Search: a245510 -id:a245510
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A245509
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Smallest m such that the first odd number after n^m is composite.
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+0
6
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3, 5, 3, 2, 3, 1, 1, 3, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 3, 1, 1, 3, 3, 2, 2, 1, 1, 5, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2
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OFFSET
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2,1
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COMMENTS
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The locution "first odd number after n^m" means n^m+1 for even n and n^m+2 for odd n.
The first few records in this sequence are a(2)=3, a(3)=5, a(909)=6, a(4995825)=7. No higher value was found up to 5500000 (see also A245510). It is not clear whether a(n) is bounded.
When n is odd, consider the numbers n+2, n^2+2, n^3+2, n^4+2, ... Then find the first term which is composite, and a(n) is the exponent of that term.
When n is even, consider the numbers n+1, n^2+1, n^3+1. Then a(n) is the exponent from the first term which is composite. For n even, we have a(n) <= 3, because n^3+1 = (n+1)(n^2-n+1) is always composite. (End)
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LINKS
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EXAMPLE
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a(2)=3 because, for k=1,2,3,..., the first odd numbers after 2^k are 3, 5, 9,... and the first one which is not prime corresponds to k=3.
a(3)=5 because the first odd numbers following 3^k are 5, 11, 29, 83, 245, ... and the first one which is not prime corresponds to k=5.
a(7)=1 because the odd number following 7^1 is 9, which is not prime.
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MATHEMATICA
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a245509[n_Integer] := Catch[
Do[
If[CompositeQ[n^m + 1 + If[OddQ[n], 1, 0]]
== True, Throw[m]],
{m, 100}]
]; Map[a245509,
f[n_] := Block[{d = If[ OddQ@ n, 2, 1], m = 1, t}, While[t = n^m + d; EvenQ@ t || PrimeQ@ t, m++]; m]; Array[f, 105, 2] (* Robert G. Wilson v, Aug 04 2014 *)
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PROG
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(PARI) avector(nmax)={my(n, k, d=2, v=vector(nmax)); for(n=2, #v+1, d=3-d; k=1; while(1, if(!isprime(n^k+d), v[n-1]=k; break, k++)); ); return(v); }
a=avector(10000) \\ For nmax=6000000 runs out of 1GB memory
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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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A245511
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Smallest m such that the largest odd number < n^m is not prime.
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+0
6
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1, 1, 2, 3, 2, 3, 2, 4, 1, 1, 2, 3, 2, 4, 1, 1, 2, 4, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 2, 3, 1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1
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OFFSET
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2,3
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COMMENTS
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The locution "largest odd number < n^m" means n^m-1 for even n and n^m-2 for odd n.
The record values in this sequence are a(2)=1, a(4)=2, a(5)=3, a(9)=4, a(279)=5, a(15331)=6, a(1685775)=7. No higher value was found up to 5500000 (see also A245512). It is not clear whether a(n) is bounded.
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LINKS
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EXAMPLE
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a(2)=1 because 2^1-1 is 1, which is not a prime.
a(5)=3 because the numbers 5^k-2, for k=1,2,3,.., are 3,23,123,... and the first nonprime among them corresponds to k=3.
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MATHEMATICA
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f[n_] := Block[{m = 1, d = If[ OddQ@ n, 2, 1]}, While[t = n^m - d; EvenQ@ t || PrimeQ@ t, m++]; m]; Array[f, 105, 2] (* Robert G. Wilson v, Aug 04 2014 *)
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PROG
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(PARI) avector(nmax)={my(n, k, d=2, v=vector(nmax)); for(n=2, #v+1, d=3-d; k=1; while(1, if(!isprime(n^k-d), v[n-1]=k; break, k++)); ); return(v); }
a=avector(10000) \\ For nmax=6000000 runs out of 1GB memory
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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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A245512
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Records in A245511: smallest m > 1 such that the largest odd number less than m^k is prime for every 0 < k < n, but not for k = n.
