cp's OEIS Frontend

A339084 Smaller term p1 of the first of two consecutive cousin prime pairs (p1,p1+4) and (p2,p2+4) such that the distance (p2-p1) is a square.

%I A339084 #11 Dec 12 2020 22:05:26
%S A339084 3,127,313,1447,2203,2437,2797,3217,4933,5653,6007,7207,7537,7603,
%T A339084 7753,8233,10627,11827,12373,20353,22027,22153,23017,23563,25303,
%U A339084 27697,27763,29023,29059,29383,31477,32323,32533,32569,32839,33199,33577,35533,36523,37273,41077
%N A339084 Smaller term p1 of the first of two consecutive cousin prime pairs (p1,p1+4) and (p2,p2+4) such that the distance (p2-p1) is a square.
%C A339084 Considering the 10^6 cousin prime pairs from (3,7) to (252115609,252115613), we note the following:
%C A339084 43617 sequence terms (4.4%) are linked to a distance between two consecutive cousin prime pairs which is a square.
%C A339084 List of the 9 classes of distances which are squares: 4,36,144,324,576,900,1296,1764,2304.
%C A339084 The distance 36 occurs with the highest frequency.
%C A339084 Distances linked to the first 50 terms of the sequence: 4,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,324,144,36,36,36,144,144,144,36,36,36,36,36,36,36,36,144,36,144,36,36,36
%C A339084 From the class 36, the frequency of the distances decreases when their size increases; the distance 4 linked to the first term of the sequence occurs only once.
%C A339084 See for comparison the sequence A338812.
%e A339084 a(3)=313 is in the sequence because the two consecutive cousin prime pairs being (313,317) and (349,353), the distance between them is 349-313=36 which is a square (6^2).
%e A339084 613 is not in the sequence because the two consecutive cousin prime pairs being (613,617) and (643,647), the distance between them is (643-613)=30 which is not a square.
%o A339084 (R)
%o A339084 Mat<-matrix(0,14000000,5)
%o A339084 primes<-generate_n_primes(14000000)
%o A339084 Mat[,1]<-c(primes)
%o A339084 a_n<-c()
%o A339084 Squares<-c()
%o A339084 Squares_sq<-c()
%o A339084 j=1
%o A339084 counter=0
%o A339084 while(j<=13999999){
%o A339084   if(is_prime((Mat[j,1])+4) & is_prime((Mat[j+1,1]))+4){
%o A339084     counter=counter+1
%o A339084     Mat[counter,2]<-(Mat[j,1])
%o A339084     Mat[counter,3]<-Mat[j,1]+4
%o A339084     Mat[counter+1,2]<-(Mat[j+1,1])
%o A339084     Mat[counter+1,3]<-Mat[j+1,1]+4
%o A339084   }
%o A339084   j=j+1
%o A339084 }
%o A339084 k=1
%o A339084 while(k<=1000000){
%o A339084   dist<- Mat[k+1,2]-Mat[k,2]
%o A339084   Mat[k,4]<-dist
%o A339084   if(sqrt(dist)%%1==0){
%o A339084     Mat[k,5]<-dist
%o A339084     a_n<-append(a_n,Mat[k,2])
%o A339084   }
%o A339084   k=k+1
%o A339084 }
%o A339084 View(Mat)
%o A339084 View(a_n)
%o A339084 (PARI) lista(nn) = {my(last=3, p=7); forprime(q=11, nn, if(q-p==4, if (issquare(p-last), print1(last, ", ")); last = p;); p = q;);} \\ _Michel Marcus_, Nov 23 2020
%Y A339084 Cf. A023200, A046132, A094343.
%Y A339084 Cf. A000290, A053320.
%Y A339084 Cf. A176130, A161002, A161533, A161534, A138198, A338812.
%K A339084 nonn,easy
%O A339084 1,1
%A A339084 _Claude H. R. Dequatre_, Nov 23 2020