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A247592 Numbers n such that A002496(n) mod A002496(n-1) is a perfect square.

%I A247592 #14 May 22 2025 10:21:40
%S A247592 2,8,10,25,42,147,160,169,238,260,491,544,869,890,923,1140,1337,1386,
%T A247592 1465,1643,1927,3371,4614,5038,5086,5225,5832,5909,5995,7118,7157,
%U A247592 8540,9859,12543,13505,13795,13841,14211,15347,17079,17263,18643,20211,21184,21245
%N A247592 Numbers n such that A002496(n) mod A002496(n-1) is a perfect square.
%C A247592 A002496 : primes of form n^2+1.
%C A247592 The prime numbers of the sequence are 2, 491, 3371, 9859, 13841,...
%C A247592 The corresponding squares A002496(n) mod A002496 (n-1) are : {1, 144, 100, 1024, 4900, 10816, 11664, 12544,...} = {1} union {A216330} minus {64}.
%H A247592 Chai Wah Wu, <a href="/A247592/b247592.txt">Table of n, a(n) for n = 1..200</a>
%e A247592 a(3)=10 because A002496(10) mod A002496(9)= 677 mod 577 = 10^2.
%p A247592 with(numtheory):nn:=360000:T:=array(1..nn):kk:=0:
%p A247592 for n from 1 to nn do:
%p A247592   if type(n^2+1,prime)=true then
%p A247592    kk:=kk+1:T[kk]:=n^2+1:
%p A247592    else
%p A247592    fi:
%p A247592 od:
%p A247592     for m from 1 to kk-1 do:
%p A247592      r:=irem(T[m+1],T[m]):z:=sqrt(r):
%p A247592       if z=floor(z)
%p A247592        then printf(`%d, `, m+1):
%p A247592        else
%p A247592       fi:
%p A247592     od:
%t A247592 lst={};lst1={};nn=400000;Do[If[PrimeQ[n^2+1],AppendTo[lst,n^2+1]],{n,1,nn}];nn1:=Length[lst];
%t A247592 Do[If[IntegerQ[Sqrt[Mod[lst[[m]],lst[[m-1]]]]],AppendTo[lst1,m]],{m,2,nn1}];lst1
%o A247592 (Python)
%o A247592 from gmpy2 import t_mod, is_square, is_prime
%o A247592 A247592_list, A002496_list, m, c = [], [2], 2, 2
%o A247592 for n in range(1, 10**7):
%o A247592     m += 2*n+1
%o A247592     if is_prime(m):
%o A247592         if is_square(t_mod(m, A002496_list[-1])):
%o A247592             A247592_list.append(c)
%o A247592         A002496_list.append(m)
%o A247592         c += 1 # _Chai Wah Wu_, Sep 20 2014
%Y A247592 Cf. A002496, A193558, A216330.
%K A247592 nonn
%O A247592 1,1
%A A247592 _Michel Lagneau_, Sep 20 2014