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A339459
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Prime numbers a(n) = floor(2^(n^d)) for all n>=1 where d=1.5039285240... is the constant defined at A339457.
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3
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2, 7, 37, 263, 2437, 28541, 414893, 7368913, 157859813, 4035572951, 122006926709, 4328504865941, 178988464493359, 8575347401843113, 473485756611713633, 29985730185033339911, 2168685169398896331137, 178419507110725228550743
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OFFSET
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1,1
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COMMENTS
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Assuming Cramer's conjecture on prime gaps is true, it can be proved that there exists at least one constant d such that all terms of the sequence are primes. The constant giving the smallest growth rate is d=1.503928524069520633527689067897583199190738...
Algorithm to generate the smallest constant d and the associated prime number sequence a(n) = floor(2^(n^d)).
0. n=1, a(1)=2, d=1
1. n=n+1
2. b=floor(2^(n^d))
3. p=smpr(b) (smallest prime >= b)
4. If p=b, then a(n)=p, go to 1.
5. d=log(log(p)/log(2))/log(n)
6. a(n)=p
7. k=1
8. b=floor(2^(k^d))
9. If b<>a(k) and b not prime, then p=smpr(b), n=k, go to 5.
10. If b is prime then a(k)=b
11. If k<n-1, then k=k+1, go to 8.
12. go to 1.
112 decimals of d are sufficient to calculate the first 50 terms of the prime sequence. The prime number given by the term of index n=50 has 109 decimal digits.
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LINKS
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FORMULA
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a(n) = floor(2^(n^d)) where d=1.5039285240...
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EXAMPLE
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This example illustrates the importance of doing full precision calculations: a(19) = floor(2^(19^d)) = floor(2^83.7826351429215150692195114432) = 16637432012996855576590853. Here, the precision required on the exponent of 2 is 28 decimals in order to obtain the correct value for a(19). And the precision required keeps increasing with the index value n.
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PROG
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\\ if precision is large enough, returns the list of first n terms of the sequence
my(curprec=default(realprecision));
default(realprecision, max(prec, curprec));
my(a=List([2]), d=1.0, c=2.0, b, p, ok, smpr(b)=my(p=b); while(!isprime(p), p=nextprime(p+1)); return(p); );
for(j=1, n-1,
b=floor(c^(j^d));
until(ok,
p=smpr(b);
ok = 1;
listput(a, p, j);
if(p!=b,
d=log(log(p)/log(c))/log(j);
for(k=1, j-2,
b=floor(c^(k^d));
if(b!=a[k],
ok=0;
j=k;
break;
);
);
);
);
);
default(realprecision, curprec);
return(a);
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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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