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A088783 Numbers n such that 10*n^k + 1 is composite for all k > 0. 2

%I

%S 12,23,32,34,45,56,65,67,78,89,98,100,111,122,131,133,144,155,164,166,

%T 177

%N Numbers n such that 10*n^k + 1 is composite for all k > 0.

%C All terms in the sequence fit the pattern 11m + 1 or 33m - 1 up to a(21) = 177. Note that 10*(11m + 1)^k + 1 is divisible by 11 for all k, and 10*(33m - 1)^k + 1 is divisible by 3 when k is odd and 11 when k is even.

%C A prime 10*173^k+1 is now known (for k=264234, see link). The next blocker is 10*185^k+1 (is 10 a base-185 SierpiƄski number?). - _Jeppe Stig Nielsen_, Apr 30 2018

%C 10*185^k+1 is composite for all k <= 10^6 (see the Barnes link). - _Eric Chen_, Jun 07 2018

%C If a prime 10*185^k+1 were found, then this sequence would continue with 188, 197, 199, 210, 221, 230, 232, 243, 254, 263, 265, 276, 287, 296, 298, 309, 320, 329, 331, ... - _Eric Chen_, Jun 07 2018

%C The prime 10*173^264234+1 is found to solve the generalized Sierpinski problem in base 173 (with conjectured smallest Sierpinski number k=28, see the Barnes link and A123159 for these problems). - _Eric Chen_, Jun 07 2018

%C All unknown terms below 1024 are 185, 338, 417, 432, 537, 614, 668, 743, 744, 773, 786, 827, 863, 929, 935, 977, 986, 1000, 1004. Search limits: 185 at 10^6, 417 at 4*10^5, 743, 773 and 935 at 2*10^5, 338, 744 and 977 at 10^5, 432 at 25000, other numbers except 1000 at 5000, 1000 is corresponding the generalized Fermat prime in base 10 and already searched to (2^24-1)/3, since the smallest prime of the form 10^n+1 greater than 101 is at least 10^(2^24)+1. - _Eric Chen_, Jun 09 2018

%C Large primes with n <= 1024 and exponent > 10^4: 10*173^264234+1, 10*198^47664+1, 10*311^314806+1, 10*341^106008+1, 10*449^18506+1, 10*492^42842+1, 10*605^12394+1, 10*708^17562+1, 10*710^31038+1, 10*800^15104+1, 10*802^149319+1, 10*879^25003+1, they are all proven primes, i.e., not merely probable primes, since they can be proved prime with the N-1 method. - _Eric Chen_, Jun 09 2018

%C All other n <= 1024 with n != 1 (mod 11) and n != 32 (mod 33) have at least one prime of the form 10*n^k+1 with k <= 10^4. - _Eric Chen_, Jun 09 2018

%H Gary Barnes, <a href="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm">Sierpinski conjectures and proofs</a>

%H Chris K. Caldwell, <a href="http://primes.utm.edu/primes/page.php?id=119526">The Prime Database: 10*173^264234 + 1</a>

%F n = 11m+1 and n = 33m-1 for m > 0.

%o (PARI) for(n=2,10^3,if(n%11==1||n%33==32,print1(n,", ");next());for(k=1,+oo,ispseudoprime(10*n^k+1)&&next(2))) \\ _Jeppe Stig Nielsen_, Apr 30 2018

%Y Cf. Indices of zero entries in A088622 & A088782.

%K more,nonn

%O 1,1

%A _Ray Chandler_, Oct 23 2003

%E a(21) from _Jeppe Stig Nielsen_, Apr 30 2018

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Last modified July 10 15:45 EDT 2020. Contains 335577 sequences. (Running on oeis4.)