A097764 -id:A097764 - cp's OEIS frontend

A255134 First differences of A097764.

12, 11, 9, 28, 36, 44, 52, 20, 40, 68, 76, 84, 92, 100, 53, 55, 116, 124, 132, 140, 148, 156, 128, 36, 172, 180, 188, 196, 204, 212, 209, 11, 228, 11, 225, 244, 252, 260, 268, 276, 284, 292, 300, 56, 252, 316, 324, 332, 340, 348, 356, 364, 372, 380, 45, 343

Offset: 1

Reinhard Zumkeller, Feb 14 2015

nonn

a(n) = A097764(n+1) - A097764(n);
conjecture: a(n) > 1; empirically: Min{a(n): 1 <= n <= 10^7} = 9;
a(A255137(n)) = A255136(n) and a(m) < A255136(n) for m < A255137(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A097764, record values and where they occur: A255136, A255137.

a255134 n = a255134_list !! (n-1)
a255134_list = zipWith (-) (tail a097764_list) a097764_list

nMax = 10000; n = 1; lst = Reap[While[p = Prime[n]; p^p <= nMax, k = 1; While[(k*p)^p <= nMax, Sow[(k*p)^p]; k++]; n++]][[2, 1]]; Union[lst] // Differences (* Jean-François Alcover, Oct 20 2016, after T. D. Noe *)

A051674 a(n) = prime(n)^prime(n).

Original entry on oeis.org

4, 27, 3125, 823543, 285311670611, 302875106592253, 827240261886336764177, 1978419655660313589123979, 20880467999847912034355032910567, 2567686153161211134561828214731016126483469, 17069174130723235958610643029059314756044734431

Offset: 1

Asher Auel

Numbers k such that bigomega(k)^(bigomega(k)) = k, where bigomega = A001222. - Lekraj Beedassy, Aug 21 2004
Positive k such that k' = k, where k' is the arithmetic derivative of k. - T. D. Noe, Oct 12 2004
David Beckwith proposes (in the AMM reference): "Let n be a positive integer and let p be a prime number. Prove that (p^p) | n! implies that (p^(p + 1)) | n!". - Jonathan Vos Post, Feb 20 2006
Subsequence of A100716; A003415(m*a(n)) = A129283(m)*a(n), especially A003415(a(n)) = a(n). - Reinhard Zumkeller, Apr 07 2007
A168036(a(n)) = 0. - Reinhard Zumkeller, May 22 2015

			a(1) = 2^2 = 4.
a(2) = 3^3 = 27.
a(3) = 5^5 = 3125.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 740 pp. 95; 312, Ellipses Paris 2004.

T. D. Noe, Table of n, a(n) for n = 1..40
David Beckwith, Problem 11158, American Mathematical Monthly, Vol. 112, No. 5 (May 2005), p. 468.
Jurij Kovic, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences, Vol. 15 (2012), #12.3.8.

Cf. A000040, A000312, A003415 (arithmetic derivative of n), A129150, A129151, A129152, A048102, A072873 (multiplicative closure), A104126.
Subsequence of A100717; A203908(a(n)) = 0.
Subsequence of A097764.
Cf. A168036, A094289 (decimal expansion of Sum(1/p^p)).

a051674_list = map (\p -> p ^ p) a000040_list
-- Reinhard Zumkeller, Jan 21 2012

[p^p: p in PrimesUpTo(30)]; // Vincenzo Librandi, Mar 27 2014

A051674:=n->ithprime(n)^ithprime(n): seq(A051674(n), n=1..10); # Wesley Ivan Hurt, Jun 25 2016

Array[Prime[ # ]^Prime[ # ] &, 12] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
#^#&/@Prime[Range[10]] (* Harvey P. Dale, May 17 2024 *)

a(n)=n=prime(n);n^n \\ Charles R Greathouse IV, Mar 20 2013

from gmpy2 import mpz
[mpz(prime(n))**mpz(prime(n)) for n in range(1,100)] # Chai Wah Wu, Jul 28 2014

a(n) = A000312(A000040(n)). - Altug Alkan, Sep 01 2016

Sum_{n>=1} 1/a(n) = A094289. - Amiram Eldar, Oct 13 2020

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

A255136 Records in A255134.

