A093324 -id:A093324 - cp's OEIS frontend

A104745 a(n) = 5^n + n.

1, 6, 27, 128, 629, 3130, 15631, 78132, 390633, 1953134, 9765635, 48828136, 244140637, 1220703138, 6103515639, 30517578140, 152587890641, 762939453142, 3814697265643, 19073486328144, 95367431640645, 476837158203146, 2384185791015647, 11920928955078148, 59604644775390649

Offset: 0

Zak Seidov, Mar 23 2005

Numbers m=5^n+n such that equation x=5^(m-x) has solution x=5^n, see A104744.
No primes of the form 5^n+n for n < 7954. - Thomas Ordowski, Oct 28 2013
a(7954) is prime (5560 digits). - Thomas Ordowski, May 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Factordb, 5^7954 + 7954.
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A006127, A058046, A081552, A093324, A104743, A104744, A158879, A181572.

I:=[1, 6, 27]; [n le 3 select I[n] else 7*Self(n-1)-11*Self(n-2) +5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

g:=1/(1-5*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)+n, n=0..31); # Zerinvary Lajos, Jan 09 2009

Table[5^n+n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)
CoefficientList[Series[(1 - x - 4 x^2) / ((1 - 5 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)
LinearRecurrence[{7,-11,5},{1,6,27},30] (* Harvey P. Dale, Dec 03 2017 *)

a(n)=5^n+n \\ Charles R Greathouse IV, Oct 07 2015

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1-x-4*x^2)/((1-5*x)*(1-x)^2).
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3). (End)
E.g.f.: exp(x)*(exp(4*x) + x). - Elmo R. Oliveira, Mar 05 2025

More terms from Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 09 2007

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

A099227 Primes of the form m^k+k, with m and k > 1.

Original entry on oeis.org

11, 37, 67, 83, 227, 443, 521, 1091, 1523, 2027, 3251, 4099, 6563, 6569, 9803, 10651, 11027, 12323, 13691, 15131, 17579, 21611, 29243, 32771, 32783, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203

Offset: 1

T. D. Noe, Oct 06 2004

nonn

It appears that primes of this form are much less common than primes of the form m^k-k (A099228).

As N increases, squares <= N outnumber all higher powers <= N by an increasingly wide margin, so the above observation is increasingly a consequence of the fact that primes of the form m^2 + 2 are less common than primes of the form m^2 - 2. Among numbers of these two forms, multiples of 3 make up 2/3 of the former, but none of the latter. - Jon E. Schoenfield, Jun 05 2021

Vincenzo Librandi and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Librandi)

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A093324 (least k such that n^k+k is prime).

Cf. A099228.

nLim=200000; lst={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Select[Union[lst], PrimeQ]

list(lim)=my(v=List()); for(e=2,logint(lim\=1,2), forstep(n=3-e%2,sqrtnint(lim-e,e),2, my(t=n^e+e); if(isprime(t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jun 23 2023

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

A084743 Smallest prime of the form n^k + k, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 11, 5

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

nonn

a(5) > 5^18 +18 = 3814697265643. Conjecture: No entry is zero.
See A093324 for the values of k. The next term, a(5) = 5^7954+7954, has 5560 digits and is too large to display here. - T. D. Noe, Oct 05 2007
a(6)-a(10) are 7, 54116956037952111668959660883, 16296287810675888690147565507275025288411747149327490005089123594835050398106693649467179109, 83, and 11, respectively. a(11) > 11^190000 + 190000. See A093324 for the k-values. - Derek Orr, Aug 08 2014

Cf. A093324.

a(n)=for(k=1,8000,s=n^k+k;if(ispseudoprime(s),return(s)))
vector(10,n,a(n)) \\ Derek Orr, Aug 08 2014

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A104745 a(n) = 5^n + n.

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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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A099227 Primes of the form m^k+k, with m and k > 1.

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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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A084743 Smallest prime of the form n^k + k, or 0 if no such prime exists.

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