A385232 -id:A385232 - cp's OEIS frontend

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).

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).

A173054 Numbers of the form x^y + y^x, 1 < x < y.

Original entry on oeis.org

17, 32, 57, 100, 145, 177, 320, 368, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 60049, 65792, 69632, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169, 1048976, 1058576

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

			17 is in the sequence because 17 = 2^3 + 3^2.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A076980, A385232, A385614.

f[a_,b_]:=a^b+b^a; Take[Union[Flatten[Table[f[a,b],{a,2,50},{b,a+1,50}]]],80]
nn=10^50; n=1; Union[Reap[While[n++; k=n+1; num=n^k+k^n; num

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.

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.

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.

A385614 Numbers of the form x^x + y^y, 1 < x < y.

Original entry on oeis.org

31, 260, 283, 3129, 3152, 3381, 46660, 46683, 46912, 49781, 823547, 823570, 823799, 826668, 870199, 16777220, 16777243, 16777472, 16780341, 16823872, 17600759, 387420493, 387420516, 387420745, 387423614, 387467145, 388244032, 404197705, 10000000004

Offset: 1

Sean A. Irvine, Jul 04 2025

Terms are all combinations of 1 < x < y ordered by increasing y then increasing x, since the largest of one y is strictly less than the smallest of the next: (y-1)^(y-1) + y^y < 2^2 + (y+1)^(y+1) for y >= 3. - Kevin Ryde, Jul 06 2025

			31 is in the sequence because 31 = 2^2 + 3^3.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A000312, A173054, A385232.

a(n) = my(r,s=sqrtint((n-1)<<1,&r), x=2 + if(r>1, y=3 + s-(rKevin Ryde, Jul 06 2025

from math import isqrt, comb
def A385614(n):
    y = (m:=isqrt(k:=n<<1))+(k>m*(m+1))+2
    x = n-comb(y-2,2)+1
    return x**x+y**y # Chai Wah Wu, Jul 07 2025

a(n) = x^x + y^y where x=A131818(n+1) and y=A133196(n). - Kevin Ryde, Jul 06 2025

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.

Original entry on oeis.org

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.

A385611 Numbers that can be written as s^w + t^x + u^y + v^z with 1 < s < t < u and {s,t,u,v} = {w,x,y,z} (the sequence of exponents can be any permutation of s,t,u,v).

Original entry on oeis.org

202, 245, 254, 322, 340, 348, 377, 383, 400, 422, 460, 465, 468, 532, 545, 548, 568, 603, 628, 688, 700, 730, 736, 738, 739, 845, 865, 876, 892, 922, 936, 961, 977, 1002, 1029, 1033, 1036, 1092, 1122, 1138, 1174, 1205, 1234, 1236, 1265, 1269, 1338, 1403, 1407, 1433

Offset: 1

Jean-Marc Rebert, Jul 04 2025

nonn

			202 = 2^5 + 3^4 + 4^3 + 5^2.
628 = 2^2 + 3^5 + 4^4 + 5^3.
936 = 2^2 + 3^5 + 4^3 + 5^4.
1234 = 2^2 + 3^4 + 4^5 + 5^3.

Cf. A001597, A385232, A385233.

cp's OEIS Frontend

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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A173054 Numbers of the form x^y + y^x, 1 < x < y.

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

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A385614 Numbers of the form x^x + y^y, 1 < x < y.

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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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A385611 Numbers that can be written as s^w + t^x + u^y + v^z with 1 < s < t < u and {s,t,u,v} = {w,x,y,z} (the sequence of exponents can be any permutation of s,t,u,v).

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