A351062 -id:A351062 - cp's OEIS frontend

A351063 Sums of four perfect powers with different exponents: m = a^x + b^y + c^z + d^t with a > 0, b > 0, c > 0, d > 0, x > 1, y > 1, z > 1, t > 1 and x, y, z, t are all different, with m not representable with fewer such addends.

7, 14, 19, 22, 30, 35, 39, 46, 54, 61, 67, 70, 78, 87, 94, 99, 103, 110, 111, 115, 119, 120, 139, 147, 167, 179, 183, 188, 195, 199, 211, 230, 237, 303, 318, 331, 335, 339, 342, 355, 399, 410, 419, 421, 429, 436, 438, 454, 461, 467, 470, 477, 483, 494, 510, 534

Offset: 1

Alberto Zanoni, Feb 22 2022

nonn

Numbers k such that A351064(k) = 4.

			7 is a term, as 7 = 2^2 + 1^3 + 1^4 + 1^5 (considering minimal possible exponents for bases equal to 1).
14 is a term, as 14 = 2^2 + 2^3 + 1^4 + 1^5 (idem).
195 is a term, as 195 = 7^2 + 1^3 + 3^4 + 2^6 or 7^2 + 4^3 + 3^4 + 1^5 or 9^2 + 1^3 + 3^4 + 2^5 (idem).

E. Garista and A. Zanoni, Somme di potenze con esponenti diversi, MatematicaMente, 317 (2024), 1-2.
E. Garista and A. Zanoni, Sums of positive integer powers with unlike exponents, Armenian Journal of Mathematics, 17 No. 3 (2025), 1-11.
Alberto Zanoni, Sums of different powers.

Cf. A351062, A351066, A351064.

Missing terms inserted by Alberto Zanoni, Jan 08 2024

A351066 Sums of three perfect powers with different exponents: m = a^x + b^y + c^z with a > 0, b > 0, c > 0, x > 1, y > 1, z > 1 and x, y, z are all different, with m not representable with fewer such addends.

Original entry on oeis.org

3, 6, 11, 18, 21, 34, 45, 56, 69, 74, 75, 84, 93, 98, 105, 112, 123, 124, 131, 138, 143, 146, 149, 154, 159, 166, 173, 178, 187, 190, 191, 194, 203, 210, 213, 214, 215, 218, 219, 222, 227, 234, 235, 236, 239, 240, 245, 254, 255, 258, 261, 262, 263, 266, 267, 274, 275, 276

Offset: 1

Alberto Zanoni, Jan 31 2022

nonn

			3 is a term as 3 = 1^2 + 1^3 + 1^4.
6 is a term as 6 = 2^2 + 1^3 + 1^4.
18 is a term. The possibilities are 1^2 + 1^3 + 2^4 or 3^2 + 2^3 + 1^4 or 4^2 + 1^3 + 1^4 (considering only minimal possible exponents for bases equal to 1).

Cf. A074499, A351062, A351063.

Definition clarified by Alberto Zanoni, Feb 28 2022

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.

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.

cp's OEIS Frontend

A351063 Sums of four perfect powers with different exponents: m = a^x + b^y + c^z + d^t with a > 0, b > 0, c > 0, d > 0, x > 1, y > 1, z > 1, t > 1 and x, y, z, t are all different, with m not representable with fewer such addends.

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A351066 Sums of three perfect powers with different exponents: m = a^x + b^y + c^z with a > 0, b > 0, c > 0, x > 1, y > 1, z > 1 and x, y, z are all different, with m not representable with fewer such addends.

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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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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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