A004050 -id:A004050 - cp's OEIS frontend

A219835 Number of terms of 2^j + 3^k <= 10^n.

7, 29, 64, 118, 181, 254, 354, 453, 565, 708, 878, 1033, 1224, 1403, 1594, 1828, 2046, 2274, 2553, 2808, 3139, 3467, 3765, 4073, 4443, 4779, 5124, 5537, 5911, 6294, 6690, 7266, 7693, 8129, 8650, 9114, 9588, 10153, 10654, 11167, 11776, 12449, 13005, 13662, 14243

Offset: 1

Zak Seidov, Nov 29 2012

nonn

As n-> infinity, a(n) -> log_2(n)*log_3(n).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A004050 (numbers of the form 2^j + 3^k).

Join[{7, 29}, Table[m = 10^x; -4 + Floor [ Log[3, m ]] + Sum[Floor @ Log[2, m - 3^i], {i, 0, Log[3, m]}], {x, 3, 100}]]

def a(n):
    s, pow3, lim = set(), 1, 10**n
    while pow3 < lim:
        for j in range((lim-pow3).bit_length()):
            s.add(2**j + pow3)
        pow3 *= 3
    return len(s)
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Jul 29 2021

A226808 Numbers of the form 2^j + 5^k, for j and k >= 0.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 13, 17, 21, 26, 27, 29, 33, 37, 41, 57, 65, 69, 89, 126, 127, 129, 133, 141, 153, 157, 189, 253, 257, 261, 281, 381, 513, 517, 537, 626, 627, 629, 633, 637, 641, 657, 689, 753, 881, 1025, 1029, 1049, 1137, 1149, 1649, 2049, 2053, 2073, 2173

Offset: 1

T. D. Noe, Jun 19 2013

nonn

Conjecture: Each integer n > 4 can be written as a_1 + ... + a_k, where a_1,...,a_k are numbers of the form 2^a + 5^b (a,b>=0) (i.e., terms of the current sequence) with no one dividing another. This has been verified for n = 5..1200. - Zhi-Wei Sun, Apr 14 2023

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 = 5; mx = 3000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226812 Numbers of the form 3^j + 6^k, for j and k >= 0.

Original entry on oeis.org

2, 4, 7, 9, 10, 15, 28, 33, 37, 39, 45, 63, 82, 87, 117, 217, 219, 225, 243, 244, 249, 279, 297, 459, 730, 735, 765, 945, 1297, 1299, 1305, 1323, 1377, 1539, 2025, 2188, 2193, 2223, 2403, 3483, 6562, 6567, 6597, 6777, 7777, 7779, 7785, 7803, 7857, 8019, 8505

Offset: 1

T. D. Noe, Jun 19 2013

nonn

Conjecture: Any positive integer not among 1, 3, 5, 6, 8, 12, 27 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), A226806-A226832 (cases to 8^j + 9^k).

a = 3; b = 6; mx = 9000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226814 Numbers of the form 5^j + 6^k, for j and k >= 0.

Original entry on oeis.org

2, 6, 7, 11, 26, 31, 37, 41, 61, 126, 131, 161, 217, 221, 241, 341, 626, 631, 661, 841, 1297, 1301, 1321, 1421, 1921, 3126, 3131, 3161, 3341, 4421, 7777, 7781, 7801, 7901, 8401, 10901, 15626, 15631, 15661, 15841, 16921, 23401, 46657, 46661, 46681, 46781, 47281

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 = 6; mx = 50000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226818 Numbers of the form 5^j + 7^k, for j and k >= 0.

Original entry on oeis.org

2, 6, 8, 12, 26, 32, 50, 54, 74, 126, 132, 174, 344, 348, 368, 468, 626, 632, 674, 968, 2402, 2406, 2426, 2526, 3026, 3126, 3132, 3174, 3468, 5526, 15626, 15632, 15674, 15968, 16808, 16812, 16832, 16932, 17432, 18026, 19932, 32432, 78126, 78132, 78174, 78468

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).

Cf. A226792 ((5^j + 7^k)/2).

a = 5; b = 7; mx = 80000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226819 Numbers of the form 6^j + 7^k, for j and k >= 0.

