This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Consecutive primes (22189, 22193, 22229) have gaps (4, 36) so 22189 is in the sequence.
Transpose[Select[Partition[Prime[Range[12300]],3,1],IntegerQ[Sqrt[#[[2]]- #[[1]]]]&&IntegerQ[Sqrt[#[[3]]-#[[2]]]]&]][[1]] (* Harvey P. Dale, Dec 21 2011 *)
Prime 9551 is a term, the gap to the previous prime 9547 is 4 and the gap to the next prime 9587 is 36 and both gaps are squares.
q:= n-> isprime(n) and andmap(issqr, [n-prevprime(n), nextprime(n)-n]): select(q, [$3..200000])[];
q[n_] := PrimeQ[n] && IntegerQ@Sqrt[n-NextPrime[n, -1]] && IntegerQ@ Sqrt[NextPrime[n]-n]; Select[Range[3, 200000], q] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *) Select[Prime[Range[2,15000]],AllTrue[{Sqrt[#-NextPrime[#,-1]],Sqrt[NextPrime[#]-#]},IntegerQ]&] (* Harvey P. Dale, Jan 22 2024 *)
from itertools import islice from sympy import nextprime, integer_nthroot def A353088_gen(): # generator of terms p, q, g, h = 3, 5, True, False while True: if g and h: yield p p, q = q, nextprime(q) g, h = h, integer_nthroot(q-p,2)[1] A353088_list = list(islice(A353088_gen(),30)) # Chai Wah Wu, Apr 22 2022
623071 is the smallest of the consecutive primes 623071, 623107, and 623171 with gaps 623107-623071 = 36, 623171-623107 = 64, and the double gap 623171-623071 = 100 each a perfect square.
PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; t = {}; n = 2; p1 = 1; p2 = 2; p3 = 3; While[Length[t] < 30, n++; p1 = p2; p2 = p3; p3 = Prime[n]; If[PerfectSquareQ[p2 - p1] && PerfectSquareQ[p3 - p2] && PerfectSquareQ[p3 - p1], AppendTo[t, p1]]]; t (* T. D. Noe, Jul 09 2013 *)
psQ[{a_,b_,c_}]:=AllTrue[{Sqrt[b-a],Sqrt[c-b],Sqrt[c-a]},IntegerQ]; Transpose[ Select[Partition[ Prime[Range[600000]],3,1],psQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 20 2014 *)
a(2) = 604441, the smallest of the consecutive primes 604441, 604477, 604481, 604517, with gaps of 36, 4 and 36, all perfect squares.
PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; t = {}; n = 3; p1 = 1; p2 = 2; p3 = 3; p4 = 5; While[Length[t] < 30, n++; p1 = p2; p2 = p3; p3 = p4; p4 = Prime[n]; If[PerfectSquareQ[p2 - p1] && PerfectSquareQ[p3 - p2] && PerfectSquareQ[p4 - p3], AppendTo[t, p1]]]; t (* T. D. Noe, Jul 09 2013 *)
Transpose[Select[Partition[Prime[Range[400000]],4,1],And@@IntegerQ/@ Sqrt[ Differences[#]]&]][[1]] (* Harvey P. Dale, Mar 24 2014 *)
7 and 11 are consecutive primes. 11-7 = 4 a square, so 11 is the second term in the table.
Select[Table[Prime[n], {n, 2, 237}], IntegerQ[Sqrt[# - Prime[PrimePi[# - 1]]]] &] (* Jayanta Basu, Apr 23 2013 *)
nn = 500; ps = Prime[Range[nn]]; t = {}; Do[If[IntegerQ[Sqrt[ps[[n]] - ps[[n-1]]]], AppendTo[t, ps[[n]]]], {n, 2, nn}]; t (* T. D. Noe, Apr 23 2013 *)
Prime[#+1]&/@Flatten[Position[Differences[Prime[Range[250]]],?(IntegerQ[ Sqrt[#]]&)]] (* _Harvey P. Dale, May 08 2019 *)
g(n) = for(x=2, n, if(issquare(prime(x)-prime(x-1)), print1(prime(x)",")))
a(2)=89 since nextprime(89)-89=97-89=8 is the first occurrence of 8 as a difference between successive primes.
with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi end: P:={}; Q:=[]; p:=2; for w to 1 do for k from 0 do # keep track if k mod 10^6 = 0 then print(k,p) fi; lastprime:=p; q:=nextprime(p); d:=q-p; x:=egcd(d); if x>1 and not d in P then P:=P union {d}; Q:=[op(Q), [p,d]]; print(p,d); print(P); print(Q); fi ; p:=q; od od; # let it run with AutoSave enabled.
NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; perfectPowerQ[x_] := GCD @@ Last /@ FactorInteger@x > 1; dd = {1}; pp = {2}; qq = {3}; p = 3; Do[q = NextPrim@p; d = q - p; If[perfectPowerQ@d && ! MemberQ[dd, d], Print@q; AppendTo[pp, p]; AppendTo[dd, d]]; p = q, {n, 10^7}]; pp (* Robert G. Wilson v, Nov 03 2006 *)
S=[];print1(p=2);forprime(q=1+p,,ispower(q-p)&& !setsearch(S,q-p)&& !print1(","p)&& S=setunion(S,[q-p]);p=q) \\ M. F. Hasler, Oct 18 2018
list_gaps(g=256,f,N=25,p=0)=for(c=1,N,while(g+p!=p=nextprime(p+1),);if(f,write(f".txt",c" ",p-g),print1(", "p-g)))
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.
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
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
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)
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). 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.
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
Mat<-matrix(0,14000000,5)
primes<-generate_n_primes(14000000)
Mat[,1]<-c(primes)
a_n<-c()
Squares<-c()
Squares_sq<-c()
j=1
counter=0
while(j<=13999999){
if(is_prime((Mat[j,1])+4) & is_prime((Mat[j+1,1]))+4){
counter=counter+1
Mat[counter,2]<-(Mat[j,1])
Mat[counter,3]<-Mat[j,1]+4
Mat[counter+1,2]<-(Mat[j+1,1])
Mat[counter+1,3]<-Mat[j+1,1]+4
}
j=j+1
}
k=1
while(k<=1000000){
dist<- Mat[k+1,2]-Mat[k,2]
Mat[k,4]<-dist
if(sqrt(dist)%%1==0){
Mat[k,5]<-dist
a_n<-append(a_n,Mat[k,2])
}
k=k+1
}
View(Mat)
View(a_n)
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