Search: a266909 -id:a266909
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A257378
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Smallest odd number k such that k*n*2^n+1 is a prime number.
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+0
4
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1, 5, 3, 3, 13, 3, 3, 9, 5, 13, 9, 3, 3, 5, 9, 7, 3, 3, 3, 5, 3, 7, 19, 5, 5, 33, 3, 7, 7, 9, 5, 15, 3, 21, 15, 7, 35, 89, 25, 15, 25, 49, 53, 45, 13, 15, 21, 31, 27, 3, 9, 33, 37, 23, 41, 41, 19, 9, 111, 7, 3, 89, 13, 39, 31, 17, 11, 101, 17, 37, 7, 51, 75
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n) exists for every n.
The conjecture is a corollary of Dirichlet's theorem on primes in arithmetic progressions. - Robert Israel, Jan 05 2016
As N increases sum {k, n=1 to N} / sum {n, n=1 to N} tends to 0.818.
If k=1 then n*2^n+1 is a Cullen prime.
Generalized Cullen primes have the form n*b^n+1, I propose to name the primes k*n*2^n-1 generalized Cullen primes of the second type.
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LINKS
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EXAMPLE
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1*1*2^1+1=3 prime so a(1)=1.
1*2*2^2+1=9 composite, 3*2*2^2+1=25 composite, 5*2*2^2+1=41 prime so a(2)=5.
1*3*2^3+1=25 composite, 3*3*2^3=73 prime so a(3)=3.
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MAPLE
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Q:= proc(m) local k;
for k from 1 by 2 do if isprime(k*m+1) then return k fi od
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MATHEMATICA
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Table[k = 1; While[!PrimeQ[k*n*2^n + 1], k += 2]; k, {n, 73}] (* Michael De Vlieger, Apr 21 2015 *)
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PROG
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(PFGW & SCRIPT)
SCRIPT
DIM n, 0
DIM k
DIMS t
OPENFILEOUT myf, a(n).txt
LABEL loop1
SET n, n+1
IF n>3000 THEN END
SET k, -1
LABEL loop2
SET k, k+2
SETS t, %d, %d\,; n; k
PRP k*n*2^n+1, t
IF ISPRP THEN GOTO a
GOTO loop2
LABEL a
WRITE myf, t
GOTO loop1
(PARI) a(n) = k=1; while(!isprime(k*n*2^n+1), k+=2); k \\ Colin Barker, Apr 21 2015
(PFGW) ABC2 $b*$a*2^$a+1 // {number_primes, $b, 1}
a: from 1 to 10000
b: from 1 to 100000 step 2
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A257379
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Smallest odd number k such that k*n*2^n - 1 is a prime number.
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+0
4
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3, 1, 1, 3, 3, 1, 3, 3, 5, 5, 9, 5, 7, 7, 3, 17, 11, 11, 7, 9, 11, 15, 3, 7, 9, 67, 3, 45, 3, 1, 33, 21, 15, 23, 17, 3, 7, 9, 19, 15, 17, 63, 51, 3, 9, 33, 53, 61, 13, 45, 75, 39, 83, 43, 7, 19, 13, 41, 5, 19, 31, 165, 13, 27, 3, 13, 135, 33, 31, 15, 33, 87
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) exists for every n.
The conjecture follows from Dirichlet's theorem on primes in arithmetic progressions. - Robert Israel, Jan 05 2016
As N increases, (Sum_{n=1..N} k) / (Sum_{n=1..N} n) approaches 0.833.
If k=1 then n*2^n-1 is a Woodall prime (see A002234).
Generalized Woodall primes have the form n*b^n-1, I propose to name the primes k*n*2^n-1 generalized Woodall primes of the second type.
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LINKS
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EXAMPLE
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1*1*2^1 - 1 = unity, 3*1*2^1 - 1 = 5, which is prime, so a(1) = 3.
1*2*2^2 - 1 = 7, which is prime, so a(2) = 1.
1*3*2^3 - 1 = 23, which is prime, so a(3) = 1.
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MAPLE
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Q:= proc(m) local k;
for k from 1 by 2 do if isprime(k*m-1) then return k fi od
end proc:
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MATHEMATICA
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Table[k = 1; While[!PrimeQ[k*n*2^n - 1], k += 2]; k, {n, 72}] (* Michael De Vlieger, Apr 21 2015 *)
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PROG
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(PFGW & SCRIPT)
SCRIPT
DIM n, 0
DIM k
DIMS t
OPENFILEOUT myf, a(n).txt
LABEL loop1
SET n, n+1
IF n>3000 THEN END
SET k, -1
LABEL loop2
SET k, k+2
SETS t, %d, %d\,; n; k
PRP k*n*2^n-1, t
IF ISPRP THEN GOTO a
GOTO loop2
LABEL a
WRITE myf, t
GOTO loop1
(PARI) a(n) = k=1; while(!isprime(k*n*2^n-1), k+=2); k \\ Colin Barker, Apr 21 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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