Puzzle 1243 Consecutive Cube root smooth numbers

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Puzzle 1243 Consecutive Cube root smooth numbers

Giorgos Kalogeropouls sent the following nice puzzle:

Cube Root Smooth (CRS) numbers are numbers k whose largest prime factor lpf(k) <= k^(1/3) (A090081)

In this challenge you are asked to find the least number k that produces a run of exactly n consecutive CRS numbers.

n=1 -> 8
n=2 -> 2400, 2401
n=3 -> 134848, 134849, 134850
n=4 -> 3678723, 3678724, 3678725, 3678726

Example:

n=2

2400 = 2⁵ × 3 × 5²& 5^3 = 125< 2400

2401 = 74 & 7^3 = 343 < 2401

Q1: Find the least k that starts a run of 5,6,7... consecutive CRS numbers.

Q2: Is this sequence infinite?

Q3: What is the longest run of consecutive CRS numbers that you can find?

From Oct 25-31, Oct, 2025, contributions came from Emmanuel Vantieghem, Gennady Gusev, Simoin Cavegn, Jeff Heleen, Fred Schneider

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Emmanuel wrote:

(a is the first of n consecutive CRS numbers, b the last)
n a b
5 84441507 84441511
6 2020271859 2020271864
7 7374557947 7374557953

The next a - if it exists - will be > 37-10^9.

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Gennady wrote:

Q1. I found the following run of numbers starting from:

run 5 - 84441507

run 6 - 2020271859

run 7 - 7374557947

run 8 - 121153257533

Q2. Unknown

Q3. Run 8 - 121153257533

I tested the numbers up to 10^13 and didn't find a run > 8.

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Simon wrote:

n=1 -> 8
n=2 -> 2400, 2401
n=3 -> 134848, 134849, 134850
n=4 -> 3678723, 3678724, 3678725, 3678726
n=5 -> 84441507, 84441508, 84441509, 84441510, 84441511
n=6 -> 2020271859, 2020271860, 2020271861, 2020271862, 2020271863, 2020271864
n=7 -> 7374557947, 7374557948, 7374557949, 7374557950, 7374557951, 7374557952, 7374557953
n=8 -> 121153257533, 121153257534, 121153257535, 121153257536, 121153257537, 121153257538, 121153257539, 121153257540
n=9 -> 28632343404756, 28632343404757, 28632343404758, 28632343404759, 28632343404760, 28632343404761, 28632343404762, 28632343404763, 28632343404764

Searched up to 4×10^13 with an algorithm that multiplies all factors up to the cube root and temporarily stores them in a large bit array.

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Jeff wrote:

I only found chains of length 3 and 4:

134848, 134849, 134850

3678723, 3678724, 3678725, 3678726

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Fred wrote:

Q1:

Here are my solutions

Length = 5

84441507 = 3 * 61 * 67 * 71 * 97,

84441508 = 2^2 * 43 * 53 * 59 * 157
84441509 = 41 * 73 * 89 * 317

84441510 = 2 * 3^2 * 5 * 19^2 * 23 * 113

84441511 = 7 * 11 * 37 * 107 * 277
(Smooth UB = 433)

Length = 6

2020271859 = 3^2 * 499 * 599 * 751

2020271860 = 2^2 * 5 * 31^2 * 257 * 409

2020271861 = 23 * 53 * 73^2 * 311

2020271862 = 2 * 3 * 7^2 * 19 * 397 * 911

2020271863 = 29 * 181 * 557 * 691

2020271864 = 2^3 * 13 * 59 * 337 * 977

(Smooth UB = 1259)

Length = 7

7374557947 = 41 * 353 * 503 * 1013

7374557948 = 2^2 * 947 * 1201 * 1621

7374557949 = 3 * 11 * 37^2 * 239 * 683

7374557950 = 2 * 5^2 * 103 * 859 * 1667

7374557951 = 19 * 577 * 599 * 1123

7374557952 = 2^8 * 3^3 * 659 * 1619

7374557953 = 7 * 719 * 1061 * 1381

(Smooth UB = 1933)

Length = 8

121153257533 = 7 * 11 * 269 * 2311 * 2531

121153257534 = 2 * 3 * 181^2 * 419 * 1471

121153257535 = 5 * 97 * 197 * 607 * 2089

121153257536 = 2^6 * 17^2 * 61 * 167 * 643

121153257537 = 3 * 13 * 59 * 131 * 277 * 1451

121153257538 = 2 * 31 * 743 * 1153 * 2281

121153257539 = 71 * 137 * 149 * 179 * 467

121153257540 = 2^2 * 3^2 * 5 * 7^2 * 3041 * 4517

(Smooth UB = 4943)

Q2:

I would suspect there is no limit to the sequence length nor the level of the root (cube root and beyond)

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