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Search: a192110 -id:a192110
Displaying 1-8 of 8 results found. page 1
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A328077 Complement of A192110. +20
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..66.
Math Overflow, 3^n - 2^m = +-41 is not possible. How to prove it?, Several contributors, Jun 29 2010.
CROSSREFS
Complement of A192110.
Cf. A192111, A173671.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 12 2019
EXTENSIONS
Edited by N. J. A. Sloane, Oct 21 2019
STATUS
approved
A192111 Monotonic ordering of nonnegative differences 3^i-2^j, for i>=0, j>=0. +10
7
0, 1, 2, 5, 7, 8, 11, 17, 19, 23, 25, 26, 49, 65, 73, 77, 79, 80 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Complement of A173671 in the nonnegative integers.
LINKS
Table of n, a(n) for n=1..18.
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. A173671 (complement), A192110, A227048 (partial unions of rows).
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 23 2011
EXTENSIONS
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
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
A192120 Monotonic ordering of nonnegative differences 2^i - 8^j, for 40 >=i >= 0, j >= 0. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..288 [truncated to 2^40-1 by Georg Fischer, Nov 16 2021]
MAPLE
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
MATHEMATICA
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 ]]
CROSSREFS
Cf. A192110, A192121.
KEYWORD
nonn,easy,fini
AUTHOR
Clark Kimberling, Jun 23 2011
STATUS
approved
A192121 Monotonic ordering of nonnegative differences 8^i - 2^j, for 40 >= i >= 0, j >= 0. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..2461 [truncated to 8^40-1 by Georg Fischer, Nov 16 2021]
MAPLE
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
MATHEMATICA
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 ]]
CROSSREFS
Cf. A192110, A192120.
KEYWORD
nonn,easy,fini
AUTHOR
Clark Kimberling, Jun 23 2011
STATUS
approved
A321671 Primes of the form 2^j - 3^k, for j >= 0, k >= 0. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..43.
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.
FORMULA
Intersection of A000040 and A192110.
EXAMPLE
7 = 2^3 - 3^0, so 7 is a term.
PROG
(PARI) forprime(p=1, 1000, k=0; x=2; y=1; while(k<p+1, while(x<y+p, x=2*x); if(x-y==p, print1(p, ", "); k=p); k++; y=3*y))
CROSSREFS
Cf. A000040, A007643, A192110.
Cf. A004051 (primes of the form 2^a + 3^b).
Cf. A063005.
KEYWORD
nonn
AUTHOR
Jinyuan Wang, Nov 16 2018
EXTENSIONS
More terms from Alois P. Heinz, Nov 16 2018
STATUS
approved
A075824 Odd numbers that cannot be expressed as 2^k - 3^m where k and m are integers. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All listed terms can be certified by considering 2^k - 3^m modulo 2552550. [Max Alekseyev, Feb 08 2010]
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, D9.
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
LINKS
Table of n, a(n) for n=1..61.
T. Metsankyla, Catalan's Conjecture : Another old Diophantine problem solved, Bull. Amer. Math. Soc. 41 (2004), 43-57.
Wikipedia, Catalan's conjecture
EXAMPLE
5 doesn't belong to the sequence because it can be expressed as 2^3 - 3^1.
CROSSREFS
Cf. A074981, A192110, A328077.
KEYWORD
nonn
AUTHOR
Felice Russo, Oct 14 2002
EXTENSIONS
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
STATUS
approved
A364001 Primes of the form |2^i - 3^j|, i >= 1, j >= 1. +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..49.
MATHEMATICA
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 *)
CROSSREFS
Cf. A014121, A192110, A192111, A363998.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 09 2023
STATUS
approved
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Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)