147,160,169,238,260,491,544,869,890,923,1140,1337,1386,

%T 1465,1643,1927,3371,4614,5038,5086,5225,5832,5909,5995,7118,7157,

%U 8540,9859,12543,13505,13795,13841,14211,15347,17079,17263,18643,20211,21184,21245

%N Numbers n such that A002496(n) mod A002496(n-1) is a perfect square.

%C A002496 : primes of form n^2+1.

%C The prime numbers of the sequence are 2, 491, 3371, 9859, 13841,...

%C The corresponding squares A002496(n) mod A002496 (n-1) are : {1, 144, 100, 1024, 4900, 10816, 11664, 12544,...} = {1} union {A216330} minus {64}.

%H Chai Wah Wu, <a href="/A247592/b247592.txt">Table of n, a(n) for n = 1..200</a>

%e a(3)=10 because A002496(10) mod A002496(9)= 677 mod 577 = 10^2.

%p with(numtheory):nn:=360000:T:=array(1..nn):kk:=0:

%p for n from 1 to nn do:

%p   if type(n^2+1,prime)=true then

%p    kk:=kk+1:T[kk]:=n^2+1:

%p    else

%p    fi:

%p od:

%p     for m from 1 to kk-1 do:

%p      r:=irem(T[m+1],T[m]):z:=sqrt(r):

%p       if z=floor(z)

%p        then printf(`%d, `, m+1):

%p        else

%p       fi:

%p     od:

%t lst={};lst1={};nn=400000;Do[If[PrimeQ[n^2+1],AppendTo[lst,n^2+1]],{n,1,nn}];nn1:=Length[lst];

%t Do[If[IntegerQ[Sqrt[Mod[lst[[m]],lst[[m-1]]]]],AppendTo[lst1,m]],{m,2,nn1}];lst1

%o (Python)

%o from gmpy2 import t_mod, is_square, is_prime

%o A247592_list, A002496_list, m, c = [], [2], 2, 2

%o for n in range(1, 10**7):

%o ....m += 2*n+1

%o ....if is_prime(m):

%o ........if is_square(t_mod(m, A002496_list[-1])):

%o ............A247592_list.append(c)

%o ........A002496_list.append(m)

%o ........c += 1 # _Chai Wah Wu_, Sep 20 2014

%Y Cf. A002496, A193558, A216330.

%K nonn

%O 1,1

%A _Michel Lagneau_, Sep 20 2014