Original entry on oeis.org

12, 28, 36, 44, 52, 68, 76, 84, 92, 100, 116, 124, 132, 140, 148, 156, 172, 180, 188, 196, 204, 212, 228, 244, 252, 260, 268, 276, 284, 292, 300, 316, 324, 332, 340, 348, 356, 364, 372, 380, 396, 404, 412, 420, 428, 436, 444, 452, 460, 476, 484, 492, 500

Offset: 1

Reinhard Zumkeller, Feb 15 2015

nonn

a(n) = A255134(A255137(n)) and A255134(m) < a(n) for m < A255137(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255137, A255134, A097764.

a255136 n = a255136_list !! (n-1)
(a255136_list, a255137_list) = unzip $ f [1..] a255134_list (-1) where
   f (x:xs) (y:ys) r = if y > r then (y, x) : f xs ys y else f xs ys r

A255137 Where records occur in A255134.

Original entry on oeis.org

1, 4, 5, 6, 7, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 33, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Feb 15 2015

nonn

A255134(a(n)) = A255136(n) and A255134(m) < A255136(n) for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255136, A255134, A097764.

a255137 n = a255137_list !! (n-1)  -- a255137_list is defined in A255136.

A239279 Smallest k such that n^k - k^n is prime, or 0 if no such number exists.

Original entry on oeis.org

5, 1, 1, 14, 1, 20, 1, 10, 273, 14, 1, 38, 1, 68, 0

Offset: 2

Derek Orr, Mar 14 2014

It is believed that for all n > 4 and not in A097764, a(n) > 0.
a(n+1) = 1 if and only if n is prime.
If a(n) > 0 then a(n) and n are coprime.
If n is in the sequence A097764, then a(n) = 0 or 1 since n^k-k^n is factorable.
33^2570 - 2570^33 is a probable prime, so a(33) is probably 2570. - Jon E. Schoenfield, Mar 20 2014
Unknown a(n) values checked for k <= 10000 using PFGW. a(97) = 6006 found by Donovan Johnson in 2005. The Lifchitz link shows some large candidates for larger n but a smaller k exists in many of those cases. - Jens Kruse Andersen, Aug 13 2014
Unknown a(n) values checked for k <= 15000 using PFGW.

			2^1-1^2 = 1 is not prime. 2^2-2^2 = 0 is not prime. 2^3-3^2 = -1 is not prime. 2^4-4^2 = 0 is not prime. 2^5-5^2 = 7 is prime. So a(2) = 5.

H. Lifchitz and R. Lifchitz, PRP Top Records. Search for x^y-y^x
Derek Orr, Table of n, a(n) for n = 2..99 (unknown k-values are marked "unknown")

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128450.

a(n)=k=1; if(n>4, forprime(p=1, 100, if(ispower(n)&&ispower(n)%p==0&&n%p==0, return(0)); if(n%p==n, break))); k=1; while(!ispseudoprime(n^k-k^n), k++); return(k)
vector(15, n, a(n+1))

import sympy
from sympy import isprime
from sympy import gcd
def Min(x):
  k = 1
  while k < 5000:
    if gcd(k,x) == 1:
      if isprime(x**k-k**x):
        return k
      else:
        k += 1
    else:
      k += 1
x = 1
while x < 100:
  print(Min(x))
  x += 1

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

A239938 a(n) = least number k > 0 such that n*k^n - 1 is prime, or 0 if no such k exists.

