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

1,1

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}.

Links

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

Example

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

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

Prog

(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

Crossrefs

Cf. A002496, A193558, A216330.

Sequence in context: A106358 A209449 A002510 * A102943 A327988 A062880

Adjacent sequences: A247589 A247590 A247591 * A247593 A247594 A247595

Keyword

nonn

Author

Michel Lagneau, Sep 20 2014

Status

approved