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 A256787 Smallest odd number k such that k*2^(2*n+1)+1 is a prime number. 4
 1, 5, 3, 5, 15, 9, 5, 5, 9, 11, 11, 45, 5, 15, 23, 35, 9, 59, 15, 5, 3, 9, 35, 27, 23, 17, 51, 5, 29, 27, 53, 9, 9, 9, 23, 39, 23, 5, 29, 249, 9, 51, 5, 75, 51, 117, 29, 77, 131, 219, 221, 29, 53, 105, 321, 95, 179, 197, 101, 51, 81, 101, 11, 5, 21, 221, 53 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS As N increases, (Sum_{n=1..N} a(n))/(Sum_{n=1..N} n) appears to tend to log(2), as can be seen by plotting the first 10000 terms. This observation is consistent with the prime number theorem as the probability that k*2^n+1 is prime is 1/(n*log(2)+log(k)) so ~ 1/(n*log(2)) as n increases, if k ~ n*log(2) then k/(n*log(2)) ~ 1. LINKS Pierre CAMI, Table of n, a(n) for n = 0..10000 EXAMPLE 1*2^(2*0+1)+1=3 is prime, so a(0)=1. 1*2^(2*1+1)+1=9 and 3*2^(2*1+1)+1=25 are composite, 5*2^(2*1+1)+1=41 is prime, so a(1)=5. MAPLE for n from 0 to 100 do R:= 2^(2*n+1); for k from 1 by 2 do    if isprime(k*R+1) then A[n]:= k; break fi od: od: seq(A[n], n=0..100); # Robert Israel, Apr 24 2015 MATHEMATICA f[n_] := Block[{g, i, k}, g[x_, y_] := y*2^(2*x + 1) + 1; Reap@ For[i = 0, i <= n, i++, k = 1; While[Nand[PrimeQ[g[i, k]] == True, OddQ@ k], k++]; Sow@ k] // Flatten // Rest]; f@ 66 (* Michael De Vlieger, Apr 18 2015 *) PROG (PFGW & SCRIPT) SCRIPT DIM i DIM n, -1 DIMS t OPENFILEOUT myf, a(n).txt LABEL loop1 SET n, n+2 SET i, -1 LABEL loop2 SET i, i+2 SETS t, %d, %d\,; n; i PRP i*2^n+1, t IF ISPRP THEN GOTO a GOTO loop2 LABEL a WRITE myf, t GOTO loop1 (PARI) vector(100, n, n--; k=1; while(!isprime(k*2^(2*n+1)+1), k+=2); k) \\ Colin Barker, Apr 10 2015 CROSSREFS Sequence in context: A128008 A265800 A144386 * A265782 A073685 A019212 Adjacent sequences:  A256784 A256785 A256786 * A256788 A256789 A256790 KEYWORD nonn AUTHOR Pierre CAMI, Apr 10 2015 STATUS approved

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Last modified January 29 05:09 EST 2022. Contains 350672 sequences. (Running on oeis4.)