Original entry on oeis.org

3, 2, 1, 1, 4, 1, 8, 1, 40, 3, 10, 1, 56, 1, 10, 0, 46, 1, 6, 1, 42, 51, 4, 1, 8, 67, 0, 18, 102, 1, 98, 1, 38, 6, 136, 0, 90, 1, 10, 3, 52, 1, 12, 1, 18, 3, 28, 1, 72, 165, 40, 657, 418, 1, 44, 205, 94, 9, 426, 1, 482, 1, 4, 0, 418, 252, 38, 1, 400, 165, 28, 1, 140

Offset: 1

Derek Orr, Mar 29 2014

nonn

a(n) = 1 iff n-1 is prime.
If a(n) = 0 then n is in A097764. Note the converse is not true: a(4) = 1, not 0.
Up to a(1000), the largest term is a(456) = 947310. The PFGW program has been used to certify all the terms up to a(1000), using the 'N+1' deterministic test. - Giovanni Resta, Mar 30 2014

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

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A066049, A097764, A239787.

nope[n_] := n > 4 && Catch@Block[{p = 2}, While[n >= p^p, If[ IntegerQ[ n^(1/p)/p], Throw@ True]; p = NextPrime@ p]; False]; a[n_] := If[nope@ n, 0, Block[{k = 1}, While[! PrimeQ[n*k^n - 1], k++]; k]]; Array[a, 80] (* Giovanni Resta, Mar 30 2014 *)
A239938[n_] := If[n != 4 && # != 1 && GCD[n, #] != 1 &[GCD @@ FactorInteger[n][[All, -1]]], 0, NestWhile[# + 1 &, 1, Not@PrimeQ[n #^n - 1] &]]; Array[A239938, 73] (* JungHwan Min, Dec 28 2015 *)

Pro(n) = for(k=1,10^4,if(ispseudoprime(n*k^n-1),return(k)));
n=1; while(n<100,print1(Pro(n), ", ");n+=1)

A281141 Least number b > 2 such that n*b^n - 1 is a prime number or 0 if no such b exists.

Original entry on oeis.org

3, 3, 4, 0, 4, 3, 8, 4, 40, 3, 10, 8, 56, 4, 10, 0, 46, 3, 6, 6, 42, 51, 4, 6, 8, 67, 0, 18, 102, 18, 98, 34, 38, 6, 136, 0, 90, 17, 10, 3, 52, 5, 12, 8, 18, 3, 28, 132, 72, 165, 40, 657, 418, 101, 44, 205, 94, 9, 426, 10, 482, 36, 4, 0, 418, 252, 38, 7

Offset: 1

Pierre CAMI, Jan 15 2017

nonn

By definition, if b < n+2 then the prime n*b^n - 1 is a generalized Woodall prime.
a(n) = 0 if n is in A097764. - Robert Israel, Jan 15 2017
From Robert G. Wilson v, Jan 20 2017: (Start)
Odd terms are about 3/14 of the total.
Records: 3, 4, 8, 40, 56, 67, 102, 136, 165, 657, 882, 1442, 4080, 5146, 6388, 8617, 9440, 13470, 19285, 22155, 947310, ..., .
Indices of prime terms: 1, 2, 6, 10, 18, 26, 38, 40, 42, 46, 54, 68, 84, 86, 110, ..., .
Indices of perfect power terms: 3, 5, 7, 8, 12, 14, 23, 25, 44, 58, 62, 63, 69, 107, ..., .
(End)

			1*3^1 - 1 = 2 prime, so a(1) = 3.
2*3^2 - 1 = 17 prime, so a(2) = 3.
3*4^3 - 1 = 191 prime, so a(3) = 4.
4*b^4 - 1 = (2*b^2)^2 - 1 = (2*b^2 + 1)*(2*b^2 - 1), which is always composite, so a(4) = 0.

Robert G. Wilson v, Table of n, a(n) for n = 1..1030 (first 500 terms from Pierre CAMI; a(456) corrected by Robert G. Wilson v)
Pierre CAMI, PFGW Script

Cf. A097764, A240235.

lst = {* the terms in A097764 *}; f[n_] := If[ MemberQ[lst, n], 0, Block[{b = 3}, While[ !PrimeQ[n*b^n - 1], b++]; b]]; Array[f, 70] (* Robert G. Wilson v, Jan 20 2017 *)

cp's OEIS Frontend

A255134 First differences of A097764.

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A051674 a(n) = prime(n)^prime(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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A303121 Least k>1 such that k^n + n is prime, 0 if no such k exists.

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A255136 Records in A255134.

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A255137 Where records occur in A255134.

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A239279 Smallest k such that n^k - k^n is prime, or 0 if no such number exists.

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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.