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6
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OFFSET
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1,1
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COMMENTS
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For more comments and a program, see A245511. a(9), if it exists, certainly exceeds 500000000. It is not clear whether this sequence is infinite, nor whether a(n) is defined for every n.
For n > 2, a(n) is always odd, because A245511(i) can exceed 2 only when i is odd. Therefore to find more terms, it suffices to find odd bases m such that m-2, m^2-2, m^3-2, m^4-2, ..., m^N-2 is a long list of primes. - Jeppe Stig Nielsen, Sep 14 2022
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LINKS
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EXAMPLE
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a(3) = 5 because the odd numbers preceding 5^k, for k = 1,2,3, are 3, 23 and 123, and the first one which is not a prime corresponds to k = 3. Moreover, 5 is the smallest natural having this property.
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MATHEMATICA
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f[n_] := Block[{d = If[ OddQ@ n, 2, 1], m = 1, t}, While[t = n^m - d; EvenQ@ t || PrimeQ@ t, m++]; m]; t = Table[0, {25}]; k = 2; While[k < 210000000, a = f@ k; If[ t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t (* Robert G. Wilson v, Aug 04 2014 *)
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PROG
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(PARI)
a(n) = for(k=1, 10^6, c=0; for(i=1, n-1, if(isprime(k^i-(k%2)-1), c++)); if(c==n-1&&!isprime(k^n-(k%2)-1), return(k)))
n=1; while(n<10, print1(a(n), ", "); n++) \\ Derek Orr, Jul 27 2014
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A245513
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Smallest m such that neither of the two odd numbers that bracket n^m is a prime.
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6
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6, 7, 3, 4, 3, 3, 2, 6, 3, 2, 2, 3, 3, 6, 3, 2, 2, 4, 3, 3, 2, 1, 3, 2, 1, 4, 2, 5, 2, 2, 2, 3, 1, 3, 3, 1, 2, 3, 3, 2, 2, 3, 2, 5, 2, 1, 2, 3, 1, 2, 2, 1, 3, 3, 1, 3, 2, 2, 2, 3, 2, 6, 1, 2, 3, 1, 2, 5, 2, 4, 2, 2, 3, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 2, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 2
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OFFSET
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2,1
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COMMENTS
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The locution "the two odd numbers which bracket n^m" indicates the pair (n^m-1,n^m+1) for even n and (n^m-2,n^m+2) for odd n.
The initial records in this sequence are a(2)=6, a(3)=7, a(2055)=8. No higher value was found up to 5500000. It is not clear whether a(n) is bounded.
Heuristically, Prob(a(n) > m) ~ (2/log n)^m/m! as n -> infinity for fixed m. The sum over n diverges, so we should expect infinitely many a(n) > m. - Robert Israel, Aug 12 2014
a(215539779) = 9 is a record and there is no higher value up to 4*10^9. a(n) <= 3 for all even n > 2, since n-1 divides n^3-1 and n+1 divides n^3+1. - Jens Kruse Andersen, Aug 14 2014
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LINKS
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EXAMPLE
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a(4)=3 because 4^1 and 4^2 are bracketed by the odd numbers (3,5) and (15,17) and each pair contains a prime, but 4^3 is bracketed by (63,65) which are both nonprimes.
a(5)=4 because 5^1, 5^2, and 5^3 are bracketed by odd pairs (3,7), (23,27) and (123,127) which all contain at least one prime. But 5^4 is bracketed by odd numbers (623,627) which are both composites.
