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

2, 8, 10, 25, 42, 147, 160, 169, 238, 260, 491, 544, 869, 890, 923, 1140, 1337, 1386, 1465, 1643, 1927, 3371, 4614, 5038, 5086, 5225, 5832, 5909, 5995, 7118, 7157, 8540, 9859, 12543, 13505, 13795, 13841, 14211, 15347, 17079, 17263, 18643, 20211, 21184, 21245

Offset: 1

Author

Michel Lagneau, Sep 20 2014

Keywords

• nonn

Comments

A002496 : primes of form n^2+1.

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

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

Examples

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

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A002496, A193558, A216330.

Programs

Maple

with(numtheory):nn:=360000:T:=array(1..nn):kk:=0:
for n from 1 to nn do:
  if type(n^2+1,prime)=true then
   kk:=kk+1:T[kk]:=n^2+1:
   else
   fi:
od:
    for m from 1 to kk-1 do:
     r:=irem(T[m+1],T[m]):z:=sqrt(r):
      if z=floor(z)
       then printf(`%d, `, m+1):
       else
      fi:
    od:

Mathematica

lst={};lst1={};nn=400000;Do[If[PrimeQ[n^2+1],AppendTo[lst,n^2+1]],{n,1,nn}];nn1:=Length[lst];
Do[If[IntegerQ[Sqrt[Mod[lst[[m]],lst[[m-1]]]]],AppendTo[lst1,m]],{m,2,nn1}];lst1

Python

from gmpy2 import t_mod, is_square, is_prime
A247592_list, A002496_list, m, c = [], [2], 2, 2
for n in range(1, 10**7):
    m += 2*n+1
    if is_prime(m):
        if is_square(t_mod(m, A002496_list[-1])):
            A247592_list.append(c)
        A002496_list.append(m)
        c += 1 # Chai Wah Wu, Sep 20 2014