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.
For A002145(4) = 19: Since (1+i)^(4k) = (-4)^k, we have (1+i)^72 == 1 (mod 19), and 72 is the smallest such exponent. Hence a(4) = 72.
forprime(p=3, 1e3, if(p%4==3, print1(4*znorder(Mod(-4,p)), ", ")))
n=1: the first prime in the form of 4k+3 is 3, 3+3=6=4*1+2, so a(1)=1; n=2: the second prime in the form of 4k+3 is 7, 7+7=14=3+11=4*3+2, and 11 is also a prime in the form of 4k+3, so a(2)!=3. 7+11=18=4*4+2=3+15, and 15 is not a prime number. So a(2)=4.
goal = 46; plst = {}; pct = 0; clst = {}; n = -1; While[pct < goal,
n = n + 4; If[PrimeQ[n], AppendTo[plst, n]; AppendTo[clst, 0];
pct++]]; n = 2; cct = 0; While[cct < goal, n = n + 4; p1 = n + 1;
While[p1 = p1 - 4; p2 = n - p1; ! ((PrimeQ[p1]) && (PrimeQ[p2]) && (Mod[p2, 4] == 3))]; If[MemberQ[plst, p2], If[id = Position[plst, p2][[1, 1]]; clst[[id]] == 0, clst[[id]] = (n - 2)/4; cct++]]]; clst
ok(n,p)=if(!isprime(n-p),return(0));forprime(q=2,p-1,if(q%4==3 && isprime(n-q),return(0)));1 a(n)=my(p,k); forprime(q=2,,if(q%4==3&&n--==0,p=q;break)); k=(p+1)/4; while(!ok(4*k+2,p),k++); k \\ Charles R Greathouse IV, Mar 19 2013
The loop beginning with 31 is {31, 43, 43 - 8i, 37 - 8i, 37 - 2i, 45 - 2i, 45 - 8i, 43 - 8i, 43, 47, 47 - 2i, 45 - 2i, 45 + 2i, 47 + 2i, 47, 43, 43 + 8i, 45 + 8i, 45 + 2i, 37 + 2i, 37 + 8i, 43 + 8i, 43, 31, 31 + 4i, 41 + 4i, 41 - 4i, 31 - 4i, 31}. But only 19 are unique.
loop2[n_] := Module[{p = n, direction = 1}, lst = {n}; While[While[p = p + direction; ! PrimeQ[p, GaussianIntegers -> True]]; direction = direction*(-I); AppendTo[lst, p]; ! (p == n && direction == 1)]; Length[Union[lst]]]; cp = Select[Range[1000], PrimeQ[#, GaussianIntegers -> True] &]; Table[loop2[p], {p, cp}]
a(4)=23 because 23,29,31,37 alternate 4*n+3,4*n+1,4*n+3,4*n+1 for exactly four primes and 23 is the least prime for a string of exactly four.
Primes:= select(isprime,[seq(2*i+1,i=1..10^7)]): Pm4:= map(`modp`,[seq((-1)^j*Primes[j],j=1..nops(Primes))],4): Starts:= [1,op(select(t -> Pm4[t-1]<> Pm4[t], [$2..nops(Pm4)]))]: Lengths:= [seq(Starts[i+1]-Starts[i],i=1..nops(Starts)-1)]: for i from 1 to max(Lengths) do A[i]:= ListTools:-Search(i,Lengths) od: R:=[seq(A[i],i=1..max(Lengths))]: seq(`if`(a=0,0,Primes[Starts[a]]),a=R); # Robert Israel, Sep 15 2014
i = 2; While[ Mod[ Prime[i] - Prime[i - 1], 4] != 0 || Mod[ Prime[i + 1] - Prime[i], 4] != 0, i++]; T = {Prime[i]}; Do[j = 2; While[! (Product[ Mod[ Prime[k + 1] - Prime[k], 4], {k, j, j + n}] != 0 && (Mod[Prime[j] - Prime[j - 1], 4] == 0 || j == 2) && Mod[ Prime[j + n + 2] - Prime[j + n + 1], 4] == 0), j++]; T = Append[T, Prime[j]], {n, 0, 13}]; T (* Jonathan Sondow, Jun 28 2017 *)
