A069353 -id:A069353 - cp's OEIS frontend

A376699 Positions of primes in the sequence of numbers of the form 2^i * 3^j - 1 (A069353).

3, 4, 5, 6, 8, 10, 11, 13, 15, 16, 18, 21, 22, 25, 31, 32, 36, 39, 40, 42, 51, 57, 61, 63, 65, 66, 71, 73, 79, 82, 94, 97, 106, 107, 110, 120, 121, 127, 128, 129, 130, 138, 142, 144, 161, 192, 204, 205, 212, 216, 232, 234, 244, 259, 264, 265, 308, 329, 346, 348

Offset: 1

Amiram Eldar, Oct 02 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586, A005105, A069353, A375906, A376700, A376701.

With[{lim = 10^10}, Position[Sort@ Flatten@ Table[2^i*3^j - 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], _?PrimeQ] // Flatten]

lista(lim) = {my(s = List()); for(i = 0, logint(lim, 2), for(j = 0, logint(lim >> i, 3), listput(s, 2^i * 3^j - 1))); s = Set(s); for(i = 1, #s, if(isprime(s[i]), print1(i, ", ")));}

from itertools import count, islice
from sympy import isprime, integer_log
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 A376699_gen(): # generator of terms
    return filter(lambda n:isprime(A069353(n)), count(1))
A376699_list = list(islice(A376699_gen(),30)) # Chai Wah Wu, Mar 31 2025

A069353(a(n)) = A003586(a(n)) - 1 = A005105(n).

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

Original entry on oeis.org

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

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 14, 15, 17, 23, 26, 27, 29, 33, 41, 44, 45, 50, 51, 53, 57, 65, 68, 77, 80, 81, 86, 87, 89, 98, 99, 101, 105, 113, 122, 129, 131, 134, 135, 149, 152, 153, 158, 159, 161, 170, 171, 173, 177, 194, 195, 197, 201, 203, 209, 225, 230, 239, 242

Offset: 1

Clark Kimberling, May 25 2011

nonn

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then hx+i and jx+k are in a, where h,i,j,k are integers." If m>1, at least one of the numbers b(m)=(a(m)-i)/h and c(m)=(a(m)-k)/j is in the set N of natural numbers. Let d(n) be the n-th b(m) in N, and let e(n) be the n-th c(m) in N. Note that a is a subsequence of both d and e.
Examples, where [A......] indicates a conjecture:
A190803: (h,i,j,k)=(2,-1,3,-1); d=A190841, e=A190842
A190804: (h,i,j,k)=(2,-1,3,0); d=[A190803], e=A190844
A190805: (h,i,j,k)=(2,-1,3,1); d=A190845, e=[A190808]
A190806: (h,i,j,k)=(2,-1,3,2); d=[A190804], e=A190848
...
A190807: (h,i,j,k)=(2,0,3,-1); d=A190849, e=A190850
A003586: (h,i,j,k)=(2,0,3,0); d=e=A003586
A190808: (h,i,j,k)=(2,0,3,1); d=A190851, e=A190852
A190809: (h,i,j,k)=(2,0,3,2); d=A190853, e=A190854
...
A190810: (h,i,j,k)=(2,1,3,-1); d=A190855, e=A190856
A190811: (h,i,j,k)=(2,1,3,0); d=A002977, e=A190857
A002977: (h,i,j,k)=(2,1,3,1); d=A190858, e=A190859
A190812: (h,i,j,k)=(2,1,3,2); d=A069353, e=[A190812]
...
For h=j=3, see A191106; for h=3 and j=4, see A191113.

			1 -> 2 -> 3,5 -> 8,9,14 -> 15,17,23,26,27,41 -> ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.

Cf. A002977, A003586, A190804-A190812, A190841-A190860.

import Data.Set (singleton, deleteFindMin, insert)
a190803 n = a190803_list !! (n-1)
a190803_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)
b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];
d = Intersection[b, r] (* A190841 *)
e = Intersection[c, r] (* A190842 *)
(* Regarding this program - useful for many choices of h,i,j,k,f,g - the depth g must be chosen with care - that is, large enough.  Assuming that h<=j, the least new terms in successive nests a are typically iterates of hx+i, starting from x=1.  If, for example, h=2 and i=0, the least terms are 2,4,8,...,2^g, so that g>=9 ensures inclusion of all the desired terms <=256. *)

a(34)=225 inserted by Reinhard Zumkeller, Jun 01 2011

A069355 Numbers of form 2^i*3^j - (i+j) with i, j >= 0.

