A226806 -id:A226806 - cp's OEIS frontend

A226821 Numbers of the form 3^j + 8^k, for j and k >= 0.

2, 4, 9, 10, 11, 17, 28, 35, 65, 67, 73, 82, 89, 91, 145, 244, 251, 307, 513, 515, 521, 539, 593, 730, 737, 755, 793, 1241, 2188, 2195, 2251, 2699, 4097, 4099, 4105, 4123, 4177, 4339, 4825, 6283, 6562, 6569, 6625, 7073, 10657, 19684, 19691, 19747, 20195, 23779

Offset: 1

T. D. Noe, Jun 19 2013

nonn

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

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).

a = 3; b = 8; mx = 25000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226822 Numbers of the form 4^j + 8^k, for j and k >= 0.

Original entry on oeis.org

2, 5, 9, 12, 17, 24, 65, 68, 72, 80, 128, 257, 264, 320, 513, 516, 528, 576, 768, 1025, 1032, 1088, 1536, 4097, 4100, 4104, 4112, 4160, 4352, 4608, 5120, 8192, 16385, 16392, 16448, 16896, 20480, 32769, 32772, 32784, 32832, 33024, 33792, 36864, 49152, 65537

Offset: 1

T. D. Noe, Jun 19 2013

nonn

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

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).

a = 4; b = 8; mx = 70000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]
With[{upto=66000},Select[Union[4^#[[1]]+8^#[[2]]&/@Tuples[Range[ 0,Log[ 4,upto]],2]],#<=upto&]] (* Harvey P. Dale, Sep 15 2019 *)

A226823 Numbers of the form 5^j + 8^k, for j and k >= 0.

Original entry on oeis.org

2, 6, 9, 13, 26, 33, 65, 69, 89, 126, 133, 189, 513, 517, 537, 626, 633, 637, 689, 1137, 3126, 3133, 3189, 3637, 4097, 4101, 4121, 4221, 4721, 7221, 15626, 15633, 15689, 16137, 19721, 32769, 32773, 32793, 32893, 33393, 35893, 48393, 78126, 78133, 78189, 78637

Offset: 1

T. D. Noe, Jun 19 2013

nonn

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

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).

a = 5; b = 8; mx = 80000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226824 Numbers of the form 6^j + 8^k, for j and k >= 0.

Original entry on oeis.org

2, 7, 9, 14, 37, 44, 65, 70, 100, 217, 224, 280, 513, 518, 548, 728, 1297, 1304, 1360, 1808, 4097, 4102, 4132, 4312, 5392, 7777, 7784, 7840, 8288, 11872, 32769, 32774, 32804, 32984, 34064, 40544, 46657, 46664, 46720, 47168, 50752, 79424, 262145, 262150, 262180

Offset: 1

T. D. Noe, Jun 19 2013

nonn

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

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).

a = 6; b = 8; mx = 300000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226826 Numbers of the form 2^j + 9^k, for j and k >= 0.

Original entry on oeis.org

2, 3, 5, 9, 10, 11, 13, 17, 25, 33, 41, 65, 73, 82, 83, 85, 89, 97, 113, 129, 137, 145, 209, 257, 265, 337, 513, 521, 593, 730, 731, 733, 737, 745, 761, 793, 857, 985, 1025, 1033, 1105, 1241, 1753, 2049, 2057, 2129, 2777, 4097, 4105, 4177, 4825, 6562, 6563

Offset: 1

T. D. Noe, Jun 19 2013

nonn

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

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).

a = 2; b = 9; mx = 7000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226828 Numbers of the form 4^j + 9^k, for j and k >= 0.

Original entry on oeis.org

2, 5, 10, 13, 17, 25, 65, 73, 82, 85, 97, 145, 257, 265, 337, 730, 733, 745, 793, 985, 1025, 1033, 1105, 1753, 4097, 4105, 4177, 4825, 6562, 6565, 6577, 6625, 6817, 7585, 10657, 16385, 16393, 16465, 17113, 22945, 59050, 59053, 59065, 59113, 59305, 60073, 63145

Offset: 1

T. D. Noe, Jun 19 2013

nonn

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

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).

a = 4; b = 9; mx = 70000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226830 Numbers of the form 6^j + 9^k, for j and k >= 0.

Original entry on oeis.org

2, 7, 10, 15, 37, 45, 82, 87, 117, 217, 225, 297, 730, 735, 765, 945, 1297, 1305, 1377, 2025, 6562, 6567, 6597, 6777, 7777, 7785, 7857, 8505, 14337, 46657, 46665, 46737, 47385, 53217, 59050, 59055, 59085, 59265, 60345, 66825, 105705, 279937, 279945, 280017

Offset: 1

T. D. Noe, Jun 19 2013

nonn

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

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).

a = 6; b = 9; mx = 300000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

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.

cp's OEIS Frontend

A226821 Numbers of the form 3^j + 8^k, for j and k >= 0.

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A226822 Numbers of the form 4^j + 8^k, for j and k >= 0.

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A226823 Numbers of the form 5^j + 8^k, for j and k >= 0.

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A226824 Numbers of the form 6^j + 8^k, for j and k >= 0.

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A226826 Numbers of the form 2^j + 9^k, for j and k >= 0.

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A226828 Numbers of the form 4^j + 9^k, for j and k >= 0.

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A226830 Numbers of the form 6^j + 9^k, for j and k >= 0.

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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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