A192110 -id:A192110 - cp's OEIS frontend

A328077 Complement of A192110.

2, 4, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

N. J. A. Sloane, Oct 12 2019

nonn

Note that, because A192110 assumes i <= 40, it is incorrect to say that the present sequence consists of "the positive integers that cannot be expressed as 2^m-3^n where m and n are integers".

This sequence is included because one way to remove the assumption i <= 40 from A192110 (and the fifty other unproved sequences of the same type) would be to show that the complements are correct, using the method used to prove the correctness of A173671.

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

Complement of A192110.

Cf. A192111, A173671.

Edited by N. J. A. Sloane, Oct 21 2019

A192111 Monotonic ordering of nonnegative differences 3^i-2^j, for i>=0, j>=0.

Original entry on oeis.org

0, 1, 2, 5, 7, 8, 11, 17, 19, 23, 25, 26, 49, 65, 73, 77, 79, 80

Offset: 1

Clark Kimberling, Jun 23 2011

nonn

Complement of A173671 in the nonnegative integers.

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.

Cf. A173671 (complement), A192110, A227048 (partial unions of rows).

Deleted unwarranted programs and b-file. Only the terms in A173671 (that is, up to 100) have been proved to be correct. - N. J. A. Sloane, Oct 21 2019

A173671 Positive integers that cannot be expressed as 3^m-2^n where m and n are integers.

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Nov 24 2010

nonn

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

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.

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

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

A192120 Monotonic ordering of nonnegative differences 2^i - 8^j, for 40 >=i >= 0, j >= 0.

Original entry on oeis.org

0, 1, 3, 7, 8, 15, 24, 31, 56, 63, 64, 120, 127, 192, 248, 255, 448, 504, 511, 512, 960, 1016, 1023, 1536, 1984, 2040, 2047, 3584, 4032, 4088, 4095, 4096, 7680, 8128, 8184, 8191, 12288, 15872, 16320, 16376, 16383, 28672, 32256, 32704, 32760, 32767, 32768

Offset: 1

Clark Kimberling, Jun 23 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..288 [truncated to 2^40-1 by _Georg Fischer_, Nov 16 2021]

Cf. A192110, A192121.

A192120:={}: for i from 0 to 15 do for j from 0 to floor(i/3) do A192120 := A192120 union {2^i-8^j}: od: od: op(A192120); # Nathaniel Johnston, Jun 23 2011

c = 2; d = 8; 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 ]]

A192121 Monotonic ordering of nonnegative differences 8^i - 2^j, for 40 >= i >= 0, j >= 0.

Original entry on oeis.org

0, 4, 6, 7, 32, 48, 56, 60, 62, 63, 256, 384, 448, 480, 496, 504, 508, 510, 511, 2048, 3072, 3584, 3840, 3968, 4032, 4064, 4080, 4088, 4092, 4094, 4095, 16384, 24576, 28672, 30720, 31744, 32256, 32512, 32640, 32704, 32736, 32752, 32760, 32764, 32766, 32767

Offset: 1

Clark Kimberling, Jun 23 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..2461 [truncated to 8^40-1 by _Georg Fischer_, Nov 16 2021]

Cf. A192110, A192120.

A192121:={}: for i from 0 to 5 do for j from 0 to 3*i do A192121 := A192121 union {8^i-2^j}: od: od: op(A192121); # Nathaniel Johnston, Jun 23 2011

c = 8; d = 2; 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 ]]

A321671 Primes of the form 2^j - 3^k, for j >= 0, k >= 0.

Original entry on oeis.org

3, 5, 7, 13, 23, 29, 31, 37, 47, 61, 101, 127, 229, 269, 431, 503, 509, 997, 1021, 1319, 2039, 3853, 4093, 7949, 8111, 8191, 14197, 16141, 16381, 32687, 45853, 65293, 130343, 130829, 131063, 131071, 347141, 502829, 524261, 524287, 1028893, 1046389, 1048549

Offset: 1

Jinyuan Wang, Nov 16 2018

nonn

The numbers in A007643 are not in this sequence.
For n > 1, a(n) is of the form 8k - 1 or 8k - 3.
In this sequence, only 3 and 7 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 2^j - 3^k = x.

			7 = 2^3 - 3^0, so 7 is a term.

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.

Cf. A000040, A007643, A192110.
Cf. A004051 (primes of the form 2^a + 3^b).
Cf. A063005.

forprime(p=1,1000,k=0;x=2;y=1;while(k

Intersection of A000040 and A192110.

More terms from Alois P. Heinz, Nov 16 2018

A075824 Odd numbers that cannot be expressed as 2^k - 3^m where k and m are integers.

Original entry on oeis.org

9, 11, 17, 19, 21, 25, 27, 33, 35, 39, 41, 43, 45, 49, 51, 53, 57, 59, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 103, 105, 107, 109, 111, 113, 115, 117, 121, 123, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Felice Russo, Oct 14 2002

nonn

All listed terms can be certified by considering 2^k - 3^m modulo 2552550. [Max Alekseyev, Feb 08 2010]

			5 doesn't belong to the sequence because it can be expressed as 2^3 - 3^1.

R. K. Guy, Unsolved Problems in Number Theory, D9.
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

T. Metsankyla, Catalan's Conjecture : Another old Diophantine problem solved, Bull. Amer. Math. Soc. 41 (2004), 43-57.
Wikipedia, Catalan's conjecture

Cf. A074981, A192110, A328077.

Inserted "odd" in definition. - N. J. A. Sloane, Jan 30 2009
Jon E. Schoenfield observed that 49 was missing, Jan 30 2009
More terms from Max Alekseyev, Feb 08 2010

A364001 Primes of the form |2^i - 3^j|, i >= 1, j >= 1.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 29, 37, 47, 61, 73, 79, 101, 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, 11491, 14197, 16141, 16381, 19427, 19681, 32687

Offset: 1

Clark Kimberling, Aug 09 2023

nonn

Cf. A014121, A192110, A192111, A363998.

z = 500;
t = Table[Abs[2^i - 3^j], {i, 1, z}, {j, 1, z}];
u = Sort[Flatten[t]];
v = Union[u] ; (* A363999 *)
w = (v - 1)/2 ;  (* A364000 *)
Intersection[v, Prime[Range[200000]]]  (* this sequence *)

cp's OEIS Frontend

A328077 Complement of A192110.

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A192111 Monotonic ordering of nonnegative differences 3^i-2^j, for i>=0, j>=0.

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A173671 Positive integers that cannot be expressed as 3^m-2^n where m and n are integers.

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A192120 Monotonic ordering of nonnegative differences 2^i - 8^j, for 40 >=i >= 0, j >= 0.

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A192121 Monotonic ordering of nonnegative differences 8^i - 2^j, for 40 >= i >= 0, j >= 0.

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Maple

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A321671 Primes of the form 2^j - 3^k, for j >= 0, k >= 0.

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A075824 Odd numbers that cannot be expressed as 2^k - 3^m where k and m are integers.

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A364001 Primes of the form |2^i - 3^j|, i >= 1, j >= 1.

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Mathematica