A072883 -id:A072883 - cp's OEIS frontend

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

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

A084047 Smallest prime p such that p - n is an n-th power, or 0 if no such number exists; i.e., smallest prime of the form k^n + n.

Original entry on oeis.org

2, 3, 11, 5, 37, 7, 268435463, 6569, 521, 11, 36028797018963979, 13, 859935929762876868984659981, 4807339234680508004200143948920808143, 32783, 17, 30491346729331195921, 19, 32064977213018365645815827, 147808829414345923316083210206383297621

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 26 2003

A072883 is the main entry for the problem of finding the smallest prime of the form k^n + n: many such k (up to n = 750 and beyond) are listed in the b-file there, but the corresponding primes are too large to list more of them here. - M. F. Hasler, Jul 07 2024

			a(2) = 3 = 1^2 + 2; a(4) = 5 = 1^4 + 4; a(6) = 7 = 1^6 + 6.

Barry Carter, Tweaks to A084047 (request), in r/OEIS on reddit.com, May 8, 2023.

Cf. A084046, A072883.

f[n_] := f[n] = Module[{m=1}, While[!PrimeQ[m^n + n], m++]; Return[m^n + n]]; (* from Barry Carter in r/OEIS, May 08 2023, cf. link. - M. F. Hasler, Jul 07 2024 *)

A084047(n, k=A072883(n))=if(k, k^n + n, 0) \\ M. F. Hasler, Jul 07 2024

In general, if n+1 is prime, then a(n) = n + 1 = 1^n + n.

a(n) = A072883(n)^n + n if A072883(n) is not 0, otherwise 0. - Michel Marcus, Mar 27 2020

Corrected and extended by Ray Chandler, Jun 16 2003

One more term from Michel Marcus, Mar 27 2020

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

A130827 Least k >= 1 such that k^n + n is semiprime, or 0 if no such k exists.

Original entry on oeis.org

3, 2, 1, 3, 1, 7, 3, 1, 1, 11, 2, 7, 1, 1, 7, 3, 5, 23, 4, 1, 1, 3, 2, 1, 1, 21, 14, 11, 12, 7, 16, 1, 1, 1, 26, 37, 1, 1, 4, 21, 6, 31, 4, 25, 1, 71, 14, 1, 10, 1, 10, 371, 36, 1, 3, 1, 1, 185, 2, 43, 1, 49, 104, 1, 18, 205, 70, 1, 2, 33, 38, 541, 1, 105, 8, 1, 24, 395, 30, 3, 1, 71, 20, 1, 1, 1

Offset: 1

Zak Seidov, Aug 18 2007

nonn

There exist values of n for which k^n + n is never prime (cf. A072883). Do there exist values of n for which k^n + n is never semiprime?

Compare with A361803, the equivalent sequence for k^n - n, where a generalized factorization (effectively a polynomial factorization) into 3 factors is given to show that k^n - n is never semiprime for certain n. - Peter Munn, Jun 19 2023

			a(1)=3 because 1^1 + 1 = 2 (prime) and 2^1 + 1 = 3 (prime) but 3^1 + 1 = 4 = 2*2 (semiprime).
a(2)=2 because 1^2 + 2 = 3 (prime) but 2^2 + 2 = 6 = 2*3 (semiprime).
a(3)=1 because 1^3 + 3 = 4 = 2*2 (semiprime).
a(4)=3 because 1^4 + 4 = 5 (prime) and 2^4 + 4 = 20 = 2^2 * 5 but 3^4 + 4 = 85 = 5*17 (semiprime).
a(5)=1 because 1^5 + 5 = 6 = 2*3 (semiprime).

Sean A. Irvine, Table of n, a(n) for n = 1..100.

Cf. A097792 (n such that x^n+n is reducible), A072883 (least k >= 1 such that k^n+n is prime, or 0 if no such k exists).

Cf. A361803.

a(n) = my(k=1); while (bigomega(k^n+n)!=2, k++); k; \\ Michel Marcus, Jun 19 2023

More terms from Sean A. Irvine, Oct 20 2009

A239474 Smallest k >= 1 such that k^n-n is prime. a(n) = 0 if no such k exists.

