cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A225765 Least k>0 such that k^3+n is prime, or 0 if there is no such k.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 0, 2, 1, 2, 1, 6, 3, 2, 1, 6, 1, 4, 3, 2, 1, 2, 5, 4, 3, 0, 1, 2, 1, 10, 3, 2, 3, 2, 1, 4, 5, 2, 1, 6, 1, 4, 3, 2, 1, 6, 5, 4, 11, 2, 1, 2, 5, 6, 3, 8, 1, 2, 1, 6, 3, 2, 0, 2, 1, 4, 5, 10, 1, 2, 1, 4, 3, 2, 3, 6, 1, 24, 3, 2, 1, 12, 13, 4
Offset: 1

Views

Author

M. F. Hasler, Jul 25 2013

Keywords

Comments

See A225768 for motivation.
a(n) = 0 for n = m^3 (m > 1) but are there other cases of a(n)=0? - Zak Seidov, Nov 10 2014

Examples

			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.

Crossrefs

See A085099, A225766, A225767, A225768 for the k^2, k^4, k^5, k^6 analog.

Programs

  • PARI
    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.
  • PARI
    a(n) = {if ((n!=1) && ispower(n, 3), return (0)); k = 1; while (! isprime(k^3+n), k++); k;} \\ Michel Marcus, Nov 10 2014

Extensions

More terms from Michel Marcus, Nov 10 2014

A225770 Least k > 0 such that k^8 + n is prime, or 0 if there is no such k.

Original entry on oeis.org

0, 1, 1, 4, 1, 12, 1, 2, 3, 110, 1, 6, 1, 2, 195, 2, 1, 6, 1, 40, 3, 2, 1, 66, 25, 2, 9, 2, 1, 180, 1, 22, 15, 58, 25, 408, 1, 2, 3, 10, 1, 12, 1, 4, 465, 4, 1, 12, 5, 10, 147, 2, 1, 6, 35, 2, 45, 2, 1, 570, 1, 2, 21, 4, 0, 6, 1, 6, 9, 100, 1
Offset: 0

Views

Author

M. F. Hasler, Jul 25 2013

Keywords

Comments

See A225768 for motivation and references.

Examples

			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.

Crossrefs

See A085099, A225765, ..., A225769 for the k^2, k^3, ..., k^7 analogs.

Programs

  • PARI
    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.

A225766 Least k>0 such that k^4+n is prime, or 0 if k^4+n is always composite.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 2, 3, 10, 1, 6, 1, 2, 165, 2, 1, 12, 1, 20, 3, 2, 1, 6, 35, 2, 3, 2, 1, 90, 1, 2, 3, 8, 5, 12, 1, 2, 9, 10, 1, 60, 1, 2, 75, 2, 1, 18, 5, 20, 3, 2, 1, 12, 85, 2, 3, 2, 1, 30, 1, 4, 21, 2, 0, 6, 1, 2, 3, 10, 1, 6, 1, 2, 255, 4, 3, 6, 1, 10, 27, 2, 1, 72, 5, 2, 3, 2, 1, 570, 11, 2, 3, 2, 5, 18, 1, 2, 3, 10
Offset: 0

Views

Author

M. F. Hasler, Jul 25 2013

Keywords

Comments

See A225768 for motivation and references.

Examples

			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.

Crossrefs

See A085099, A225765--A225770 for the k^2, k^3, ..., k^8 analogs.

Programs

  • PARI
    {(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.

A225767 Least k>0 such that k^5+n is prime, or 0 if k^5+n is never prime.

Original entry on oeis.org

0, 1, 1, 8, 1, 2, 1, 4, 3, 2, 1, 2, 1, 6, 3, 2, 1, 6, 1, 10, 3, 2, 1, 14, 7, 4, 3, 2, 1, 2, 1, 22, 0, 8, 3, 2, 1, 4, 3, 2, 1, 2, 1, 10, 5, 4, 1, 2, 13, 10, 3, 2, 1, 6, 17, 12, 5, 2, 1, 12, 1, 12, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 7, 2, 1, 4, 63, 2, 1, 18, 5, 4, 11, 32, 1, 14, 11, 6, 5, 4, 3, 2, 1, 6, 11, 2
Offset: 0

Views

Author

M. F. Hasler, Jul 25 2013

Keywords

Comments

See A225768 for motivation and references.
By the theorem of Brillhart, Filaseta and Odlyzko (see link), if a(n) > n > 1 then x^5 + n must be irreducible. If x^5 + n is irreducible, the Bunyakovsky conjecture implies a(n) is finite. - Robert Israel, Apr 25 2016

Examples

			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.

Links

Crossrefs

See A085099, A225765--A225770 for the k^2, k^3, ..., k^8 analogs.

Programs

  • Maple
    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
  • Mathematica
    {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 *)
  • PARI
    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.

A225769 Least k>0 such that k^7+n is prime, or 0 if there is no such k.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 16, 9, 2, 1, 2, 1, 6, 5, 22, 1, 8, 1, 10, 3, 2, 1, 2, 17, 12, 3, 4, 1, 2, 1, 6, 5, 4, 3, 2, 1, 4, 5, 2, 1, 6, 1, 4, 23, 2, 1, 24, 5, 4, 3, 2, 1, 2, 5, 6, 3, 28, 1, 8, 1, 22, 39, 2, 3, 2, 1, 4, 5, 2, 1, 2, 1, 6, 9, 34, 7, 6, 1, 10, 3, 16, 1, 2, 23, 22, 3, 14, 1, 14, 23, 12, 9, 4, 3, 2, 1, 4, 9, 2, 1, 2, 1, 4, 5, 2, 1, 8, 1, 4, 3, 2, 1, 2, 23, 12, 5, 40, 31, 92, 7
Offset: 0

Views

Author

M. F. Hasler, Jul 25 2013

Keywords

Comments

See A225768 for motivation and references.

Crossrefs

See A085099, A225765--A225770 for the k^2, k^3, ..., k^8 analogs.

Programs

  • PARI
    (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.
Showing 1-5 of 5 results.

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