A239666 -id:A239666 - 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

A239735 Least number k such that n*k^n +/- 1 are twin primes, or a(n) = 0 if no such number exists.

Original entry on oeis.org

4, 3, 4, 1, 570, 1, 1464, 54, 60, 14025, 1932, 1, 7194, 15, 3612, 0, 4746, 1, 540, 150, 7060, 138, 80094, 6160, 33480, 93135, 0, 366618, 26058, 1, 90510, 16836, 9824, 418875, 57246, 0, 182394, 64077, 14178, 943410, 36078, 1, 314520, 15870, 194942, 15044700, 241944, 3871, 308730

Offset: 1

Derek Orr, Mar 30 2014

nonn

a(n) = 1 iff n is in A014574.
If a(n) = 0, then n is in A097764.
If a(n) > 1 then A367566(n) divides a(n). - Jon E. Schoenfield, Nov 23 2023

			1*1^1+1 (2) and 1*1^1-1 (0) are not both prime. 1*2^1+1 (3) and 1*2^1-1 (1) are not both prime. 1*3^1+1 (4) and 1*3^1-1 (2) are not both prime. 1*4^1+1 (5) and 1*4^1-1 (3) are both prime. So, a(1) = 4.

Jon E. Schoenfield, Table of n, a(n) for n = 1..200 (first 77 terms from Michel Marcus)

Cf. A097764, A014574, A239938, A239666, A367566.

zeroQ[n_] := Module[{f = FactorInteger[n]}, pow = GCD @@ f[[;; , 2]]; n > 4 && AnyTrue[Divisors[pow], # > 1 && Divisible[n, #] &]];
a[n_, kmax_] := Module[{k = 1}, If[zeroQ[n], 0, While[k <= kmax && ! And @@ PrimeQ[n*k^n + {-1, 1}], k++]; If[k < kmax, k, -1]]]; Table[a[n, 10^6], {n, 1, 25}] (* Amiram Eldar, Nov 18 2023, returns -1 if the search limit should exceed kmax *)

bot(n) = for(k=1, 10^5, if(ispseudoprime(n*k^n-1), if(ispseudoprime(n*k^n+1), return(k))));
n=1; while(n<100, print1(bot(n), ", "); n+=1)

a(n) = if ((n==16) || (n==27) || (n==36) || (n==64) /* || (n== ... */, return(0)); my(k=1); while (!(ispseudoprime(n*k^n-1) && ispseudoprime(n*k^n+1)), k++); k; \\ Michel Marcus, Nov 18 2023

a(46) from Giovanni Resta, Mar 31 2014

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

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

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A239735 Least number k such that n*k^n +/- 1 are twin primes, or a(n) = 0 if no such number exists.

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Crossrefs

Programs

Mathematica

PARI

PARI

Extensions