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Odd numbers that cannot be expressed as 2^k - 3^m where k and m are integers.
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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
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.
EXAMPLE
5 doesn't belong to the sequence because it can be expressed as 2^3 - 3^1.
Positive integers that cannot be expressed as 3^m-2^n where m and n are integers.
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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
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
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.
Monotonic ordering of nonnegative differences 3^i-2^j, for i>=0, j>=0.
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7
0, 1, 2, 5, 7, 8, 11, 17, 19, 23, 25, 26, 49, 65, 73, 77, 79, 80
COMMENTS
Complement of A173671 in the nonnegative integers.
LINKS
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.
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
Monotonic ordering of nonnegative differences 2^i - 8^j, for 40 >=i >= 0, j >= 0.
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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
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 ]]
Monotonic ordering of nonnegative differences 8^i - 2^j, for 40 >= i >= 0, j >= 0.
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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
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 ]]
Primes of the form 2^j - 3^k, for j >= 0, k >= 0.
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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
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
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.
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. A004051 (primes of the form 2^a + 3^b).
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
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.
Primes of the form |2^i - 3^j|, i >= 1, j >= 1.
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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
MATHEMATICA
z = 500;
t = Table[Abs[2^i - 3^j], {i, 1, z}, {j, 1, z}];
u = Sort[Flatten[t]];
Intersection[v, Prime[Range[200000]]] (* this sequence *)
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