v=vector(100);v[1]=7;cur=1;p=3;forprime(q=5, 1e10, if((q-p)%4==0,if(!v[cur],v[cur]=back(p,cur);print("a("cur") = "v[cur]));cur=1,cur++);p=q) \\ Charles R Greathouse IV, Sep 15 2014
The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4: n | Equation | x_min | y_min 1 | x^2 - 3*y^2 = -2 | 1 | 1 2 | x^2 - 7*y^2 = +2 | 3 | 1 3 | x^2 - 11*y^2 = -2 | 3 | 1 4 | x^2 - 19*y^2 = -2 | 13 | 3 5 | x^2 - 23*y^2 = +2 | 5 | 1 6 | x^2 - 31*y^2 = +2 | 39 | 7 7 | x^2 - 43*y^2 = -2 | 59 | 9 8 | x^2 - 47*y^2 = +2 | 7 | 1 9 | x^2 - 59*y^2 = -2 | 23 | 3
b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((x^2 - 2*(-1)^((p+1)/4))/p), x++); x) forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))
The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4: n | Equation | x_min | y_min 1 | x^2 - 3*y^2 = -2 | 1 | 1 2 | x^2 - 7*y^2 = +2 | 3 | 1 3 | x^2 - 11*y^2 = -2 | 3 | 1 4 | x^2 - 19*y^2 = -2 | 13 | 3 5 | x^2 - 23*y^2 = +2 | 5 | 1 6 | x^2 - 31*y^2 = +2 | 39 | 7 7 | x^2 - 43*y^2 = -2 | 59 | 9 8 | x^2 - 47*y^2 = +2 | 7 | 1 9 | x^2 - 59*y^2 = -2 | 23 | 3
b(p) = if(isprime(p)&&p%4==3, y=1; while(!issquare(p*y^2 + 2*(-1)^((p+1)/4)), y++); y) forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))
The smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4) for the first primes congruent to 3 modulo 4: n | Equation | x_min | y_min 1 | 2*x^2 - 3*y^2 = -1 | 1 | 1 2 | 2*x^2 - 7*y^2 = +1 | 2 | 1 3 | 2*x^2 - 11*y^2 = -1 | 7 | 3 4 | 2*x^2 - 19*y^2 = -1 | 3 | 1 5 | 2*x^2 - 23*y^2 = +1 | 78 | 23 6 | 2*x^2 - 31*y^2 = +1 | 4 | 1 7 | 2*x^2 - 43*y^2 = -1 | 51 | 11 8 | 2*x^2 - 47*y^2 = +1 | 732 | 151 9 | 2*x^2 - 59*y^2 = -1 | 277 | 51
b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((2*x^2 - (-1)^((p+1)/4))/p), x++); x) forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))
The smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4) for the first primes congruent to 3 modulo 4: n | Equation | x_min | y_min 1 | 2*x^2 - 3*y^2 = -1 | 1 | 1 2 | 2*x^2 - 7*y^2 = +1 | 2 | 1 3 | 2*x^2 - 11*y^2 = -1 | 7 | 3 4 | 2*x^2 - 19*y^2 = -1 | 3 | 1 5 | 2*x^2 - 23*y^2 = +1 | 78 | 23 6 | 2*x^2 - 31*y^2 = +1 | 4 | 1 7 | 2*x^2 - 43*y^2 = -1 | 51 | 11 8 | 2*x^2 - 47*y^2 = +1 | 732 | 151 9 | 2*x^2 - 59*y^2 = -1 | 277 | 51
b(p) = if(isprime(p)&&p%4==3, y=1; while(!issquare((p*y^2 + (-1)^((p+1)/4))/2), y++); y) forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))
Select[Range[150], AllTrue[FactorInteger[4*# - 1][[;; , 1]], Mod[#1, 4] == 3 &] &] (* Amiram Eldar, Apr 01 2021 *)
{for(n=1,140,fac=factor(4*n-1);v=vector(matsize(fac)[1],j,fac[j,1])%4;if(vecmin(v)==3,print1(n,",")))} \\ Klaus Brockhaus, May 08 2004
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