A303122 -id:A303122 - cp's OEIS frontend

A072883 Least k >= 1 such that k^n + n is prime, or 0 if no such k exists.

1, 1, 2, 1, 2, 1, 16, 3, 2, 1, 32, 1, 118, 417, 2, 1, 14, 1, 22, 81, 76, 1, 12, 55, 28, 15, 0, 1, 110, 1, 232, 117, 230, 3, 12, 1, 4, 375, 2, 1, 48, 1, 46, 15, 218, 1, 78, 7, 100, 993, 28, 1, 624, 13, 252, 183, 226, 1, 104, 1, 1348, 777, 1294, 0, 1806, 1, 306, 1815, 10, 1, 30, 1

Offset: 1

Benoit Cloitre, Aug 13 2002

nonn

Because the polynomial x^n + n is reducible for n in A097792, a(27) and a(64) are 0. Although x^4 + 4 is reducible, the factor x^2 - 2x + 2 is 1 for x=1. - T. D. Noe, Aug 24 2004

Hugo Pfoertner, Table of n, a(n) for n = 1..756

Cf. A097792 (n such that x^n + n is reducible).

Cf. A239666, A303121, A303122.

Table[If[MemberQ[{27, 64}, n], 0, k=1; While[ !PrimeQ[k^n+n], k++ ]; k], {n, 100}]
(* Second program: *)
okQ[n_] := n == 4 || IrreduciblePolynomialQ[x^n + n];
a[n_] := If[!okQ[n], 0, s = 1; While[!PrimeQ[s^n + n], s++]; s];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 15 2019, from PARI *)

isok(n) = (n==4) || polisirreducible(x^n+n);
a(n) = if (!isok(n), 0, my(s=1); while(!isprime(s^n+n), s++); s); \\ adapted by Michel Marcus, Jan 15 2019

apply( {A072883(n)=if(is_A097792(n), n==4, for(k=1, oo, ispseudoprime(k^n+n) && return(k)))}, [1..99]) \\ M. F. Hasler, Jul 07 2024

from sympy import isprime
def A072883(n):
    if is_A097792(n): return int(n==4)
    for k in range(1,99**9):
        if isprime(k**n+n): return k # M. F. Hasler, Jul 07 2024

More terms from T. D. Noe, Aug 24 2004

A097792 Numbers of the form 4k^4 or (kp)^p for prime p > 2 and k = 1, 2, 3, ....

Original entry on oeis.org

4, 27, 64, 216, 324, 729, 1024, 1728, 2500, 3125, 3375, 5184, 5832, 9261, 9604, 13824, 16384, 19683, 26244, 27000, 35937, 40000, 46656, 58564, 59319, 74088, 82944, 91125, 100000, 110592, 114244, 132651, 153664, 157464, 185193, 202500, 216000

Offset: 1

T. D. Noe, Aug 24 2004

nonn

A result of Vahlen shows that the polynomial x^n + n is reducible over the integers for n in this sequence and no other n.

David A. Corneth, Table of n, a(n) for n = 1..10000
A. Schinzel, Problems and results on polynomials, Algorithms Seminar, INRIA, 1992-1993.
K. T. Vahlen, Über reductible Binome, Acta Mathematica 19:1 (December 1895), pp. 195-198.

Cf. A093324 (least k such that n^k+k is prime), A097764 (numbers of the form (kp)^p).

Cf. A072883, A239666, A303121, A303122.

nMax=500000; lst={}; k=1; While[4k^4<=nMax, AppendTo[lst, 4k^4]; k++ ]; n=2; While[p=Prime[n]; p^p<=nMax, k=1; While[(k*p)^p<=nMax, AppendTo[lst, (k*p)^p]; k++ ]; n++ ]; Union[lst]

upto(n) = {my(res = List()); for(i = 1, sqrtnint(n \ 4, 4), listput(res, 4*i^4) ); forprime(p = 3, log(n), pp = p^p; for(k = 1, sqrtnint(n \ pp, p), listput(res, pp * k ^ p); ) ); listsort(res); res } \\ David A. Corneth, Jan 12 2019

select( {is_A097792(n, p=0)= n%4==0 && ispower(n\4,4) || ((2 < p = ispower(n,,&n)) && if(isprime(p), n%p==0, foreach(factor(p)[,1], q, q%2 && n%q==0 && return(1))))}, [1..10^4]) \\ M. F. Hasler, Jul 07 2024

from sympy import isprime, perfect_power, primefactors
def is_A097792(n):
    return n%4==0 and (perfect_power(n//4,[4]) or n==4) or (
        p := perfect_power(n)) and p[1] > 2 and (p[0]%p[1]==0 if isprime(p[1])
        else any(p[0]%q==0 for q in primefactors(p[1]) if q > 2))
# M. F. Hasler, Jul 07 2024

Is a(n) ~ c * n^3? - David A. Corneth, Jan 12 2019

A303121 Least k>1 such that k^n + n is prime, 0 if no such k exists.

Original entry on oeis.org

2, 3, 2, 0, 2, 175, 16, 3, 2, 539, 32, 221, 118, 417, 2, 85, 14, 133, 22, 81, 76, 115, 12, 55, 28, 15, 0, 2465, 110, 31, 232, 117, 230, 3, 12, 851, 4, 375, 2, 1599, 48, 5461, 46, 15, 218, 6815, 78, 7, 100, 993, 28, 901, 624, 13, 252, 183, 226, 43247, 104, 5063, 1348, 777, 1294, 0, 1806

Offset: 1

Hugo Pfoertner, Apr 23 2018

nonn

The values of n for which k^n + n is reducible over the integers are given in A097792. - Joseph Myers, Allan C. Wechsler Apr 16 2018

			a(2) = 3 because 3^2 + 3 = A303122(2) = 11 is prime, whereas 2^2 + 2 = 6 is composite.

Hugo Pfoertner, Table of n, a(n) for n = 1..365

Cf. A072883, A093324, A097764, A097792, A303122.

If n + 1 is composite, then a(n) = A072883(n). - Altug Alkan, Apr 23 2018

cp's OEIS Frontend

A072883 Least k >= 1 such that k^n + n is prime, or 0 if no such k exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Extensions

A097792 Numbers of the form 4k^4 or (kp)^p for prime p > 2 and k = 1, 2, 3, ....

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

A303121 Least k>1 such that k^n + n is prime, 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula