A351066 -id:A351066

Search: a351066 -id:a351066

Displaying 1-3 of 3 results found.

page 1

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers k such that A351064(k) = 4.

REFERENCES

E. Garista and A. Zanoni, Somme di potenze con esponenti diversi, MatematicaMente 317 (2024), 1-2.

LINKS

Table of n, a(n) for n=1..56.

Alberto Zanoni, Sums of different powers.

EXAMPLE

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).

CROSSREFS

Cf. A351062, A351066, A351064.

KEYWORD

nonn

AUTHOR

Alberto Zanoni, Feb 22 2022

EXTENSIONS

Missing terms inserted by Alberto Zanoni, Jan 08 2024

STATUS

approved

A351064

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..106.

EXAMPLE

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.

KEYWORD

nonn

AUTHOR

Alberto Zanoni, Feb 22 2022

STATUS

approved

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).

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,16

COMMENTS

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

LINKS

Table of n, a(n) for n=1..107.

EXAMPLE

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.

KEYWORD

nonn

AUTHOR

Alberto Zanoni, Feb 22 2022

STATUS

approved

page 1

Search completed in 0.006 seconds