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A299145 Primes of the form j^k + (j-1)^k + ... + 2^k, for j > 1 and k > 0. 1
2, 5, 13, 29, 97, 139, 353, 4889, 72353, 353815699, 42065402653, 84998999651, 102769130749, 15622297824266188673, 28101527071305611527, 20896779938941631284493075599148668795944697935466419104293, 105312291668560568089831550410013687058921146068446092937783402353 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Except for the terms 2, 5, 13, 29, 139, the exponent k satisfies k >= 4. More generally, if Q(j) = j^k + (j-1)^k + ... + 2^k is a term, then j-1 is a divisor of A064538(k). This is because (j-1) is a factor of Q(j) and thus Q(j) is prime only if j-1 is a divisor of the denominator of Q(j), i.e. A064538(k). Thus for each k there is only a finite number of values of j to check. This provides an efficient algorithm to find terms of this sequence by looking only for primes in the numbers H_{j,-k) - 1 = j^k + (j-1)^k + ... + 2^k for j-1 a divisor of A064538(k). - Chai Wah Wu, Mar 06 2018

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..45 (all terms < 10^1000).

EXAMPLE

2 = 2^1;

5 = 3^1 + 2^1;

13 = 3^2 + 2^2;

29 = 4^2 + 3^2 + 2^2;

97 = 3^4 + 2^4;

139 = 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2;

353 = 4^4 + 3^4 + 2^4;

4889 = 4^6 + 3^6 + 2^6;

72353 = 4^8 + 3^8 + 2^8;

MATHEMATICA

With[{nn = 350}, Sort@ Flatten@ Map[Select[#, PrimeQ] &, Table[Total[Range[j, 1, -1]^k] - 1, {j, 2, nn}, {k, nn - j}]]] (* Michael De Vlieger, Feb 03 2018 *)

PROG

(PARI) limit=100000; v=vector(limit); for(n=1, ceil((-1+(1+8*limit)^(1/2))/2), for(k=1, logint(limit, n+0^(n-1)), a=sum(i=1, n, i^k)-1; if(isprime(a)&&a<limit+1, v[a]=1, ))); for(a=1, limit, if(v[a], print1(a", ")))

CROSSREFS

Cf. A007689, A074526, A064538.

Sequence in context: A289843 A242080 A178444 * A122025 A236414 A057873

Adjacent sequences:  A299142 A299143 A299144 * A299146 A299147 A299148

KEYWORD

nonn,more

AUTHOR

Gionata Neri, Feb 03 2018

EXTENSIONS

a(10)-a(15) from Michael De Vlieger, Feb 03 2018

a(16)-a(17) from Chai Wah Wu, Mar 07 2018

STATUS

approved

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Last modified October 27 18:16 EDT 2021. Contains 348287 sequences. (Running on oeis4.)