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 12, 15, 20, 24, 27, 32, 43, 50, 58, 67, 77, 90, 103, 121, 138, 157, 185, 210, 238, 248, 281, 318, 376, 425, 480, 503, 568, 641, 723, 759, 856, 965, 1014, 1143, 1288, 1451, 1526, 1719, 1936, 2037, 2180, 2294, 2583, 2908, 3061, 3446, 3879, 4084

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Distinct values of A003586(m)-A069352(m) or of A069345(A003586(m)). - Michel Marcus, Apr 09 2018

			1 is a term because 2^0*3^0 - (0+0) = 2^1*3^0 - (1+0) = 1.
2 is a term because 2^2*3^0 - (2+0) = 2^0*3^1 - (0+1) = 2.
4 is a term because 2^1*3^1 - (1+1) = 4.

Cf. A003586, A069345, A069352, A069353, A069356, A069357, A022328, A022329.

With[{nn=20},Take[Flatten[Table[2^i 3^j-i-j,{i,0,nn},{j,0,nn}]]//Union,60]] (* Harvey P. Dale, Aug 29 2022 *)

Duplicated term 2 and incorrect formula removed by Altug Alkan, Apr 09 2018

A190812 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x+1 and 3x+2 are in a.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 23, 31, 35, 47, 53, 63, 71, 95, 107, 127, 143, 161, 191, 215, 255, 287, 323, 383, 431, 485, 511, 575, 647, 767, 863, 971, 1023, 1151, 1295, 1457, 1535, 1727, 1943, 2047, 2303, 2591, 2915, 3071, 3455, 3887, 4095, 4373, 4607, 5183, 5831, 6143, 6911, 7775, 8191, 8747, 9215, 10367, 11663

Offset: 1

Clark Kimberling, May 20 2011

nonn

See A190803.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A190803, A069353, A190812.

import Data.Set (singleton, deleteFindMin, insert)
a190812 n = a190812_list !! (n-1)
a190812_list = f $ singleton 1
   where f s = m : (f $ insert (2*m+1) $ insert (3*m+2) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 2; i = 1; j = 3; k = 2; f = 1; g = 20 ;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190812 *)
b = (a - 1)/2; c = (a - 2)/3; r = Range[1, 30000];
d = Intersection[b, r] (* A069353 *)
e = Intersection[c, r] (* A190812 conjectured *)

a(41)=2047 inserted by Reinhard Zumkeller, Jun 01 2011

A264164 3-smooth numbers whose number of divisors is not 3-smooth.

Original entry on oeis.org

16, 48, 64, 81, 144, 162, 192, 324, 432, 512, 576, 648, 729, 1024, 1296, 1458, 1536, 1728, 2592, 2916, 3072, 3888, 4096, 4608, 5184, 5832, 8192, 9216, 10368, 11664, 12288, 13824, 15552, 16384, 19683, 20736, 23328, 24576, 27648, 34992, 36864, 39366, 41472

Offset: 1

Reinhard Zumkeller, Nov 19 2015

nonn

			a(12) = 648 = 2^3*3^4 = A003586(36) and A000005(648) = 20 = 2^2*5.
a(13) = 729 = 3^6 = A003586(37) and A000005(729) = 7.
A003586(38) = 768 = 2^8*3 is not a term, as A000005(768) = 18 = 2*3^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A003586, A065333, A069353, A264165 (complement with respect to A003586).

a264164 n = a264164_list !! (n-1)
a264164_list = filter ((== 0) . a065333 . a000005') a003586_list

smQ[n_] := n == Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]);
seq[max_] := Sort@ Flatten@ Table[If[smQ[i + 1] && smQ[j + 1], Nothing, 2^i * 3^j], {i, 0 , Log2[max]}, {j, 0, Log[3, max/2^i]}]; seq[42000] (* Amiram Eldar, Sep 03 2023 *)

A065333(a(n)) * (1 - A065333(A000005(a(n)))) = 1.