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MAPLE
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f:= proc(n) local m, nm;
for m from 1 do
nm:= n^m;
if n::odd then if not isprime(nm+2) and not isprime(nm-2) then return(m) fi
elif not isprime(nm+1) and not isprime(nm-1) then return(m)
fi
od
end proc:
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MATHEMATICA
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a245513Q[n_Integer] := Module[{i},
Catch[For[i = 0, i <= 20, i++,
If[EvenQ[n],
If[! PrimeQ[n^i + 1] && ! PrimeQ[n^i - 1], Throw[i]],
If[! PrimeQ[n^i + 2] && ! PrimeQ[n^i - 2], Throw[i]]
]]]]; a245513[n_Integer] := a245513Q /@ Range[2, n]; a245513[120] (* Michael De Vlieger, Aug 12 2014 *)
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PROG
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(PARI) avector(nmax)={my(n, k, d=2, v=vector(nmax)); for(n=2, #v+1, d=3-d; k=1; while(1, if((!isprime(n^k-d))&&(!isprime(n^k+d)), v[n-1]=k; break, k++)); ); return(v); }
a=avector(10000) \\ For nmax=6000000 runs out of 1GB memory
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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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A245514
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Smallest m such that at least one of the two odd numbers which bracket n^m is not a prime.
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+0
6
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1, 1, 2, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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2,3
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COMMENTS
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The locution "the two odd numbers which bracket n^m" indicates the pair (n^m-1,n^m+1) for even n and (n^m-2,n^m+2) for odd n.
The initial records in this sequence are a(2)=1, a(4)=2, a(9)=3, a(102795)=4. No higher value was found up to 5500000. It is not clear whether a(n) is bounded.
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LINKS
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EXAMPLE
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a(2)=1 because one of the two odd numbers (1,3) which bracket 2^1 is not a prime. a(5)=2 because 5^1 is bracketed by the odd numbers (3,7) which are both prime, while 5^2 is bracketed by the odd numbers (23,27), one of which is not a prime.
The number c=102795 is the smallest one whose powers c^1, c^2, c^3 are all odd-bracketed by primes, while c^4 is not.
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PROG
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(PARI) avector(nmax)={my(n, k, d=2, v=vector(nmax)); for(n=2, #v+1, d=3-d; k=1; while(1, if((!isprime(n^k-d))||(!isprime(n^k+d)), v[n-1]=k; break, k++)); ); return(v); }
a=avector(10000) \\ For nmax=6000000 runs out of 1GB memory
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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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A216930
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Numbers k such that k + 2, k^2 + 2, k^3 + 2, k^4 + 2 and k^5 + 2 are all prime.
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+0
1
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1, 909, 11925, 358875, 959595, 1047585, 3673089, 3925635, 3973971, 4995825, 5519241, 6516015, 6832245, 7217805, 7422381, 9145809, 10929765, 11038071, 11477235, 11721291, 12015555, 12262791, 12280935, 13454349, 13508475, 14625849, 15320829, 15321489, 15332745
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OFFSET
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1,2
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COMMENTS
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k^6 + 2 is also prime for k = 4995825, 11038071, ...
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LINKS
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FORMULA
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MATHEMATICA
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Select[Range[16000000], And@@PrimeQ/@(Table[n^i+2, {i, 1, 5}]/.n->#)&]
Select[Range[16*10^6], AllTrue[2 + #^Range[5], PrimeQ] &] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 24 2015 *)
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PROG
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(Python)
from sympy import isprime
def ok(n): return all(isprime(n**i+2) for i in range(1, 6))
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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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A357280
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Smallest m such that m^k-2 and m^k+2 are prime for k=1..n.
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+0
0
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(3) = 102795 because:
for k = 1; 102795^1-2 = 102793 and 102795^1+2 = 102797, both of which are prime, and
for k = 2; 102795^2-2 = 10566812023 and 102795^2+2 = 10566812027, both of which are prime, and
for k = 3; 102795^3-2 = 1086215442109873 and 102795^3+2 = 1086215442109877, both of which are prime, and
102795 is the smallest number with this property.
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PROG
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(PARI) isok(m, n) = for (k=1, n, if(!isprime(m^k-2) || !isprime(m^k+2), return(0)); ); return(1);
a(n) = my(m=1); while(!isok(m, n), m++); m; \\ Michel Marcus, Nov 14 2022
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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STATUS
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approved
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