Alberto Zanoni - cp's OEIS frontend

A386966 Numbers that can be written in exactly two different ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

545, 1407, 1492, 2409, 3370, 3605, 3718, 4516, 4523, 4684, 5441, 6348, 7346, 7737, 7865, 7922, 8122, 8538, 9046, 10010, 10037, 10298, 10458, 10554, 10651, 10891, 10953, 11047, 11653, 11853, 11986, 12025, 12449, 13621, 14078, 14098, 14535, 14970, 16138, 16449, 16705, 16905, 17401, 18149, 18161, 18509

Offset: 1

Alberto Zanoni, Aug 12 2025

			 545 = 2^6  + 3^2 + 4^4 + 6^3  = 2^7 + 3^5 + 5^3 + 7^2.
1407 = 2^10 + 3^3 + 4^4 + 10^2 = 2^7 + 3^4 + 4^5 + 5^3 + 7^2.
1492 = 2^10 + 3^5 + 5^3 + 10^2 = 2^7 + 3^6 + 4^4 + 6^2 + 7^3.

Subsequence of A385969.

Cf. A385232, A385233, A385970, A386967, A386968.

A386967 Numbers that can be written in exactly three different ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

Original entry on oeis.org

23506, 23778, 41682, 50261, 53554, 76754, 78289, 92030, 96981, 99913, 101559, 105885, 109094, 114097, 117538, 125943, 132867, 133116, 135697, 143154, 150041, 158539, 160161, 197547, 198333, 204359, 225138, 225530, 265685, 269986, 277243, 280063, 286299, 291016, 295391, 306251, 312341, 313323

Offset: 1

Alberto Zanoni, Aug 12 2025

			23506 = 2^9 + 4^7 + 7^2 + 9^4
      = 2^3 + 3^4 + 4^2 + 5^6 + 6^5
      = 2^4 + 3^8 + 4^5 + 5^6 + 6^3 + 8^2.
23778 = 2^11 + 3^8 + 4^2 + 8^3 + 11^4
      = 2^12 + 3^2 + 4^5 + 5^6 + 6^4 + 12^3
      = 2^10 + 3^8 + 4^6 + 5^3 + 6^5 + 8^4 + 10^2.
41682 = 2^3 + 3^9 + 4^7 + 5^5 + 7^4 + 9^2
      = 2^9 + 3^8 + 4^7 + 5^4 + 7^5 + 8^2 + 9^3
      = 2^14 + 3^8 + 4^6 + 5^2 + 6^5 + 8^4 + 14^3.

Subsequence of A385969.

Cf. A385232, A385233, A385970, A386966, A386968.

A386968 Numbers that can be written in exactly four ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

Original entry on oeis.org

331979, 536134, 602342, 848707, 1007017, 1360430, 1484182, 1767157, 1891086, 2024074, 2036922, 2095031, 2097159, 2231826, 2257754, 2292303, 2293830, 2320578, 2440812, 2503676, 2590739, 2744591, 2852016, 2890344, 2914526, 2944901, 2951290, 3019920, 3020295, 3053910, 3121946, 3157114, 3310022

Offset: 1

Alberto Zanoni, Aug 12 2025

nonn

			331979 = 2^15 + 4^5  + 5^6 + 6^7 + 7^4 + 15^2
       = 2^2 + 3^8  + 4^9 + 5^3 + 8^4 + 9^5
       = 2^5  + 3^10 + 4^9 + 5^2 + 9^3 + 10^4
       = 2^6 + 3^10 + 4^9 + 5^5 + 6^2 + 9^4 + 10^3.
536134 = 2^6 + 3^12 + 4^3 + 5^5 + 6^4 + 12^2
       = 2^16 + 3^10 + 4^9 + 5^7 + 6^6 + 7^5  + 9^4
       = 2^16 + 4^8  + 5^5 + 6^7 + 7^6 + 8^4  + 16^2
       = 2^18 + 3^11 + 4^4 + 5^7 + 7^5 + 11^3 + 18^2.
602342 = 2^16 + 3^12 + 4^4 + 5^5 + 12^3 + 16^2
       = 2^8  + 3^12 + 4^3 + 5^5 + 6^6 + 8^2 + 12^4
       = 2^13 + 3^8  + 4^2 + 5^3 + 6^7 + 7^5 + 8^6 + 13^4
       = 2^15 + 3^10 + 4^8 + 5^7 + 6^4 + 7^2 + 8^6 + 10^5 + 15^3.
5260225 is not in the sequence as it can be written in exactly five ways a such. - _David A. Corneth_, Aug 16 2025