Sum_{n>=1} 1/a(n) = 3 - (Sum_{n>=1} 1/2^A069353(n)) * (Sum_{n>=1} 1/3^A069353(n)) = 0.14870906483739045152... . - Amiram Eldar, Apr 17 2025

A264165 3-smooth numbers whose number of divisors is 3-smooth.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 32, 36, 54, 72, 96, 108, 128, 216, 243, 256, 288, 384, 486, 768, 864, 972, 1152, 1944, 2048, 2187, 2304, 3456, 4374, 6144, 6561, 6912, 7776, 8748, 13122, 17496, 18432, 26244, 31104, 32768, 52488, 55296, 62208, 69984

Offset: 1

Reinhard Zumkeller, Nov 19 2015

nonn

			a(25) = 768 = 2^8*3 = A003586(38) and A000005(768) = 18 = 2*3^2;
a(26) = 864 = 2^5*3^3 = A003586(39) and A000005(864) = 24 = 2^3*3;
a(27) = 972 = 2^2*3^5 = A003586(40) and A000005(972) = 18 = 2*3^2;
but A003586(41) = 1024 = 2^10 is not a term, as A000005(1024) = 11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..2000

Cf. A000005, A003586, A065333, A069353, A264164 (complement with respect to A003586).

a264165 n = a264165_list !! (n-1)
a264165_list = filter ((== 1) . a065333 . a000005') a003586_list

smQ[n_] := n == Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]);
seq[max_] := Sort@ Flatten@ Table[2^i * 3^j, {i, Select[Range[0, Floor[Log2[max]]], smQ[# + 1] &]}, {j, Select[Range[0, Floor[Log[3, max/2^i]]], smQ[# + 1] &]}]; seq[70000] (* Amiram Eldar, Sep 03 2023 *)

A065333(a(n)) * A065333(A000005(a(n))) = 1.

Sum_{n>=1} 1/a(n) = (Sum_{n>=1} 1/2^A069353(n)) * (Sum_{n>=1} 1/3^A069353(n)) = 2.85129093516260954847... . - Amiram Eldar, Apr 17 2025

A327315 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(x^2-x+1)).

Original entry on oeis.org

-2, 1, -1, 4, -1, 1, 3, -6, 1, 2, -4, -6, 8, -1, 1, -10, 10, 10, -10, 1, -1, -6, 30, -20, -15, 12, -1, -2, 7, 21, -70, 35, 21, -14, 1, -1, 16, -28, -56, 140, -56, -28, 16, -1, 1, 9, -72, 84, 126, -252, 84, 36, -18, 1, 2, -10, -45, 240, -210, -252, 420, -120

Offset: 0

Clark Kimberling, Nov 01 2019

Conjecture: The numbers n for which the n-th polynomial is irreducible are given by A069353.

			First eight rows:
  -2,   1;
  -1,   4,  -1;
   1,   3,  -6,   1;
   2,  -4,  -6,   8,  -1;
   1, -10,  10,  10, -10,   1;
  -1,  -6,  30, -20, -15,  12,  -1;
  -2,   7,  21, -70,  35,  21, -14,  1;
  -1,  16, -28, -56, 140, -56, -28, 16, -1;
First eight polynomials:
  -2 + x
  -1 + 4 x - x^2
   1 + 3 x - 6 x^2 + x^3
   2 - 4 x - 6 x^2 + 8 x^3 - x^4
   (1 + x) (1 - 11 x + 21 x^2 - 11 x^3 + x^4)
  -1 - 6 x + 30 x^2 - 20 x^3 - 15 x^4 + 12 x^5 - x^6
   (-2 + x) (1 - 3 x - 12 x^2 + 29 x^3 - 3 x^4 - 12 x^5 + x^6)
  -1 + 16 x - 28 x^2 - 56 x^3 + 140 x^4 - 56 x^5 - 28 x^6 + 16 x^7 - x^8

Cf. A069353, A328646.

g[x_, n_] := Numerator[ Factor[D[(x - 2)/(x^2 - x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* A327315 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}]  (* A327315 sequence *)
Column[%]   (* A327315 array *)

cp's OEIS Frontend

A376699 Positions of primes in the sequence of numbers of the form 2^i * 3^j - 1 (A069353).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A069355 Numbers of form 2^i*3^j - (i+j) with i, j >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A190812 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x+1 and 3x+2 are in a.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A264164 3-smooth numbers whose number of divisors is not 3-smooth.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A264165 3-smooth numbers whose number of divisors is 3-smooth.

Views

Author

Keywords

Examples

Links