cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

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Author

Alberto Zanoni, Feb 22 2022

Keywords

Comments

Numbers k such that A351064(k) = 4.

Examples

			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).

Links

Crossrefs

Cf. A351062, A351066, A351064.

Extensions

Missing terms inserted by Alberto Zanoni, Jan 08 2024

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

Views

Author

Alberto Zanoni, Feb 22 2022

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Views

Author

Alberto Zanoni, Feb 22 2022

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Views

Author

Alberto Zanoni, Apr 17 2025

Keywords

Comments

The sequence is infinite.

Examples

			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

Links

Crossrefs

Cf. A351062, A351066, A351063, A351065, A351064.
Showing 1-4 of 4 results.

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