Original entry on oeis.org

2, 7, 8, 13, 37, 43, 50, 55, 85, 217, 223, 265, 344, 349, 379, 559, 1297, 1303, 1345, 1639, 2402, 2407, 2437, 2617, 3697, 7777, 7783, 7825, 8119, 10177, 16808, 16813, 16843, 17023, 18103, 24583, 46657, 46663, 46705, 46999, 49057, 63463, 117650, 117655, 117685

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 = 7; mx = 120000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226825 Numbers of the form 7^j + 8^k, for j and k >= 0.

Original entry on oeis.org

2, 8, 9, 15, 50, 57, 65, 71, 113, 344, 351, 407, 513, 519, 561, 855, 2402, 2409, 2465, 2913, 4097, 4103, 4145, 4439, 6497, 16808, 16815, 16871, 17319, 20903, 32769, 32775, 32817, 33111, 35169, 49575, 117650, 117657, 117713, 118161, 121745, 150417, 262145

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 = 7; b = 8; mx = 300000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226827 Numbers of the form 3^j + 9^k, for j and k >= 0.

Original entry on oeis.org

2, 4, 10, 12, 18, 28, 36, 82, 84, 90, 108, 162, 244, 252, 324, 730, 732, 738, 756, 810, 972, 1458, 2188, 2196, 2268, 2916, 6562, 6564, 6570, 6588, 6642, 6804, 7290, 8748, 13122, 19684, 19692, 19764, 20412, 26244, 59050, 59052, 59058, 59076, 59130, 59292, 59778

Offset: 1

T. D. Noe, Jun 19 2013

nonn

If every number 3^j + 9^k is considered, then there are duplicates of 10, 82, 90, 730, 738, 810, 6562, 6570, 6642, 7290, 59050, 59058, 59130, 59778, 65610,....

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).

Cf. A226793 ((3^j + 9^k)/2).

a = 3; b = 9; mx = 60000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A226829 Numbers of the form 5^j + 9^k, for j and k >= 0.

Original entry on oeis.org

2, 6, 10, 14, 26, 34, 82, 86, 106, 126, 134, 206, 626, 634, 706, 730, 734, 754, 854, 1354, 3126, 3134, 3206, 3854, 6562, 6566, 6586, 6686, 7186, 9686, 15626, 15634, 15706, 16354, 22186, 59050, 59054, 59074, 59174, 59674, 62174, 74674, 78126, 78134, 78206

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).

Cf. A226794 ((5^j + 9^k)/2).

a = 5; b = 9; mx = 80000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]

A216442 Numbers of the form 2^i + 3^j + 5^k, where i, j, k >= 0.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 53, 54, 56, 58, 60, 64, 66, 68, 70, 72, 74, 78, 83, 84, 86, 87, 88, 90, 92, 94, 96, 98, 102, 107, 108, 110, 114, 116, 118, 122, 127

Offset: 1

Robert G. Wilson v, Sep 20 2012

nonn

Conjecture: Unlike A004050, which has a limited set of integers expressible in more than one way, this set has no such limit.

Number of terms less than or equal to 10^k, k > 0: 8, 56, 238, 615, 1304, 2169, 3606, 5280, 7196, 10414, ....

T. D. Noe, Table of n, a(n) for n = 1..10000
Diego Marques and Alain Togbé, Fibonacci and Lucas numbers of the form 2^a + 3^b + 5^c, Proc. Japan Acad. Ser. A Math. Sci. 89(3): 47-50 (March 2013).

Cf. A004050.

mx = 140; Union@ Flatten@ Table[2^i + 3^j + 5^k, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx - 2^i]}, {k, 0, Log[5, mx - 2^i - 3^j]}]

There are O(log^3 x) terms of the sequence up to x. - Charles R Greathouse IV, Oct 28 2022

cp's OEIS Frontend

A219835 Number of terms of 2^j + 3^k <= 10^n.

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A226808 Numbers of the form 2^j + 5^k, for j and k >= 0.

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A226812 Numbers of the form 3^j + 6^k, for j and k >= 0.

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A226814 Numbers of the form 5^j + 6^k, for j and k >= 0.

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A226818 Numbers of the form 5^j + 7^k, for j and k >= 0.

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A226819 Numbers of the form 6^j + 7^k, for j and k >= 0.

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A226825 Numbers of the form 7^j + 8^k, for j and k >= 0.

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A226827 Numbers of the form 3^j + 9^k, for j and k >= 0.

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A226829 Numbers of the form 5^j + 9^k, for j and k >= 0.

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A216442 Numbers of the form 2^i + 3^j + 5^k, where i, j, k >= 0.

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