A056577 -id:A056577 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A063003 Difference between 3^n and the next larger or equal power of 2.

Original entry on oeis.org

0, 1, 7, 5, 47, 13, 295, 1909, 1631, 13085, 6487, 84997, 517135, 502829, 3605639, 2428309, 24062143, 5077565, 149450423, 985222181, 808182895, 6719515981, 2978678759, 43295774645, 267326277407, 252223018333, 1856180682775

Offset: 0

Jens Voß, Jul 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..200

Cf. A056577, A063004, A063005.

Table[2^Ceiling@ Log2@ # - # &[3^n], {n, 0, 26}]  (* Michael De Vlieger, Sep 30 2017 *)

{ default(realprecision, 50); t=1/log(2); for (n=0, 200, write("b063003.txt", n, " ", 2^ceil(t*log(3^n)) - 3^n) ) } \\ Harry J. Smith, Aug 15 2009

a(n) = 2^(ceiling(log_2(3^n))) - 3^n.

More terms from Marc LeBrun, Jul 11 2001

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

A098232 Largest power of 2 <= 3^n.

Original entry on oeis.org

1, 2, 8, 16, 64, 128, 512, 2048, 4096, 16384, 32768, 131072, 524288, 1048576, 4194304, 8388608, 33554432, 67108864, 268435456, 1073741824, 2147483648, 8589934592, 17179869184, 68719476736, 274877906944, 549755813888, 2199023255552, 4398046511104, 17592186044416

Offset: 0

Henry Bottomley, Oct 25 2004

			a(4)=64 since 3^4=81 and 64 <= 81 < 128.

Cf. A000079, A000244, A020857, A022921, A056576, A056577.

A098232:=n->2^floor(n*log[2](3)): seq(A098232(n), n=0..30); # Wesley Ivan Hurt, Mar 12 2015

a(n) = 2^logint(3^n, 2); \\ Michel Marcus, Sep 06 2022

a(n) = 2^floor(n*log_2(3)) = A000079(A056576(n)) = A000244(n)-A056577(n).

A196486 Irregular triangle 3^n-2^m.

Original entry on oeis.org

1, 7, 5, 1, 25, 23, 19, 11, 79, 77, 73, 65, 49, 17, 241, 239, 235, 227, 211, 179, 115, 727, 725, 721, 713, 697, 665, 601, 473, 217, 2185, 2183, 2179, 2171, 2155, 2123, 2059, 1931, 1675, 1163, 139, 6559, 6557, 6553, 6545, 6529, 6497, 6433, 6305, 6049, 5537

Offset: 1

Zak Seidov, Oct 03 2011

Each n-th row consists of A056576(n) terms, the first term is 3^n-2, n-th term is 3^n-2^n=A001047(n), and the last term is A056577(n).

T(n,k) = A227048(n,A056576(n)-k) for k = 1..A056576(n). - Reinhard Zumkeller, Jun 30 2013

			Rows are:
n=1: 1
n=2: 7,5,1
n=3: 25,23,19,11
n=4: 79,77,73,65,49,17
n=5: 241,239,235,227,211,179,115.

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A001047, A056576, A056577.

a196486 n k = a196486_tabf !! (n-1) !! (k-1)
a196486_row n = a196486_tabf !! (n-1)
a196486_tabf = map (tail . reverse) $ tail a227048_tabf
-- Reinhard Zumkeller, Jun 30 2013

Flatten[Table[3^k - 2^m, {k, 10}, {m, Floor[Log[2, 3^k]]}]]

A292410 Difference between (2n+1)^2 and highest power of 2 less than or equal to (2n+1)^2.

Original entry on oeis.org

0, 1, 9, 17, 17, 57, 41, 97, 33, 105, 185, 17, 113, 217, 329, 449, 65, 201, 345, 497, 657, 825, 1001, 161, 353, 553, 761, 977, 1201, 1433, 1673, 1921, 129, 393, 665, 945, 1233, 1529, 1833, 2145, 2465, 2793, 3129, 3473, 3825, 89, 457, 833, 1217, 1609, 2009, 2417, 2833, 3257

Offset: 0

Zhandos Mambetaliyev, Sep 15 2017

			a(0) = 1^2 - 2^0 =  0.
a(1) = 3^2 - 2^3 =  1.
a(2) = 5^2 - 2^4 =  9.
a(3) = 7^2 - 2^5 = 17.
a(4) = 9^2 - 2^6 = 17.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000079 (2^n), A016754 (odd squares), A053645 (distance to power of 2), A056577.

seq((2*n+1)^2 - 2^ilog2((2*n+1)^2), n=0..100); @ Robert Israel, Oct 19 2017

Table[# - 2^Floor@ Log2@ # &[(2 n + 1)^2], {n, 0, 53}] (* Michael De Vlieger, Sep 18 2017 *)

a(n) = my(k = 0); while(2^k < (2*n+1)^2, k++); if (k, k--); (2*n+1)^2 - 2^k; \\ Michel Marcus, Sep 16 2017

a(n) = A053645(A016754(n)). - Michel Marcus, Sep 16 2017

More terms from Michel Marcus, Sep 16 2017

A292425 The smallest positive number of the form 3^n-2^a-2^b.

Original entry on oeis.org

0, 0, 3, 1, 51, 89, 11, 417, 1251, 9897, 13307, 3057, 21459, 64377, 1765995, 1103681, 28476867, 51876169, 21410779, 265558929, 796676787, 5611255833, 8243832907, 3256662241, 22654888611, 67964665833, 1028527718331, 886559899441, 15853819231635, 29969271650489

Offset: 1

Tomohiro Yamada, Sep 29 2017

nonn

It follows from a work of Vojta that a(n) tends to infinity as n tends to infinity.

P. Vojta, Integral points on varieties. Thesis, Harvard, 1983.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Mo Deze and R. Tijdeman, Exponential diophantine equations with four terms, Indag. Math. N.S. 3 (1992), 47--57.
R. Tijdeman and Lian Xiang Wang, Sums of products of powers of given prime numbers, Pacific J. Math. 132, (1988), 177--193.
Tomohiro Yamada, On the diophantine equation x^m=y^n1+y^n2+...+y^nk, Glasgow Math. J. 51 (2009), 143--148.

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

f(x)=x-2^floor(log(x)/log(2)); g(x)=f(f(x)); a(n)=g(3^n)

a(n)={my(t=3^n); t-=1<Andrew Howroyd, Dec 23 2019

Terms a(15) and beyond from Andrew Howroyd, Dec 23 2019

cp's OEIS Frontend

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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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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A063003 Difference between 3^n and the next larger or equal power of 2.

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

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A098232 Largest power of 2 <= 3^n.

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Maple

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A196486 Irregular triangle 3^n-2^m.

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A292410 Difference between (2n+1)^2 and highest power of 2 less than or equal to (2n+1)^2.