A338812 Smaller term of a pair of sexy primes (A023201) such that the distance to next pair (A227346) is a square.

7, 13, 37, 97, 103, 223, 307, 331, 457, 541, 571, 853, 877, 1087, 1297, 1423, 1483, 1621, 1867, 1993, 2683, 3457, 3511, 3691, 3761, 3847, 4513, 4657, 4783, 4951, 5227, 5521, 5647, 5861, 6337, 6547, 6823, 7481, 7541, 7681, 7717, 7753, 7873, 8287, 8521, 8887, 9007, 9397, 10267, 10453

Offset: 1

Author

Claude H. R. Dequatre, Nov 10 2020

Keywords

• nonn
• easy

Comments

Considering the 10^6 sexy prime pairs from (5,11) to (115539653,115539659), we note the following:

65340 sequence terms (6.5%) are linked to a distance between two consecutive sexy prime pairs which is a square.

List of the 16 classes of distances which are squares: 4,16,36,64,100,144,196,256,324,400,484,576,676,784,900,1024.

The frequency of the distances which are squares decreases when their size increases, with a noticeable higher frequency for the distance 36.

First 20 distances which are squares with in parentheses the subtraction of the smallest members of the related two consecutive sexy prime pairs: 4 (11-7), 4 (17-13),4 (41-37),4 (101-97),4 (107-103),4 (227-223),4 (311-307), 16 (347-331),4 (461-457),16 (557-541),16 (587-571),4 (857-853), 4 (881-877), 4 (1091-1087),4 (1301-1297),4 (1427-1423),4 (1487-1483),36 (1657-1621), 4 (1871-1867),4 (1997-1993).

Examples

			a(2)=13 is in the sequence because the two consecutive sexy prime pairs being (13,19) and (17,23),the distance between them is 17-13=4 which is a square (2^2).
73 is not in the sequence because the two consecutive sexy prime pairs being (73,79) and (83,89),the distance between them is 83-73=10 which is not a square.
		

Links

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

Crossrefs

Cf. A023201, A046117,A227346, A000290, A161002, A161533, A161534, A138198.

Programs

Maple

count:= 0: sp:= 5: R:= NULL:
p:= sp;
while count < 100  do
    p:= nextprime(p);
    if isprime(p+6) then
      d:= p - sp;
      if issqr(d) then
        count:= count+1; R:= R, sp;
      fi;
      sp:= p;
    fi;
od:
R; # Robert Israel, May 08 2024

PARI

lista(nn) = {my(vs = select(x->(isprime(x) && isprime(x+6)), [1..nn]), vd = vector(#vs-1, k, vs[k+1] - vs[k]), vk = select(issquare, vd, 1)); vector(#vk, k, vs[vk[k]]);} \\ Michel Marcus, Nov 14 2020

R

primes<-generate_n_primes(7000000)
Matrix_1<-matrix(c(primes),nrow=7000000,ncol=1,byrow=TRUE)
p1<-c(0)
p2<-c(0)
k<-c(0)
distance<-c(0)
distance_square<-(0)
Matrix_2<-cbind(Matrix_1,p1,p2,k,distance,distance_square)
counter=0
j=1
while(j<= 7000000){
  p<-(Matrix_2[j,1])+6
  if(is_prime(p)){
    counter=counter+1
    Matrix_2[counter,2]<-(p-6)
    Matrix_2[counter,3]<-p
  }
  j=j+1
}
a_n<-c()
k=1
while(k<=1000000){
  Matrix_2[k,4]<-k
  dist<-Matrix_2[k+1,2]-Matrix_2[k,2]
  Matrix_2[k,5]<-dist
  if(sqrt(dist)%%1==0){
    Matrix_2[k,6]<-dist
    a_n<-append(a_n,Matrix_2[k,2])
  }
  k=k+1
}
View(Matrix_2)
View(a_n)