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.
n=5: 5+(1..5)^2=(6,9,13,21,30) all composite and 5+6^2=41 is prime hence a(5)=6.
[NextPrime(n^2) - n^2: n in [0..100]]; // Vincenzo Librandi, Jul 06 2015
A053000 := n->nextprime(n^2)-n^2;
nxt[n_]:=Module[{n2=n^2},NextPrime[n2]-n2]
nxt/@Range[0,100] (* Harvey P. Dale, Dec 20 2010 *)
A053000(n)=nextprime(n^2)-n^2 \\ M. F. Hasler, Mar 23 2013
from sympy import nextprime def a(n): nn = n*n; return nextprime(nn) - nn print([a(n) for n in range(94)]) # Michael S. Branicky, Feb 17 2022
a(7)=4 because 1^3+7=8, 2^3+7=15, 3^3+7=34 are all composite, but 4^3+7=71 is prime. a(8)=0 because x^3+8 = (x+2)(x^2-2x+4) is composite for all integer values x>0.
A225765(a,b=3)={#factor(x^b+a)~==1&for(n=1,9e9,ispseudoprime(n^b+a)&return(n));a==1&return(1);print1("/*"factor(x^b+a)"*/")} \\ For illustrative purpose only: the polynomial is factored to avoid an infinite search loop when it is composite. But this does not exclude that all but one factors might equal 1, therefore the factorization is printed for control before 0 is returned.
a(n) = {if ((n!=1) && ispower(n, 3), return (0)); k = 1; while (! isprime(k^3+n), k++); k;} \\ Michel Marcus, Nov 10 2014
f:= proc(n) local exact, x,k,F,nf,F1,C;
iroot(n,3,exact);
if exact and n > 1 then return 0 fi;
if irreduc(x^6+n) then
for k from 1+(n mod 2) by 2 do if isprime(k^6+n) then return k fi od
else
F:= factors(x^6+n)[2]; #
F1:= map(t -> t[1],F);
nf:= nops(F);
C:= map(t -> op(map(rhs@op,{isolve(t^2-1)})),F1);
for k in sort(convert(select(type,C,positive),list)) do
if isprime(k^6+n) then return k fi
od:
0
fi
end proc:
map(f, [$0..100]); # Robert Israel, Apr 25 2016
{0, 1}~Join~Table[If[IrreduciblePolynomialQ[x^6 + n], SelectFirst[Range[1 + Mod[n, 2], 10^3, 2], PrimeQ[#^6 + n] &], 0], {n, 2, 120}] (* Michael De Vlieger, Apr 25 2016, Version 10 *)
{(a,b=6)->#factor(x^b+a)~==1 & for(n=1, 9e9, ispseudoprime(n^b+a)&return(n)); a==1&return(1);print1("/*"a":", factor(x^b+a)"*/")} /* For illustrative purpose only. The polynomial x^6+a is factored to avoid an infinite loop when it is composite. But there could be x such that this is prime, when all factors but one are 1 (not for exponent b=6, but, e.g., x=4 for exponent b=4), see A225766. */
n=6: a(6)=11 and 11^2+6 is 127, a prime; n=97: a(97) = 2144 and 2144^2+97 = 4596833, the least prime of the form m^2+97.
for m from 1 to 10^5 do
r:= nextprime(m^2)-m^2;
if not assigned(R[r]) then R[r]:= m end if;
end do:
J:= map(op,{indices(R)}):
N:= min({$1..J[-1]} minus J)-1:
[seq(R[j],j=1..N)]; # Robert Israel, Aug 10 2012
nn = 100; t = Table[0, {nn}]; found = 0; m = 0; While[found < nn, m++; k = NextPrime[m^2] - m^2; If[k <= nn && t[[k]] == 0, t[[k]] = m; found++]]; t (* T. D. Noe, Aug 10 2012 *)
R = {} # After Robert Israel's Maple script.
for m in (1..2^12) :
r = next_prime(m^2) - m^2
if r not in R : R[r] = m
L = sorted(R.keys())
for i in (1..len(L)-1) :
if L[i] != L[i-1]+1 : break
[R[k] for k in (1..i)] # Peter Luschny, Aug 11 2012
a(0) = 0 since k^8 is not prime for any k > 0. a(4) = 1 since k^8 + 4 is prime for k = 1, although k^8 + 4 = (k^4 - 2k^2 + 2)(k^4 + 2k^2 + 2), but the first factor equals 1 for k = 1. a(64) = 0 since k^8 + 64 = (k^4 - 4*k^2 + 8)(k^4 + 4k^2 + 8) which is composite for all integers k > 1.