David A. Corneth, Table of n, a(n) for n = 1..3353 (terms <= 10^8)

Subsequence of A385969.

Cf. A385232, A385233, A385970, A386966, A386967.

A385969 Numbers that can be written as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

Original entry on oeis.org

4, 17, 27, 31, 32, 57, 59, 84, 89, 100, 105, 127, 145, 149, 166, 177, 202, 204, 245, 254, 256, 260, 273, 276, 283, 287, 289, 313, 320, 322, 340, 347, 348, 356, 368, 372, 377, 383, 400, 422, 433, 460, 465, 468, 480, 532, 545, 548, 568, 576, 593, 603, 620, 624, 628, 673, 688, 700

Offset: 1

Alberto Zanoni, Jul 13 2025

nonn

The value 1 is excluded because for t = 2 one would obtain 1^x_1 + x_1^1 = 1 + x_1, which for x_1 = 2,3,4,... would give the trivial sequence 3,4,5,6,...

			a(1) =  4 : 2^2 = 4 (t = 1)
a(2) = 17 : 2^3 + 3^2 = 8 + 9 = 17 (t = 2)
a(3) = 27 : 3^3 = 27 (t = 1)
a(4) = 31 : 2^2 + 3^3 = 4 + 27 = 31 (t = 2)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A385232, A385233, A385970.

A385970 a(n) is the minimum number of powers needed to obtain the n-th term of sequence A385969.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 4, 3, 4, 4, 1, 2, 3, 3, 2, 3, 3, 3, 2, 4, 4, 3, 4, 3, 2, 3, 4, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 4, 4, 5, 4, 4, 4, 4, 3, 3, 5, 5, 4, 3, 4, 4, 4, 4, 3, 4, 5, 2, 3, 4, 4, 5, 4, 4, 5, 4, 4, 3, 4, 5, 4, 2, 4, 3, 5, 4, 5

Offset: 1

Alberto Zanoni, Jul 14 2025

nonn

The sequence is unbounded (conjectured).

			a(1) = 1, as  4 = 2^2
a(2) = 2, as 17 = 2^3 + 3^2
a(3) = 1, as 27 = 3^3
a(4) = 2, as 31 = 2^2 + 3^3
a(7) = 3, as 59 = 2^4 + 3^3 + 4^2

Cf. A385969, A385232, A385233.

A385233 Numbers that can be written as s^x + t^y + u^z with 1 < s < t < u and {s,t,u} = {x,y,z} (the sequence of exponents can be any permutation of s,t,u).

Original entry on oeis.org

59, 84, 89, 105, 127, 149, 166, 204, 273, 276, 287, 289, 313, 347, 356, 372, 433, 480, 576, 620, 624, 673, 773, 777, 849, 932, 949, 1065, 1151, 1201, 1230, 1250, 1376, 1380, 1653, 1676, 2033, 2089, 2196, 2244, 2425, 2534, 2545, 2786, 3142, 3156, 3157, 3225, 3270, 3302, 3385, 3388, 3408, 3445, 3718, 4070, 4148, 4249

Offset: 1

Alberto Zanoni, Jun 28 2025

nonn

			a(1) = 2^4 + 3^3 + 4^2 = 16 + 27 + 16 =  59.
a(2) = 2^5 + 3^3 + 5^2 = 32 + 27 + 25 =  84.
a(3) = 2^4 + 3^2 + 4^3 = 16 +  9 + 64 =  89.
a(4) = 2^3 + 3^4 + 4^2 =  8 + 81 + 16 = 105.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001597, A385232 (two addends).

