A005109 -id:A005109 - cp's OEIS frontend

A005105 Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0.

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

Offset: 1

N. J. A. Sloane, Simon Plouffe

nonn

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

			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.

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).

Ray Chandler, Table of n, a(n) for n = 1..7170 (terms < 10^1000; terms 1..691 from T. D. Noe, terms 692..5000 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).
R. J. Mathar, Maple programs to generate b-files for b005105 to b005108, b081633 etc.
Index entries for sequences related to the Erdos-Selfridge classification

Cf. A069353, A069356, A005109, A005108, A019434, A000668, A000040, A003586, A129469.

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

[p: p in PrimesUpTo(6*10^6) | forall{d: d in PrimeDivisors(p+1) | d le 3}]; // Bruno Berselli, Sep 24 2012

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

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 &]]

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

from itertools import count, islice
from sympy import integer_log, isprime
def A069353(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x+1)//3**i).bit_length() for i in range(integer_log(x+1,3)[0]+1))
    return bisection(f,n-1,n-1)
def A005105_gen(): # generator of terms
    return filter(lambda n:isprime(n), map(A069353,count(1)))
A005105_list = list(islice(A005105_gen(),30)) # Chai Wah Wu, Mar 31 2025

{primes p : A126433(PrimePi(p)) = 1 }. - R. J. Mathar, Sep 24 2012

More terms from Benoit Cloitre, Feb 22 2002

Edited and extended by Robert G. Wilson v, Mar 20 2003

A056637 a(n) is the least prime of class n-, according to the Erdős-Selfridge classification of primes.

Original entry on oeis.org

2, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107, 1432349099, 22111003847, 110874748763

Offset: 1

Robert G. Wilson v, Jan 31 2001

A prime p is in class 1- if p-1 has no prime factor larger than 3. If p-1 has other prime factors, p is in class (c+1)-, where c- is the largest class of its prime factors. See also A005109.
1432349099 < a(16) <= 25782283783.
a(18) <= 619108107719, a(19) <= 19811459447009, a(20) <= 152772264735359. These upper limits can be found by generating class (n+1)- primes from a list of n- class primes; if the latter is sufficiently complete, one can deduce that there is no smaller (n+1)- prime. - M. F. Hasler, Apr 05 2007

Cf. A005113, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430.

Cf. A082449, A129246, A081640, A129248.

PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; 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]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; a = Table[0, {15}]; a[[1]] = 2; k = 5; Do[c = ClassMinusNbr[ k]; If[ a[[c]] == 0, a[[c]] = k]; k = NextPrime[k], {n, 3, 7223000}]; a

a(n+1) >= 2*a(n)+1, since a(n+1)-1 is even and must have a factor of class n- which is odd (n>1) and >= a(n). a(n+1) <= min { p = 2*k*a(n)+1 | k=1,2,3... such that p is prime }, since a(n) is a prime of class n-. - M. F. Hasler, Apr 05 2007

Extended by Robert G. Wilson v, Mar 20 2003
More terms from Don Reble, Apr 11 2003
a(16) and a(17) from M. F. Hasler, Apr 21 2007

A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

Original entry on oeis.org

7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993

Offset: 1

Labos Elemer, Dec 20 2000

nonn

Prime numbers n such that cos(2*Pi/n) is an algebraic number of a 3-smooth degree, but not a 2-smooth degree. - Artur Jasinski, Dec 13 2006
From Antonio M. Oller-Marcén, Sep 24 2009: (Start)
In this case gcd(a,b) is a power of 2.
A regular polygon of n sides is constructible by paper folding if and only if n=2^r3^sp_1...p_t with p_i being distinct primes of this kind. (End)
Primes in A005109 but not in A092506. - R. J. Mathar, Sep 28 2012
Conjecture: these are the only solutions >=7 to the equation A000010(x) + A000010(x-1) = floor((4*x-3)/3). - Benoit Cloitre, Mar 02 2018
These are also called Pierpont primes. - Harvey P. Dale, Apr 13 2019

Ray Chandler, Table of n, a(n) for n = 1..8378 (terms < 10^1000, first 1000 terms from T. D. Noe)

Cf. A033845, A000423, A125866, A217035.

