A192110 - OEIS

%I #46 Jan 22 2022 23:45:54

%S 0,1,3,5,7,13,15,23,29,31,37,47,55,61,63,101,119,125,127,175,229,247,

%T 253,255,269,295,431,485,503,509,511,781,943,997,1015,1021,1023,1319,

%U 1631,1805,1909,1967,2021,2039,2045,2047,3367,3853,4015,4069,4087,4093

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

%C Comments from _N. J. A. Sloane_, Oct 21 2019: (Start)

%C Warning: Note the definition assumes i <= 40.

%C Because of this assumption, it is not true that this is (except for a(1)=0) the complement of A075824 in the odd integers.

%C However, by definition, it is the complement of A328077.

%C (End)

%C All 52 sequences in this set are finite. - _Georg Fischer_, Nov 16 2021

%H Rok Cestnik, <a href="/A192110/b192110.txt">Table of n, a(n) for n = 1..534</a> [truncated to 2^40-1 by _Georg Fischer_, Nov 16 2021]

%H H. Gauchman and I. Rosenholtz (Proposers), R. Martin (Solver), <a href="https://www.jstor.org/stable/2691457">Difference of prime powers, Problem 1404</a>, Math. Mag., 65 (No. 4, 1992), 265; <a href="https://www.jstor.org/stable/2690747">Solution</a>, Math. Mag., 66 (No. 4, 1993), 269.

%H Math Overflow, <a href="https://mathoverflow.net/questions/29926/3n-2m-pm-41-is-not-possible-how-to-prove-it/29956#29956">3^n - 2^m = +-41 is not possible. How to prove it?</a>, Several contributors, Jun 29 2010.

%e The differences accrue like this:

%e 1-1

%e 2-1

%e 4-3.....4-1

%e 8-3.....8-1

%e 16-9....16-3....16-1

%e 32-27...32-9....32-3....32-1

%e 64-27...64-9....64-3....64-1

%t c = 2; d = 3; t[i_, j_] := c^i - d^j;

%t u = Table[t[i, j], {i, 0, 40}, {j, 0, i*Log[d, c]}];

%t v = Union[Flatten[u ]]

%Y Cf. A075824, A173671, A192111, A328077 (complement).

%Y For primes, see A007643, A007644, A321671.

%Y This is the first of a set of 52 similar sequences:

%Y 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,

%Y 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,

%Y 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,

%Y 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,

%Y 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,

%Y 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,

%Y A192199: 5^i-9^j, A192200: 9^i-5^j, A192201: 5^i-10^j, A192202: 10^i-5^j.

%K nonn,fini

%O 1,3

%A _Clark Kimberling_, Jun 23 2011