Original entry on oeis.org

3, 2, 2, 0, 4, 5, 60, 3, 2, 21, 28, 5, 2, 199, 28, 0, 234, 11, 2, 3, 2, 159, 10, 31, 68, 145, 0, 69, 186, 163, 32, 253, 26, 261, 4, 0, 8, 11, 62, 3, 22, 43, 6, 7, 8, 945, 76, 7, 116, 129, 382, 93, 330, 361, 2, 555, 224, 1359, 78, 29, 62, 39, 110, 0, 1032, 37, 462, 29

Offset: 1

Derek Orr, Mar 20 2014

nonn

If n is of the form (pk)^p for some k and some prime p, then a(n) = 0 (See A097764).

			1^1-1 = 0 is not prime. 2^1-1 = 1 is not prime. 3^1-1 = 2 is prime. Thus, a(1) = 3.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418.

import sympy
from sympy import isprime
def TwoMin(x):
  for k in range(1,5000):
    if isprime(k**x-x):
      return k
x = 1
while x < 100:
  print(TwoMin(x))
  x += 1

a(A097764(n)) = 0 for all n.

A097794 Least k such that the absolute value of k^n-n is prime or zero if no such k exists.

Original entry on oeis.org

3, 2, 1, 1, 4, 1, 60, 1, 2, 21, 28, 1, 2, 1, 28, 0, 234, 1, 2, 1, 2, 159, 10, 1, 68, 145, 0, 69, 186, 1, 32, 1, 26, 261, 4, 0, 8, 1, 62, 3, 22, 1, 6, 1, 8, 945, 76, 1, 116, 129, 382, 93, 330, 1, 2, 555, 224, 1359, 78, 1, 62, 1, 110, 0, 1032, 37, 462, 1, 100, 9, 88, 1, 1416, 1, 218

Offset: 1

T. D. Noe, Aug 24 2004

nonn

Because the polynomial x^n - n is reducible for n in A097764, a(n) is 0 for n=16, 27, 36, 64, 100,.... Although x^4-4 is reducible, the factor x^2-2 is -1 for x=1.

Cf. A097764 (n such that x^n-n is reducible), A072883 (least k such that k^n+n is prime).

Table[If[MemberQ[{16, 27, 36, 64, 100}, n], 0, k=1; While[ !PrimeQ[k^n-n], k++ ]; k], {n, 100}]

A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

Original entry on oeis.org

4, 3, 2, 0, 42, 175, 66, 3, 2, 4983, 1770, 55055, 28686, 18765, 8456, 0, 594, 128345, 136080, 81, 92, 1163409, 18810, 10415, 11754, 3855, 0, 86043, 38880, 17639, 26088, 37293, 5540, 612015, 6876, 0, 44220, 130425, 110, 9292527, 1004850, 1812149, 442404, 1007445, 570658

Offset: 1

Derek Orr, Mar 20 2014

nonn

a(n) = 0 iff n is of the form (pk)^p for some k and some prime p (See A097764).

gcd(n,a(n)) = 1 for all a(n) > 0.

			1^1 +/- 1 = 2 and 0 are not both primes. 2^1 +/- 1 = 3 and 1 are not both primes. 3^1 +/- 1 = 4 and 2 are not both primes. 4^1 +/- 1 = 5 and 3 are both primes. Thus a(1) = 4.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418, A239474.

a(n)=for(k=1,10^7,if(ispseudoprime(k^n-n)&&ispseudoprime(k^n+n),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

import sympy
from sympy import isprime
def TwoBoth(x):
  for k in range(1,10**7):
    if isprime(k**x+x) and isprime(k**x-x):
      return k
x = 1
while x < 100:
  if TwoBoth(x) != None:
    print(TwoBoth(x))
  else:
    print(0)
  x += 1

a(A097764(n)) = 0 for all n.

cp's OEIS Frontend

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

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A084047 Smallest prime p such that p - n is an n-th power, or 0 if no such number exists; i.e., smallest prime of the form k^n + n.

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

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A130827 Least k >= 1 such that k^n + n is semiprime, or 0 if no such k exists.

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A239474 Smallest k >= 1 such that k^n-n is prime. a(n) = 0 if no such k exists.

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A097794 Least k such that the absolute value of k^n-n is prime or zero if no such k exists.

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Mathematica

A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

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