N:= 10^10: # to get all terms <= N+1
sort(select(isprime, [seq(seq(1+2^a*3^b, a=1..ilog2(N/3^b)), b=1..floor(log[3](N)))])); # Robert Israel, Mar 02 2018

Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {3} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
mx = 1500000; s = Sort@ Flatten@ Table[1 + 2^j*3^k, {j, Log[2, mx]}, {k, Log[3, mx/2^j]}]; Select[s, PrimeQ] (* Robert G. Wilson v, Sep 28 2012 *)
Select[Prime[Range[114000]],FactorInteger[#-1][[All,1]]=={2,3}&] (* Harvey P. Dale, Apr 13 2019 *)

Primes of the form 1 + A033845(n).

A055600 Numbers of form 2^i*3^j + 1 with i, j >= 0.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 10, 13, 17, 19, 25, 28, 33, 37, 49, 55, 65, 73, 82, 97, 109, 129, 145, 163, 193, 217, 244, 257, 289, 325, 385, 433, 487, 513, 577, 649, 730, 769, 865, 973, 1025, 1153, 1297, 1459, 1537, 1729, 1945, 2049, 2188, 2305, 2593, 2917, 3073, 3457, 3889

Offset: 1

Henry Bottomley, Jun 01 2000

If X is an n-set and Y a fixed (n-5)-subset of X then a(n-5) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007

			a(7) = 13 since 13 = 2^2 * 3^1 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Graham Everest, Peter Rogers, and Thomas Ward, A higher-rank Mersenne problem, Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7-12, 2002 Proceedings 5, Lect. Notes Computer Sci. 2369, Springer Berlin Heidelberg, 2002, pp. 95-107; alternative link.
Milan Janjić, Two Enumerative Functions.

Primes in this sequence give A005109 (Class 1- or Pierpoint primes).

Cf. A003586.

mx = 4000; Sort@ Flatten@ Table[ 2^i*3^j + 1, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

is(k) = if(k == 2, 1, k > 2 && vecmax(factor(k - 1, 5)[, 1]) < 5); \\ Amiram Eldar, Sep 02 2024

from sympy import integer_log
def A055600(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return bisection(f,n,n)+1 # Chai Wah Wu, Sep 15 2024

a(n) = A003586(n) + 1.

Offset corrected by Amiram Eldar, Sep 02 2024

A081426 Class 7- primes.

Original entry on oeis.org

1439, 8629, 10067, 14683, 17257, 19577, 20389, 22643, 23743, 27103, 28219, 29399, 31657, 32633, 33107, 33113, 33863, 34259, 34513, 35951, 36137, 36887, 37379, 40127, 40637, 40759, 42179, 42209, 42767, 44519, 44579, 45139, 49019, 49669

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

R. K. Guy, Unsolved Problems in Number Theory, A18.

R. J. Mathar, Table of n, a(n) for n = 1..20000

Cf. A005113, A056637, A005109, A005110, A005111, A005112, A081424, A081425, A081427, A081428, A081429, A081430.

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]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[5200], ClassMinusNbr[ Prime[ # ]] == 7 &]]

A081427 Class 8- primes.

Original entry on oeis.org

2879, 20147, 25903, 34537, 46049, 58733, 63317, 65267, 69029, 69073, 74759, 80537, 86291, 86341, 103549, 106487, 108413, 112877, 120877, 131687, 135859, 138053, 140939, 141023, 147647, 155413, 157427, 165527, 172681, 187163, 189949, 207079

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

R. K. Guy, Unsolved Problems in Number Theory, A18.

R. J. Mathar, Table of n, a(n) for n = 1..20000

Cf. A005113, A056637, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081428, A081429, A081430.

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]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[20000], ClassMinusNbr[ Prime[ # ]] == 8 &]]

A216148 Primes of the form 2*k^k + 1 = A216147(k).

Original entry on oeis.org

3, 17832200896513, 78692816150593075150849

Offset: 1

M. F. Hasler, Sep 02 2012

The sequence should be extended through A110932, which lists the corresponding values of k: The next term, 2*251^251 + 1 = A216147(A110932(4)) ~ 4.16*10^602, is too large to include here.

