From Newsgroup: rec.puzzles


Date: 11 Oct 2006
"jonnie303" <john.grint1@btinternet.com> posted:

Many numbers can be expressed as the sum of consecutive positive
integers, eg. 12=3+4+5, 22=4+5+6+7, 40=6+7+8+9+10. How can you
characterize all the numbers which can be expressed in this way?

---------- Omg... What a great problem!!!


How can you characterize all the numbers that can be
thus expressed in exactly 1 way?

How can you characterize all the numbers that can be
thus expressed in exactly 2 ways?

How can you characterize all the numbers that can be
thus expressed in exactly 3 ways?
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From Newsgroup: rec.puzzles


HenHanna@NewsGrouper <user4055@newsgrouper.org.invalid> posted:

Date: 11 Oct 2006
"jonnie303" <john.grint1@btinternet.com> posted:

Many numbers can be expressed as the sum of consecutive positive
integers, eg. 12=3+4+5, 22=4+5+6+7, 40=6+7+8+9+10. How can you characterize all the numbers which can be expressed in this way?

---------- Omg... What a great problem!!!

How can you characterize all the numbers that can be
thus expressed in exactly 1 way?

How can you characterize all the numbers that can be
thus expressed in exactly 2 ways?

How can you characterize all the numbers that can be
thus expressed in exactly 3 ways?

See https://mathblag.wordpress.com/2011/11/13/sums-of-consecutive-integers/
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From Newsgroup: rec.puzzles


Ilan Mayer <user4643@newsgrouper.org.invalid> posted:

HenHanna@NewsGrouper <user4055@newsgrouper.org.invalid> posted:

Date: 11 Oct 2006
"jonnie303" <john.grint1@btinternet.com> posted:

Many numbers can be expressed as the sum of consecutive positive integers, eg. 12=3+4+5, 22=4+5+6+7, 40=6+7+8+9+10. How can you characterize all the numbers which can be expressed in this way?

---------- Omg... What a great problem!!!

How can you characterize all the numbers that can be
thus expressed in exactly 1 way?

How can you characterize all the numbers that can be
thus expressed in exactly 2 ways?

How can you characterize all the numbers that can be
thus expressed in exactly 3 ways?

See https://mathblag.wordpress.com/2011/11/13/sums-of-consecutive-integers/

Thank you....

thus expressed in exactly 2 ways?
thus expressed in exactly 3 ways?

This reminds me of the Taxicab numbers........

______________

The sixth taxicab number, which is the smallest number expressible as the sum of two positive cubes in six distinct ways, remains the latest confirmed major find in the sequence and is

24153319581254312065344

_____________________________

Cabtaxi numbers are defined as the smallest positive integers that can be expressed as the sum or difference of two cubes in n distinct ways. Unlike taxicab numbers, which only consider sums of positive cubes, cabtaxi numbers allow for negative cubes and zero.

Known cabtaxi numbers with their representations:

Cabtaxi(1) = 1
= 1^3 + 0^3

Cabtaxi(2) = 91
= 3^3 + 4^3
= 6^3 - 5^3

Cabtaxi(3) = 728
= 6^3 + 8^3
= 9^3 - 1^3
= 12^3 - 10^3

Cabtaxi(4) = 2741256
= 108^3 + 114^3 ... (four distinct ways)

Cabtaxi(5) = 6017193
Cabtaxi(6) = 1412774811
Cabtaxi(7) = 11302198488

The sequence continues with larger numbers having multiple ways to be written as sums or differences of cubes. Cabtaxi numbers generalize taxicab numbers by allowing negative cubes, which often results in smaller numbers for the same n compared to classical taxicab numbers.

***


As of 2025, only 10 cabtaxi numbers have been discovered and verified. They are listed in the OEIS sequence A047696. The history of discoveries is as follows:

Cabtaxi(2) was known since the late 16th century.

Cabtaxi(3) was known to Euler but solved in 1902.

Cabtaxi(4) through Cabtaxi(7) were found by Randall L. Rathbun in 1992.

Cabtaxi(8) was found by Daniel J. Bernstein in 1998.

Cabtaxi(9) was discovered by Duncan Moore in 2005.

Cabtaxi(10) was first reported as an upper bound by Christian Boyer in 2006 and verified by Uwe Hollerbach in 2008.
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