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A005105
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Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0.
(Formerly M0665)
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51
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2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8191, 8747, 13121, 15551, 23327, 27647, 62207, 73727, 131071, 139967, 165887, 294911, 314927, 442367, 472391, 497663, 524287, 786431, 995327
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OFFSET
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1,1
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COMMENTS
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The definition is given by Guy: a prime p is in class 1+ if the only prime divisors of p + 1 are 2 or 3; and p is in class r+ if every prime factor of p + 1 is in some class <= r+ + 1, with equality for at least one prime factor. - N. J. A. Sloane, Sep 22 2012
See A005109 for the definition of class r- primes.
Odd terms are primes satisfying p==-1 (mod phi(p+1)). - Benoit Cloitre, Feb 22 2002
These are the primes p for which p+1 is 3-smooth. Primes for which either p+1 or p-1 have many small factors are more easily proved prime, so most of the largest primes found have this property. - Michael B. Porter, Feb 19 2013
For n>1, x=2*a(n) is a solution to the equation phi(sigma(x)) = x-phi(x). Also all Mersenne primes are in the sequence. - Jahangeer Kholdi, Sep 28 2014
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, A18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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EXAMPLE
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23 is in the sequence since 23 is prime and 23 + 1 = 24 = 2^3 * 3 has all prime factors less than or equal to 3.
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MAPLE
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For Maple program see Mathar link.
# Alternative:
N:= 10^6: # to get all terms <= N
select(isprime, {seq(seq(2^i*3^j-1, i=0..ilog2(N/3^j)), j=0..floor(log[3](N)))});
# if using Maple 11 or earlier, uncomment the following line
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MATHEMATICA
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mx = 10^6; Select[ Sort@ Flatten@ Table[2^i*3^j - 1, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}], PrimeQ] (* or *)
Prime[ Select[ Range[78200], Mod[ Prime[ # ] + 1, EulerPhi[ Prime[ # ] + 1]] == 0 &]] (* or *)
PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] + 1]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[3, 78200], ClassPlusNbr[ Prime[ # ]] == 1 &]]
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PROG
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(PARI) list(lim)=my(v=List(), N); lim=1+lim\1; for(n=0, logint(lim, 3), N=3^n; while(N<=lim, if(ispseudoprime(N-1), listput(v, N-1)); N<<=1)); Set(v) \\ Charles R Greathouse IV, Jul 15 2011; corrected Sep 22 2015
(Magma) [p: p in PrimesUpTo(6*10^6) | forall{d: d in PrimeDivisors(p+1) | d le 3}]; // Bruno Berselli, Sep 24 2012
(GAP)
A:=Filtered([1..10^7], IsPrime);; I:=[3];;
B:=List(A, i->Elements(Factors(i+1)));;
C:=List([0..Length(I)], j->List(Combinations(I, j), i->Concatenation([2], i)));;
A005105:=Concatenation([2], List(Set(Flat(List([1..Length(C)], i->List([1..Length(C[i])], j->Positions(B, C[i][j]))))), i->A[i])); # Muniru A Asiru, Sep 28 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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