A321671 -id:A321671

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A192110

Monotonic ordering of nonnegative differences 2^i - 3^j, for 40 >= i >= 0, j >= 0.

+10
9

0, 1, 3, 5, 7, 13, 15, 23, 29, 31, 37, 47, 55, 61, 63, 101, 119, 125, 127, 175, 229, 247, 253, 255, 269, 295, 431, 485, 503, 509, 511, 781, 943, 997, 1015, 1021, 1023, 1319, 1631, 1805, 1909, 1967, 2021, 2039, 2045, 2047, 3367, 3853, 4015, 4069, 4087, 4093

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Comments from N. J. A. Sloane, Oct 21 2019: (Start)

Warning: Note the definition assumes i <= 40.

Because of this assumption, it is not true that this is (except for a(1)=0) the complement of A075824 in the odd integers.

However, by definition, it is the complement of A328077.

(End)

All 52 sequences in this set are finite. - Georg Fischer, Nov 16 2021

LINKS

Rok Cestnik, Table of n, a(n) for n = 1..534 [truncated to 2^40-1 by Georg Fischer, Nov 16 2021]

H. Gauchman and I. Rosenholtz (Proposers), R. Martin (Solver), Difference of prime powers, Problem 1404, Math. Mag., 65 (No. 4, 1992), 265; Solution, Math. Mag., 66 (No. 4, 1993), 269.

Math Overflow, 3^n - 2^m = +-41 is not possible. How to prove it?, Several contributors, Jun 29 2010.

EXAMPLE

The differences accrue like this:

1-1

2-1

4-3.....4-1

8-3.....8-1

16-9....16-3....16-1

32-27...32-9....32-3....32-1

64-27...64-9....64-3....64-1

MATHEMATICA

c = 2; d = 3; t[i_, j_] := c^i - d^j;

u = Table[t[i, j], {i, 0, 40}, {j, 0, i*Log[d, c]}];

v = Union[Flatten[u ]]

CROSSREFS

Cf. A075824, A173671, A192111, A328077 (complement).

For primes, see A007643, A007644, A321671.

This is the first of a set of 52 similar sequences:

A192110: 2^i-3^j, A192111: 3^i-2^j, A192112: 2^i-4^j, A192113: 4^i-2^j, A192114: 2^i-5^j, A192115: 5^i-2^j, A192116: 2^i-6^j, A192117: 6^i-2^j,

A192118: 2^i-7^j, A192119: 7^i-2^j, A192120: 2^i-8^j, A192121: 8^i-2^j, A192122: 2^i-9^j, A192123: 9^i-2^j, A192124: 2^i-10^j, A192125: 10^i-2^j,

A192147: 3^i-4^j, A192148: 4^i-3^j, A192149: 3^i-5^j, A192150: 5^i-3^j, A192151: 3^i-6^j, A192152: 6^i-3^j, A192153: 3^i-7^j, A192154: 7^i-3^j,

A192155: 3^i-8^j, A192156: 8^i-3^j, A192157: 3^i-9^j, A192158: 9^i-3^j, A192159: 3^i-10^j, A192160: 10^i-3^j, A192161: 4^i-5^j, A192162: 5^i-4^j,

A192163: 4^i-6^j, A192164: 6^i-4^j, A192165: 4^i-7^j, A192166: 7^i-4^j, A192167: 4^i-8^j, A192168: 8^i-4^j, A192169: 4^i-9^j, A192170: 9^i-4^j,

A192171: 4^i-10^j, A192172: 10^i-4^j, A192193: 5^i-6^j, A192194: 6^i-5^j, A192195: 5^i-7^j, A192196: 7^i-5^j, A192197: 5^i-8^j, A192198: 8^i-5^j,

A192199: 5^i-9^j, A192200: 9^i-5^j, A192201: 5^i-10^j, A192202: 10^i-5^j.

KEYWORD

nonn,fini

AUTHOR

Clark Kimberling, Jun 23 2011

STATUS

approved

A173671

Positive integers that cannot be expressed as 3^m-2^n where m and n are integers.