A385232 Numbers that can be written as s^x + t^y, with 1 < s < t and {s,t} = {x,y}; that is, are of the form s^s + t^t or s^t + t^s.

Original entry on oeis.org

17, 31, 32, 57, 100, 145, 177, 260, 283, 320, 368, 593, 945, 1124, 1649, 2169, 2530, 3129, 3152, 3381, 4240, 5392, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 46660, 46683, 46912, 49781, 60049, 65792, 69632, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169, 823547, 823570

Offset: 1

Alberto Zanoni, Jun 28 2025

			a(1) = 2^3 + 3^2 =  8 +  9 = 17.
a(2) = 2^2 + 3^3 =  4 + 27 = 31.
a(3) = 2^4 + 4^2 = 16 + 16 = 32.
a(4) = 2^5 + 5^2 = 32 + 25 = 57.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000312, A001597, A385233 (three addends).

Union of A173054 and A385614.

A383122 a(n) is the smallest number that can be expressed as the sum of the smallest number of powers with different exponents greater than one in n different ways (for unitary bases, the smallest possible exponents are considered).

Original entry on oeis.org

1, 16, 17, 65, 80, 105, 139, 193, 329, 313, 336, 410, 477, 273, 553, 461, 436, 1219, 942, 10153, 1595, 1038, 722, 636, 1769, 1344, 2045, 2381, 1805, 2379, 3683, 2365, 1611, 3319, 3815, 4416, 4838, 4029, 3531, 5606, 5789, 4411, 4341, 5849, 7392, 1642, 4885, 8246, 3074, 5251, 5774, 3165, 2498, 12347, 9987, 5405, 8075, 11101, 2346, 6749

Offset: 1

Alberto Zanoni, Apr 17 2025

nonn

The sequence is infinite.

			For n = 1 the sum (1 addend) is 1^2
For n = 2 the sums (1 addend) are 4^2, 2^4
For n = 3 the sums are (2 addends) 1^2 + 2^4, 3^2 + 2^3, 4^2 + 1^3
For n = 4 the sums are (2 addends) 1^2 + 2^6, 1^2 + 4^3, 7^2 + 2^4, 8^2 + 1^3
For n = 5 the sums are (2 addends) 2^4 + 2^6, 4^3 + 2^4, 4^2 + 2^6, 4^2 + 4^3, 8^2 + 2^4
For n = 6 the sums are (3 addends) 3^2 + 2^5 + 2^6, 3^2 + 4^3 + 2^5, 4^2 + 2^3 + 3^4, 5^2 + 2^4 + 2^6, 5^2 + 4^3 + 2^4, 9^2 + 2^3 + 2^4

Eugenio Garista and Alberto Zanoni, Somme di potenze con esponenti diversi, MatematicaMente, 317 (2024), 1-2.
Eugenio Garista and Alberto Zanoni, Sums of Positive Integer Powers with Unlike Exponents, Armenian Journal of Mathematics, 17 No. 3 (2025), 1-11.
Alberto Zanoni, Sum of unlike powers for integers

Cf. A351062, A351066, A351063, A351065, A351064.

A351064 Minimal number of positive perfect powers, with different exponents, whose sum is n (considering only minimal possible exponents for bases equal to 1).

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 3, 2, 3, 4, 5, 1, 2, 3, 4, 2, 3, 4, 5, 2, 1, 2, 1, 2, 3, 4, 2, 1, 2, 3, 4, 1, 2, 3, 4, 2, 2, 3, 2, 2, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 5, 3, 2, 3, 2, 3, 4, 5, 2, 1, 2, 3, 4, 2, 3, 4, 5, 2, 2, 3, 3, 2, 3, 4, 3, 2, 1, 2, 3, 3, 2, 3, 4, 3, 2, 2, 2, 3, 3, 4, 3, 2, 2, 3, 4, 1, 2, 3, 4, 3, 3, 2

Offset: 1

Alberto Zanoni, Feb 22 2022

nonn

Conjecture: the only numbers for which 5 addends are needed are 15, 23, 55, 62, 71.