M. F. Hasler, Table of n, a(n) for n = 1..4
C. Caldwell, G.L. Honaker (Eds), Prime Curios!: 78692816150593075150849.
"Jim", Topic: The Lost Proof of Fermat, on mathforum.org, Jan 31 2004

Cf. A110932.

A subsequence of A133663, with b=a and c=1.

Select[Table[2n^n+1,{n,20}],PrimeQ] (* Harvey P. Dale, Mar 27 2016 *)

for(n=1,999, ispseudoprime(p=n^n*2+1) & print1(p","))

a(2) = A216147(12) = A005109(95) = A070855(12) = A058383(89) = A133663(18).

a(3) = A216147(18) = A005109(183)= A070855(18) = A058383(177)= A133663(36).

A081429 Class 10- primes.

Original entry on oeis.org

138197, 207227, 621679, 621883, 633383, 760079, 829177, 863711, 898253, 978863, 1035499, 1036471, 1209191, 1451059, 1566179, 1658309, 1658353, 1761407, 1794229, 1796503, 1827479, 1900147, 2015303, 2029439, 2070997, 2072893

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

R. K. Guy, Unsolved Problems in Number Theory, A18.

R. J. Mathar, Table of n, a(n) for n = 1..1188

Cf. A005113, A056637, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081430.

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]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[200000], ClassMinusNbr[ Prime[ # ]] == 10 &]]

A081430 Class 11- primes.

Original entry on oeis.org

1266767, 1520159, 2486717, 3316619, 4144541, 4512947, 4836779, 5389519, 5638379, 6218827, 6448979, 6633457, 6771419, 6907247, 7460149, 7462639, 7600597, 7739033, 7874627, 8153567, 8291573, 9110639, 9112319, 9121003

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

R. K. Guy, Unsolved Problems in Number Theory, A18.

M. F. Hasler, Table of n, a(n) for n=1..1000

Cf. A005113, A056637, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081429.

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]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[300000, 1000000], ClassMinusNbr[ Prime[ # ]] == 1 &]]

A077497 Primes of the form 2^r*5^s + 1.

Original entry on oeis.org

2, 3, 5, 11, 17, 41, 101, 251, 257, 401, 641, 1601, 4001, 16001, 25601, 40961, 62501, 65537, 160001, 163841, 16384001, 26214401, 40960001, 62500001, 104857601, 167772161, 256000001, 409600001, 655360001, 2441406251, 2500000001, 4194304001, 10485760001

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

These are also the prime numbers p for which there is an integer solution x to the equation p*x = p*10^p + x, or equivalently, the prime numbers p for which (p*10^p)/(p-1) is an integer. - Vicente Izquierdo Gomez, Feb 20 2013

For n > 2, all terms are congruent to 5 (mod 6). - Muniru A Asiru, Sep 03 2017

			101 is in the sequence, since 101 = 2^2*5^2 + 1 and 101 is prime.

Ray Chandler, Table of n, a(n) for n = 1..3150

Cf. A005109, A077497, A077498, A077500, A003592, A077313, A019434.

K:=10^7;; # to get all terms <= K.
A:=Filtered(Filtered([1..K],i-> i mod 6=5),IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,5]  then Add(C,Position(B,i)); fi; od;
A077497:=Concatenation([2,3],List(C,i->A[i])); # Muniru A Asiru, Sep 03 2017

Do[p=Prime[k];s=FindInstance[p x == p 10^p+x,x,Integers];If[s!={},Print[p]],{k,10000}] (* Vicente Izquierdo Gomez, Feb 20 2013 *)

list(lim)=my(v=List(),t);for(r=0,log(lim)\log(5),t=5^r;while(t<=lim,if(isprime(t+1),listput(v,t+1)); t<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 29 2013

Corrected and extended by Reinhard Zumkeller, Nov 19 2002

More terms from Ray Chandler, Aug 02 2003

cp's OEIS Frontend

A005105 Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0.

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A056637 a(n) is the least prime of class n-, according to the Erdős-Selfridge classification of primes.

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A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

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A055600 Numbers of form 2^i*3^j + 1 with i, j >= 0.

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A081426 Class 7- primes.

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A081427 Class 8- primes.

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A216148 Primes of the form 2*k^k + 1 = A216147(k).

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