+10
4

3, 4, 6, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 78, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The complement of this set, i.e., integers of the form 3^m-2^n, is A192111. - M. F. Hasler, Nov 24 2010

LINKS

Table of n, a(n) for n=1..83.

H. Gauchman and I. Rosenholtz (Proposers), R. Martin (Solver), Difference of prime powers, Problem 1404, Math. Mag., 65 (No. 4, 1992), 265; Solution, Math. Mag., 66 (No. 4, 1993), 269.

Math Overflow, 3^n - 2^m = +-41 is not possible. How to prove it?, Several contributors, Jun 29 2010.

CROSSREFS

Cf. A075824, A074981, A014121, A192110, A192111, A227048, A321671.

KEYWORD

nonn

AUTHOR

Max Alekseyev, Nov 24 2010

EXTENSIONS

Deleted unwarranted programs and b-file. - N. J. A. Sloane, Oct 21 2019

STATUS

approved

A363998

Primes of the form |2^i - 3^j|, for i >= 0, j >= 0.

+10
4

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 47, 61, 73, 79, 101, 127, 139, 179, 211, 227, 229, 239, 241, 269, 431, 503, 509, 601, 727, 997, 1021, 1163, 1319, 1931, 2039, 2179, 3299, 3853, 4093, 4513, 6529, 6553, 7949, 8111, 8191, 11491, 14197, 16141, 16381

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..51.

EXAMPLE

As in A014121, numbers of the form |2^i - 3^j|, for i >=0, j>=0 are 0,1,2,3,5,7,8,11,..., in which the primes are 2,3,5,7,11,... .

MATHEMATICA

z = 500;

t = Table[Abs[2^i - 3^j], {i, 0, z}, {j, 0, z}];

v = Union[Sort[Flatten[t]]]; (* A014121*)

Intersection[v, Prime[Range[200000]]] (* A363998 *)

CROSSREFS

Complement of A007643.

Cf. A000040, A000079, A000244, A014121, A321671, A363999, A364000.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 30 2023

STATUS

approved

A323698

Primes of the form 3^j - 2^k, for j>=0, k>=0.

+10
0

2, 5, 7, 11, 17, 19, 23, 73, 79, 139, 179, 211, 227, 239, 241, 601, 727, 1163, 1931, 2179, 3299, 4513, 6529, 6553, 11491, 19427, 19681, 50857, 58537, 58921, 111611, 144379, 176123, 177019, 177131, 529393, 545747, 1593299, 1594259, 2685817, 4782961, 9492289, 14346859

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

In this sequence, only 5 and 17 make both j and k even numbers.

Generally, the way to prove that a number is not in this sequence is to successively take residues modulo 3, 8, 5, and 16 on both sides of the equation 3^j - 2^k = x.

LINKS

Table of n, a(n) for n=1..43.

FORMULA

Intersection of A000040 and A192111.

EXAMPLE

11 = 3^3 - 2^4, so 11 is a term.

41 == 1 (mod 8), 41 == 2 (mod 3), so j = 2*l, k = 2*m. 41 == 1 (mod 5), but 3^(2*l) - 2^(2*m) mod 5 is 0, 2 or 3. So 41 is not in this sequence.

MATHEMATICA

c = 3; d = 2; t[i_, j_] := c^i - d^j;

u = Table[If[PrimeQ[t[i, j]] == True, u = t[i, j]], {i, 0, 20}, {j, 0, i*Log[d, c]}];

v = Union[Flatten[u]]

PROG

(PARI) forprime(p=1, 1000, k=0; x=3; y=1; while(k<p+1, while(x<y+p, x=3*x); if(x-y==p, print1(p, ", "); k=p); k++; y=2*y))

CROSSREFS

Cf. A000040, A192111.

Cf. A007643 (Primes not of form |3^x - 2^y|).

Cf. A321671 (Primes of the form 2^j - 3^k).

KEYWORD

nonn

AUTHOR

Jinyuan Wang, Jan 24 2019

STATUS

approved

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