A226807 -id:A226807 - cp's OEIS frontend

A226806 Numbers of the form 2^j + 4^k, for j and k >= 0.

2, 3, 5, 6, 8, 9, 12, 17, 18, 20, 24, 32, 33, 36, 48, 65, 66, 68, 72, 80, 96, 128, 129, 132, 144, 192, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 516, 528, 576, 768, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048, 2049, 2052, 2064

Offset: 1

T. D. Noe, Jun 19 2013

nonn

Conjecture: Any integer n > 1 not equal to 4 can be written as a sum of distinct terms of the current sequence with no summand dividing another. - Zhi-Wei Sun, May 01 2023

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A004050 (2^j + 3^k), A226807-A226832 (cases to 8^j + 9^k).

a = 2; b = 4; mx = 3000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

ispow2(n)=n>>valuation(n,2)==1
is(n)=my(h=hammingweight(n)); if(h>2, 0, h==2, valuation(n,2)%2==0 || logint(n,2)%2==0, h==1 && valuation(n,2)%2) \\ Charles R Greathouse IV, Aug 29 2016

A253208 a(n) = 4^n + 3.

Original entry on oeis.org

4, 7, 19, 67, 259, 1027, 4099, 16387, 65539, 262147, 1048579, 4194307, 16777219, 67108867, 268435459, 1073741827, 4294967299, 17179869187, 68719476739, 274877906947, 1099511627779, 4398046511107, 17592186044419, 70368744177667, 281474976710659

Offset: 0

Vincenzo Librandi, Dec 29 2014

Subsequence of A226807.

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A141725, A226807.

Cf. Numbers of the form k^n+k-1: A000057 (k=2), A168607 (k=3), this sequence (k=4), A242329 (k=5), A253209 (k=6), A253210 (k=7), A253211 (k=8), A253212 (k=9), A253213 (k=10).

Magma
```
[4^n+3: n in [0..30]];
```

Table[4^n + 3, {n, 0, 30}] (* or *) CoefficientList[Series[(4 - 13 x) / ((1 - x) (1 - 4 x)), {x, 0, 40}], x]

a(n)=4^n+3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (4 - 13*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
From Elmo R. Oliveira, Nov 14 2023: (Start)
a(n) = 4*a(n-1) - 9 with a(0) = 4.
E.g.f.: exp(4*x) + 3*exp(x). (End)

A362743 Positive integers which cannot be written as a sum of distinct numbers of the form 4^a + 5^b (a,b >= 0).

Original entry on oeis.org

1, 3, 4, 10, 12, 18

Offset: 1

Zhi-Wei Sun, May 01 2023

If a(7) exists, it will be greater than 2750.
Conjecture 1: The only terms of the current sequence are 1, 3, 4, 10, 12, 18. Moreover, any positive integer not among 1, 3, 4, 8, 10, 12, 13, 18, 25, 39, 42 can be written as a sum of numbers of the form 4^a + 5^b (a,b>=0) with no one summand dividing another.
Conjecture 2: Let k and m be positive integers greater than one with k*m even. Then, any sufficiently large integer n can be written as a sum of distinct numbers of the form k^a + m^b with a and b nonnegative integers.
Conjecture 3: Let k and m be positive integers greater than one with k*m even. Then, any sufficiently large integer n can be written as a sum of numbers of the form k^a + m^b (a,b >= 0) with no summand dividing another.
Clearly, Conjecture 3 is stronger than Conjecture 2.
See also A362861 for similar conjectures.
a(7) > 50000. - Martin Ehrenstein, May 16 2023

			a(1) = 1 since 4^a + 5^b > 1 for all a,b >= 0.
a(2) = 3 since 2 = 4^0 + 5^0, and 3 cannot be written as a sum of distinct numbers of the form 4^a + 5^b with a,b >= 0.

B. J. Birch, Note on a problem of Erdos, Proc. Cambridge Philos. Soc. 55 (1959), 370-373.
P. Erdos and M. Lewin, d-complete sequences of integers, Math. Comp. 65 (1996), 837--840.

Cf. A226806, A226807, A226808, A226810, A226812, A362861.

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A226806 Numbers of the form 2^j + 4^k, for j and k >= 0.

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A253208 a(n) = 4^n + 3.

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A362743 Positive integers which cannot be written as a sum of distinct numbers of the form 4^a + 5^b (a,b >= 0).

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