cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 23 2011

Keywords

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

Examples

			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

Links

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.

Programs

  • 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 ]]

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

Views

Author

Jinyuan Wang, Nov 16 2018

Keywords

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.

Examples

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

Links

Crossrefs

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

Programs

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

Formula

Intersection of A000040 and A192110.

Extensions

More terms from Alois P. Heinz, Nov 16 2018

A363998 Primes of the form |2^i - 3^j|, for i >= 0, j >= 0.

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Jul 30 2023

Keywords

Examples

			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,... .

Crossrefs

Complement of A007643.
Cf. A000040, A000079, A000244, A014121, A321671, A363999, A364000.

Programs

  • 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 *)

A007644 Primes not of form | 3^a +- 2^b | where a, b are nonnegative integers.

Original entry on oeis.org

53, 71, 103, 107, 109, 149, 151, 157, 163, 167, 173, 181, 191, 193, 197, 199, 223, 233, 263, 271, 277, 281, 293, 311, 313, 317, 331, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

Comments

Contains all primes congruent to 53, 77 or 103 modulo 120. - Jianing Song, Feb 15 2020

Examples

			127 = 2^7 - 3^0, so 127 is not a member.
257 = 3^0 + 2^8, so 257 is not a member.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A007643, A004051, A103146.

Extensions

Minor edits by N. J. A. Sloane, Jun 27 2010
More terms from Jon E. Schoenfield, Jun 30 2010

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

Views

Author

Jinyuan Wang, Jan 24 2019

Keywords

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.

Examples

			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.

Crossrefs

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

Programs

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

Formula

Intersection of A000040 and A192111.
Showing 1-5 of 5 results.

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