A309809 -id:A309809 - cp's OEIS frontend

A309808 Primes formed by concatenating k and 2k+1.

13, 37, 613, 919, 1021, 1123, 1327, 1429, 1531, 2143, 2347, 2551, 2857, 3061, 3163, 3469, 3571, 3673, 3877, 4591, 4999, 50101, 56113, 59119, 63127, 70141, 71143, 74149, 78157, 79159, 81163, 88177, 91183, 93187, 95191, 101203, 105211, 106213, 108217, 110221, 113227, 114229

Offset: 1

J. M. Bergot and Robert Israel, Aug 17 2019

Primes in A309809.

			a(3)=613 is the concatenation of 6 and 2*6+1=13 and is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A030458, A309809.

[a:m in [1..130]|IsPrime(a) where a is 10^(#Intseq(2*m+1))*m+2*m+1]; // Marius A. Burtea, Aug 18 2019

select(isprime, [seq(n*10^(1+ilog10(2*n+1))+2*n+1,n=1..200)]);

A309828 Squares formed by concatenating k and 2*k+1.

Original entry on oeis.org

25, 49, 1225, 4489, 112225, 444889, 11122225, 44448889, 816416329, 1111222225, 1451229025, 3832476649, 4444488889, 111112222225, 444444888889, 10185602037121, 11111122222225, 44444448888889, 46355849271169, 997230019944601, 1111111222222225, 1231148024622961

Offset: 1

Marius A. Burtea, Aug 18 2019

The sequence is infinite. The squares of the form 66...67^2 = 4..48..89 are terms.

Another infinite family is the squares 33...35^2 = 1...122...25. - Robert Israel, Aug 20 2019

			5^2 = 25 = 2_(2 * 2 + 1);
7^2 = 49 = 4_(2 * 4 + 1);
35^2 = 1225 = 12_(2 * 12 + 1);
61907^2 = 3832476649 = 38324_(2 * 38324 + 1).

Ion Cucurezeanu, Perfect squares and cubes of integers, Ed. Gil, Zalău, (2007), ch. 4, p. 25, pr. 211, 212 (in Romanian).

Chai Wah Wu, Table of n, a(n) for n = 1..286

Cf. A000290, A030466, A054215, A109344, A181719, A309808, A309809.

[a:n in [1..30000000]|IsSquare(a) where a is 10^(#Intseq(2*n+1))*n+2*n+1];

F:= proc(m) local x,X,A;
  X:= [numtheory:-rootsunity(2,10^m+2)];
  A:= map(x -> (x^2-1)/(10^m+2), X);
  A:= sort(select(x -> 2*x+1>=10^(m-1) and 2*x+1<10^m, A));
  op(map(x -> x*10^m+2*x+1, A))
end proc:
subsop(1=NULL, [seq(F(m),m=1..10)]); # Robert Israel, Aug 20 2019

Select[Array[FromDigits@ Flatten@ IntegerDigits[{#, 2 # + 1}] &, 10^5],
IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Aug 19 2019 *)

def Test(n):
    s = str(n)
    ps, ss = s[0:len(s)//2], s[len(s)//2:len(s)]
    return int(ss) == 2*int(ps)+1 and s[len(s)//2] != "0"
n, a = 1, 4
while n < 23:
    if Test(a*a):
        print(n,a*a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Aug 19 2019

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A309828_gen(): # generator of terms
    return filter(is_square,(int(str(k)+str((k<<1)+1)) for k in count(1)))
A309828_list = list(islice(A309828_gen(),20)) # Chai Wah Wu, Feb 20 2023

cp's OEIS Frontend

A309808 Primes formed by concatenating k and 2k+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple