A192111 -id:A192111 - cp's OEIS frontend

A192110 Monotonic ordering of nonnegative differences 2^i - 3^j, for 40 >= i >= 0, j >= 0.

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

Offset: 1

Clark Kimberling, Jun 23 2011

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

			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

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.

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.

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 ]]

A227048 Irregular triangle read by rows: row n, for n >= 0, lists the nonnegative differences 3^n - 2^m, m >= 0, in increasing order.

Original entry on oeis.org

0, 1, 2, 1, 5, 7, 8, 11, 19, 23, 25, 26, 17, 49, 65, 73, 77, 79, 80, 115, 179, 211, 227, 235, 239, 241, 242, 217, 473, 601, 665, 697, 713, 721, 725, 727, 728, 139, 1163, 1675, 1931, 2059, 2123, 2155, 2171, 2179, 2183, 2185, 2186, 2465, 4513, 5537, 6049, 6305

Offset: 0

Reinhard Zumkeller, Jun 29 2013

A020914(n) = length of n-th row;
T(n,1) = A056577(n);
T(n,A098294(n)) = A001047(n);
T(n,A020914(n)) = A024023(n);
T(n,k) = A196486(n,A020914(n)-k) for n > 0, k = 1..A056576(n).

			Initial rows:
0:  0
1:  1,2
2:  1,5,7,8
3:  11,19,23,25,26 (= 27-16, 27-8, 27-4, 27-2, 27-1)
4:  17,49,65,73,77,79,80
5:  115,179,211,227,235,239,241,242
6:  217,473,601,665,697,713,721,725,727,728
7:  139,1163,1675,1931,2059,2123,2155,2171,2179,2183,2185,2186
8:  2465,4513,5537,6049,6305,6433,6497,6529,6545,6553,6557,6559,6560
...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Cf. A000079, A000244, A192111.

See also A001047, A020914, A024023, A056576, A056577, A098294, A196486.

a227048 n k = a227048_tabf !! n !! (k-1)
a227048_row n = a227048_tabf !! n
a227048_tabf = map f a000244_list  where
   f x = reverse $ map (x -) $ takeWhile (<= x) a000079_list

Definition revised by N. J. A. Sloane, Oct 11 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

A328077 Complement of A192110.

Original entry on oeis.org

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

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 *)

A175832 Number of solutions of equation 3^q - 2^p = n over integer variables p, q.

Original entry on oeis.org

1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Vladimir Reshetnikov, Sep 15 2010

nonn

a(n)=0 iff n is in A173671; a(n)>0 iff n is in A192111.

Edited and extended by Max Alekseyev, Jan 28 2012

A323698 Primes of the form 3^j - 2^k, for j>=0, k>=0.

Original entry on oeis.org

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

Offset: 1

Jinyuan Wang, Jan 24 2019

nonn

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.

			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.

Cf. A000040, A192111.
Cf. A007643 (Primes not of form |3^x - 2^y|).
Cf. A321671 (Primes of the form 2^j - 3^k).

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]]

forprime(p=1, 1000, k=0; x=3; y=1; while(k

Intersection of A000040 and A192111.

cp's OEIS Frontend

A192110 Monotonic ordering of nonnegative differences 2^i - 3^j, for 40 >= i >= 0, j >= 0.

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A227048 Irregular triangle read by rows: row n, for n >= 0, lists the nonnegative differences 3^n - 2^m, m >= 0, in increasing order.

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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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A328077 Complement of A192110.

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

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A175832 Number of solutions of equation 3^q - 2^p = n over integer variables p, q.

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A323698 Primes of the form 3^j - 2^k, for j>=0, k>=0.

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