A063003 -id:A063003 - cp's OEIS frontend

A063005 Difference between 2^n and the next smaller or equal power of 3.

0, 1, 1, 5, 7, 5, 37, 47, 13, 269, 295, 1319, 1909, 1631, 9823, 13085, 6487, 72023, 84997, 347141, 517135, 502829, 2599981, 3605639, 2428309, 19205525, 24062143, 5077565, 139295293, 149450423, 686321335, 985222181, 808182895, 5103150191, 6719515981, 2978678759

Offset: 0

Jens Voß, Jul 02 2001

Sequence generalized : a(n) = A^n - B^(floor(log_B (A^n))) where A, B are integers. This sequence has A = 2, B = 3; A056577 has A = 3, B = 2. - Ctibor O. Zizka, Mar 03 2008

Alois P. Heinz, Table of n, a(n) for n = 0..3324

Cf. A056577, A063003, A063004.

Cf. A000079 (2^n), A000244 (3^n), A136409.

a:= n-> (t-> t-3^ilog[3](t))(2^n):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 11 2019

a[n_] := 2^n - 3^Floor[Log[3, 2] * n]; Array[a, 36, 0] (* Amiram Eldar, Nov 19 2021 *)

for(n=0,50,print1(2^n-3^floor(log(2^n)/log(3))","))

def a(n):
    m, p, target = 0, 1, 2**n
    while p <= target:  m += 1; p *= 3
    return target - 3**(m-1)
print([a(n) for n in range(36)]) # Michael S. Branicky, Nov 19 2021

a(n) = 2^n - 3^(floor (log_3 (2^n))).

a(n) = A000079(n) - 3^A136409(n). - Michel Marcus, Nov 19 2021

More terms from Ralf Stephan, Mar 20 2003

A056850 Minimal absolute difference of 3^n and 2^k.

Original entry on oeis.org

0, 1, 1, 5, 17, 13, 217, 139, 1631, 3299, 6487, 46075, 7153, 502829, 588665, 2428309, 9492289, 5077565, 118985033, 88519643, 808182895, 1870418611, 2978678759, 25423702091, 7551629537, 252223018333, 342842572777, 1170495537221, 5284606410545, 1738366812781

Offset: 0

Robert G. Wilson v, Aug 30 2000

nonn

Except for 3^0 - 2^0, 3^1 - 2^1 and 3^2 - 2^3, there are no cases where the differences are less than 4.
It is known that a(n) tends to infinity as n tends to infinity. Indeed, Tijdeman showed that there exists an effectively computable constant c > 0 such that |2^x - 3^y| > 2^x/x^c. - Tomohiro Yamada, Sep 29 2017
Empirical observation: For at least values a(0) through a(6308), k-2 < n*log_2(3) < k+2. - Matthew Schuster, Mar 28 2021
For all n >= 0, the lower and upper limits on n*log_2(3) - k are log_2(3/4) = -0.4150374... and log_2(3/2) = 0.5849625..., respectively; i.e., 0 <= n*log_2(3) - k - log_2(3/4) < 1. - Jon E. Schoenfield, Apr 21 2021

			For n = 4, the closest power of 2 to 3^n = 81 is 2^6 = 64, so a(4) = |3^4 - 2^6| = |81 - 64| = 17. - _Jon E. Schoenfield_, Sep 30 2017

Michael De Vlieger, Table of n, a(n) for n = 0..2097
R. Tijdeman, On integers with many small prime factors, Compos. Math. 26 (1973), 319--330.

Cf. A056577 (smallest 3^n-2^k), A063003 (smallest 2^k-3^n).

Table[Min[# - 2^Floor@ Log2@ # &[3^n], 2^Ceiling@ Log2@ # - # &[3^n]], {n, 0, 27}]

a(28)-a(29) from Jon E. Schoenfield, Mar 31 2021

A063004 Difference between 2^n and the next larger or equal power of 3.

Original entry on oeis.org

0, 1, 5, 1, 11, 49, 17, 115, 473, 217, 1163, 139, 2465, 11491, 3299, 26281, 111611, 46075, 269297, 7153, 545747, 2685817, 588665, 5960299, 26269505, 9492289, 62031299, 253202761, 118985033, 625390555, 88519643, 1339300753

Offset: 0

Jens Voß, Jul 02 2001

Cf. A056577, A063003, A063005.

dp23[n_]:=Module[{t=2^n},3^Ceiling[Log[3,t]]-t]; Array[dp23,40,0] (* Harvey P. Dale, Nov 20 2015 *)

for(n=1,50,print1(3^ceil(log(2^n)/log(3))-2^n","))

a(n) = 3^ceiling(log_3(2^n)) - 2^n.

More terms from Ralf Stephan, Mar 21 2003

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A063005 Difference between 2^n and the next smaller or equal power of 3.

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A056850 Minimal absolute difference of 3^n and 2^k.

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A063004 Difference between 2^n and the next larger or equal power of 3.

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