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A085053 Number of primes of the form nk+1, where k=1 to n; 0 if no such number exists. 3


%S 1,2,1,3,1,5,2,2,3,6,3,6,4,6,5,6,3,10,2,7,6,9,4,10,5,10,7,11,4,17,3,

%T 10,9,12,9,16,4,9,11,14,5,21,7,11,10,16,8,19,6,18,13,17,5,24,10,19,9,

%U 16,8,27,7,15,13,16,13,30,9,18,13,27,9,26,10,20,18,17,11,29,11,23,18,22,11

%N Number of primes of the form nk+1, where k=1 to n; 0 if no such number exists.

%C Conjecture: no entry is zero; i.e. for every n there exists a prime of the form nk+1, k<=n.

%C The conjecture is essentially the same as the one in A034693, which has a long history in the study of primes in arithmetic progression. - _T. D. Noe_, Jun 29 2003

%H T. D. Noe, <a href="/A085053/b085053.txt">Table of n, a(n) for n = 1..5000</a>

%e When formatted as an array of primes of the form nk+1 up to n^2+1:

%e 2

%e 3,5

%e 7

%e 5,13,17

%e 11

%e 7,13,19,31,37

%e 29,43

%e 17,41

%e 19,37,73

%e 11,31,41,61,71,101

%e 23,67,89

%e 13,37,61,73,97,109

%e 53,79,131,157

%e 29,43,71,113,127,197

%e The sequence contains the number of terms in the n-th row.

%t Table[cnt=0; Do[If[PrimeQ[k*n+1], cnt++ ], {k, n}]; cnt, {n, 100}]

%t Table[Count[n*Range[n]+1,_?PrimeQ],{n,90}] (* _Harvey P. Dale_, Jan 24 2014 *)

%o (PARI) a(m)=local(c); for(n=1,m,c=0; for(k=1,n,if(isprime(n*k+1),c++; )); print1(c","))

%Y Cf. A034693 (smallest k such that kn+1 is prime).

%K nonn

%O 1,2

%A _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 26 2003

%E Edited, corrected and extended by _T. D. Noe_, _Ray Chandler_ and _Jason Earls_, Jun 28 2003

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Last modified July 26 11:35 EDT 2021. Contains 346294 sequences. (Running on oeis4.)