A225770(a,b=8)={#factor(x^b+a)~==1&for(n=1,9e9,ispseudoprime(n^b+a)&return(n));a==0 || a==64 || print1("/*"factor(x^b+a)"*/")} \\ For illustrative purpose only. The polynomial is factored to avoid an infinite search loop when it is composite. But a factored polynomial can yield a prime when all factors but one equal 1. This happens for n=4, cf. example.
a(4)=1 because 1^4+4=5 is prime. (Although x^4+4 = (x^2-2*x+2)(x^2+2x+2), this is prime for x=1 when the first factor equals 1.) a(5)=6 because 1^4+5=6, 2^4+5=21, 3^4+5=86, 4^4+5=261 and 5^4+5 are all composite, but 6^4+5=1301 is prime. a(64)=0 because x^4+64 = (x^2-4*x+8)(x^2+4x+8) is composite for all integer values of x>0. Indeed, x^2-4x+8=(x-2)^2+4 > 1 for all x.
{(a,b=4)->#factor(x^b+a)~==1&for(n=1,9e9,ispseudoprime(n^b+a)&return(n));a==1 || a==4 || print1("/*"factor(x^b+a)"*/")} \\ For illustrative purpose only. The polynomial is factored to avoid an infinite search loop when it is composite. But a factored polynomial can yield a prime when all but one factors equal 1. This happens for n=4, cf. Example.
a(3)=8 because 1^5+3, 2^5+3, ..., 7^5+3 are all composite, but 8^5+3=32771 is prime. a(32)=0 because x^5+32 = (x + 2)(x^4 - 2x^3 + 4x^2 - 8x + 16) is composite for all integer values of x>0.
f:= proc(n) local x,k,F,nf,F1,C;
if irreduc(x^5+n) then
for k from 1+(n mod 2) by 2 do if isprime(k^5+n) then return k fi od
else
F:= factors(x^5+n)[2]; #
F1:= map(t -> t[1],F);
nf:= nops(F);
C:= map(t -> op(map(rhs@op,{isolve(t^2-1)})),F1);
for k in sort(convert(select(type,C,positive),list)) do
if isprime(k^5+n) then return k fi
od:
0
fi
end proc:
map(f, [$0..100]); # Robert Israel, Apr 25 2016
{0, 1}~Join~Table[If[IrreduciblePolynomialQ[x^5 + n], SelectFirst[Range[1 + Mod[n, 2], 10^2, 2], PrimeQ[#^5 + n] &], 0], {n, 2, 120}] (* Michael De Vlieger, Apr 25 2016, Version 10 *)
A225767(a,b=5)={#factor(x^b+a)~==1&for(n=1,9e9,ispseudoprime(n^b+a)&return(n));a==1 || print1("/*"factor(x^b+a)"*/")} \\ For illustrative purpose only. The polynomial is factored to avoid an infinite search loop when it is composite. But a factored polynomial can yield a prime when all factors but one equal 1. This happens for b=4, n=4, cf. A225766.
Table[m = n; While[! PrimeQ[m^2 + n], m++]; m, {n, 100}] (* T. D. Noe, Aug 12 2012 *)
(a,b=5)->{#factor(x^b+a)~==1&for(n=1,9e9,ispseudoprime(n^b+a)&return(n));a==1 || print1("/*"factor(x^b+a)"*/")} \\ For illustrative purpose only. The polynomial is factored to avoid an infinite search loop when it is composite. But a factored polynomial can yield a prime when all factors but one equal 1. This happens for b=4, n=4, cf. A225766.
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