A386968 -id:A386968 - 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.

A387099 Numbers that can be written in exactly five 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

5260225, 7923882, 11054875, 11224211, 11870046, 15466174, 16859617, 16911017, 17276523, 17326946, 18664520, 18668302, 18908170, 19375153, 19706896, 19854394, 20050965, 20757873, 21468249, 24723272, 26689657, 26925803, 26974782, 27214122, 27336893, 28055974

Offset: 1

David A. Corneth, Aug 16 2025

			5260225 = 2^22 + 3^8 + 4^5 + 5^2 + 7^7 + 8^3 + 22^4
        = 2^21 + 3^8 + 4^10 + 7^3 + 8^7 + 10^4 + 21^2
        = 2^7 + 3^14 + 4^5 + 5^8 + 6^6 + 7^3 + 8^2 + 14^4
        = 2^15 + 3^10 + 4^9 + 5^5 + 6^4 + 7^6 + 9^7 + 10^3 + 15^2
        = 2^11 + 3^7 + 4^10 + 5^9 + 6^8 + 7^3 + 8^5 + 9^6 + 10^4 + 11^2.
7923882 = 2^8 + 3^5 + 4^11 + 5^9 + 6^3 + 8^4 + 9^2 + 11^6
        = 2^12 + 3^9 + 4^5 + 5^6 + 6^2 + 7^8 + 8^7 + 9^3 + 12^4
        = 2^14 + 3^13 + 4^4 + 5^5 + 6^7 + 7^8 + 8^6 + 13^2 + 14^3
        = 2^18 + 3^14 + 4^6 + 5^9 + 6^3 + 7^7 + 9^5 + 14^4 + 18^2
        = 2^19 + 3^14 + 4^8 + 5^9 + 6^6 + 8^2 + 9^4 + 14^5 + 19^3.
11054875 = 2^3 + 3^6 + 4^10 + 5^5 + 6^2 + 7^4 + 10^7
         = 2^15 + 3^8 + 4^6 + 5^2 + 6^9 + 7^7 + 8^3 + 9^5 + 15^4
         = 2^22 + 3^12 + 4^2 + 5^6 + 6^7 + 7^8 + 8^5 + 12^3 + 22^4
         = 2^9 + 3^13 + 4^10 + 5^3 + 6^5 + 7^8 + 8^7 + 9^6 + 10^4 + 13^2
         = 2^11 + 3^12 + 4^8 + 5^7 + 6^9 + 7^6 + 8^3 + 9^2 + 11^5 + 12^4.

David A. Corneth, Table of n, a(n) for n = 1..152

Cf. A385969, A386968.

A387100 a(n) is the least number that can be written in exactly n 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}.

Original entry on oeis.org

4, 545, 23506, 331979, 5260225, 10630307

Offset: 1

David A. Corneth, Aug 16 2025

			a(5) = 5260225 via
5260225 = 2^22 + 3^8 + 4^5 + 5^2 + 7^7 + 8^3 + 22^4
        = 2^21 + 3^8 + 4^10 + 7^3 + 8^7 + 10^4 + 21^2
        = 2^7 + 3^14 + 4^5 + 5^8 + 6^6 + 7^3 + 8^2 + 14^4
        = 2^15 + 3^10 + 4^9 + 5^5 + 6^4 + 7^6 + 9^7 + 10^3 + 15^2
        = 2^11 + 3^7 + 4^10 + 5^9 + 6^8 + 7^3 + 8^5 + 9^6 + 10^4 + 11^2,
and no positive integer smaller than 5260225 can be written as such in exactly five ways.

Cf. A385969, A386968.

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.

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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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A387099 Numbers that can be written in exactly five 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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A387100 a(n) is the least number that can be written in exactly n 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}.

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