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 A005105 Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0. (Formerly M0665) 50
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 REFERENCES 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). LINKS Ray Chandler, Table of n, a(n) for n = 1..7170 (terms < 10^1000, first 691 terms from T. D. Noe, up to 5000 terms from Charles R Greathouse IV) C. K. Caldwell, The Prime Pages G. Everest, P. Rogers and T. Ward, A higher-rank Mersenne problem, pp. 95-107 of ANTS 2002, Lect. Notes Computer Sci. 2369 (2002). FORMULA {primes p : A126433(PrimePi(p)) = 1 }. - R. J. Mathar, Sep 24 2012 EXAMPLE 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. MAPLE 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 # sort(convert(%, list));  # Robert Israel, Sep 28 2014 MATHEMATICA 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 &]] PROG (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 CROSSREFS Cf. A069353, A069356, A005109, A005113, A005106, A005107, A005108, A019434, A000668, A000040, A003586, A081633-A081639, A084071, A090468, A129474, A129475, A129469. Sequence in context: A038953 A237288 A293074 * A086566 A235213 A188552 Adjacent sequences:  A005102 A005103 A005104 * A005106 A005107 A005108 KEYWORD nonn AUTHOR EXTENSIONS More terms from Benoit Cloitre, Feb 22 2002 Edited and extended by Robert G. Wilson v, Mar 20 2003 STATUS approved

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Last modified October 14 14:06 EDT 2019. Contains 328017 sequences. (Running on oeis4.)
The text passage is a detailed description of a specific integer sequence from the Online Encyclopedia of Integer Sequences (OEIS), including its definition, properties, examples, and code implementations in various programming languages.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it is primarily an expository text providing information about a particular sequence of numbers. while it includes mathematical concepts and formulas, these are presented as part of the sequence's definition and properties rather than as questions to be answered.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A005105 Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0. (Formerly M0665) 50
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 REFERENCES 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). LINKS Ray Chandler, Table of n, a(n) for n = 1..7170 (terms < 10^1000, first 691 terms from T. D. Noe, up to 5000 terms from Charles R Greathouse IV) C. K. Caldwell, The Prime Pages G. Everest, P. Rogers and T. Ward, A higher-rank Mersenne problem, pp. 95-107 of ANTS 2002, Lect. Notes Computer Sci. 2369 (2002). FORMULA {primes p : A126433(PrimePi(p)) = 1 }. - R. J. Mathar, Sep 24 2012 EXAMPLE 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. MAPLE 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 # sort(convert(%, list));  # Robert Israel, Sep 28 2014 MATHEMATICA 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 &]] PROG (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 CROSSREFS Cf. A069353, A069356, A005109, A005113, A005106, A005107, A005108, A019434, A000668, A000040, A003586, A081633-A081639, A084071, A090468, A129474, A129475, A129469. Sequence in context: A038953 A237288 A293074 * A086566 A235213 A188552 Adjacent sequences:  A005102 A005103 A005104 * A005106 A005107 A005108 KEYWORD nonn AUTHOR EXTENSIONS More terms from Benoit Cloitre, Feb 22 2002 Edited and extended by Robert G. Wilson v, Mar 20 2003 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 14 14:06 EDT 2019. Contains 328017 sequences. (Running on oeis4.)
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The text passage is a detailed description of a specific integer sequence from the Online Encyclopedia of Integer Sequences (OEIS), including its definition, properties, examples, and code implementations in various programming languages.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It is primarily an expository text providing information about a particular sequence of numbers. While it includes mathematical concepts and formulas, these are presented as part of the sequence's definition and properties rather than as questions to be answered.