The numbers mentioned in the conjecture are also the first five terms of A111151. - Omar E. Pol, Mar 01 2022

			a(1) = 1 because 1 can be represented with a single positive perfect power: 1 = 1^2.
a(2) = 2 because 2 can be represented with two (and not fewer) positive perfect powers with different exponents: 2 = 1^2 + 1^3.
a(6) = 3 because 6 can be represented with three (and not fewer) positive perfect powers with different exponents: 6 = 2^2 + 1^3 + 1^4.
a(7) = 4 because 7 can be represented with four (and not fewer) positive perfect powers with different exponents: 7 = 2^2 + 1^3 + 1^4 + 1^5.
a(15) = 5 because 15 can be represented with five (and not fewer) positive perfect powers with different exponents: 15 = 2^2 + 2^3 + 1^4 + 1^5 + 1^6.

Cf. A111151, A351062, A351063, A351066.

A351065 Number of different ways to obtain n as a sum of the minimal possible number of positive perfect powers with different exponents (considering only minimal possible exponents for bases equal to 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 5, 2, 2, 2, 4, 1, 1, 1, 3, 4, 1, 2, 3, 1, 1, 2, 3, 2, 3, 3, 1, 1, 1, 1, 2, 6, 1, 4

Offset: 1

Alberto Zanoni, Feb 22 2022

nonn

Every positive integer k appears in the sequence, as a(2^(2^k)) = k.

			a(4) = 1, because 4 = 2^2 is its only possible representation, and similarly for every power a^p, with a > 1 and p prime.
a(16) = 2, because 16 = 2^4 = 4^2. More generally, a^(p^2) -- with a > 1 and p prime -- can be written in exactly two ways.
a(17) = 3, because 17 = 1^2 + 2^4 = 3^2 + 2^3 = 4^2 + 1^3.
a(313) = 10, because 313 can be written in exactly 10 different ways (with three perfect powers): 4^2 + 6^3 + 3^4 = 5^2 + 2^5 + 2^8 = 5^2 + 4^4 + 2^5 = 7^2 + 2^3 + 2^8 = 7^2 + 2^3 + 4^4 = 9^2 + 6^3 + 2^4 = 11^2 + 2^6 + 2^7 = 11^2 + 4^3 + 2^7 = 13^2 + 2^4 + 2^7 = 17^2 + 2^3 + 2^4.

Cf. A351062, A351063, A351064, A351066.

cp's OEIS Frontend

User: Alberto Zanoni

A386966 Numbers that can be written in exactly two different ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

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A386967 Numbers that can be written in exactly three different ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

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A386968 Numbers that can be written in exactly four ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

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A385969 Numbers that can be written as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

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A385970 a(n) is the minimum number of powers needed to obtain the n-th term of sequence A385969.

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A385233 Numbers that can be written as s^x + t^y + u^z with 1 < s < t < u and {s,t,u} = {x,y,z} (the sequence of exponents can be any permutation of s,t,u).

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A385232 Numbers that can be written as s^x + t^y, with 1 < s < t and {s,t} = {x,y}; that is, are of the form s^s + t^t or s^t + t^s.

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A383122 a(n) is the smallest number that can be expressed as the sum of the smallest number of powers with different exponents greater than one in n different ways (for unitary bases, the smallest possible exponents are considered).

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A351064 Minimal number of positive perfect powers, with different exponents, whose sum is n (considering only minimal possible exponents for bases equal to 1).

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A351065 Number of different ways to obtain n as a sum of the minimal possible number of positive perfect powers with different exponents (considering only minimal possible exponents for bases equal to 1).

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