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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

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Last modified December 10 20:23 EST 2019. Contains 329909 sequences. (Running on oeis4.)