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Hardy–Ramanujan number

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1729 is the Hardy–Ramanujan number (taxi-cab number or taxicab number), the smallest [positive] integer that is the sum of 2 cubes in two different ways, viz.

Other properties of 1729

Some properties of 1729:

Roots and powers of 1729

In the table below, irrational numbers are given truncated to eight decimal places.

Roots Value A-number Powers Value A-number
41.58124577 1729 2 2989441  
12.00231436 A215889 1729 3 5168743489  
      1729n, n ≥ 0.   A138130

Sequences pertaining to 1729

Multiples of 1729 0, 1729, 3458, 5187, 6916, 8645, 10374, 12103, 13832, 15561, ... A138129
1729-gonal numbers 1, 1729, 5184, 10366, 17275, 25911, 36274, 48364, 62181, 77725, ... A051871
sequence starting at 1729 1729, 5188, 2594, 1297, 3892, 1946, 973, 2920, 1460, 730, 365, ... A245671

Values for number theoretic functions with 1729 as an argument

–1 See Möbius function.
–8 See Mertens function.
269 See prime counting function.
8 See number of divisors function.
2240 See sum of divisors function.
1296 See totient function.
3 See number of prime factors (with multiplicity) function.
3 See number of distinct prime factors function.
36 See Carmichael lambda function.
–1 See Liouville lambda function.
1 + 3.30474152... × 10 –521 See Riemann zeta function. (Requires more than five hundred decimal places to distinguish from 1.)
1729! 1.86377... × 10 4849 See factorial.
1.0779473... × 10 4846 See Gamma function.

Factorization of 1729 in some quadratic integer rings

INCOMPLETE. SOME CELLS IN THE TABLE BELOW ARE BLANK FOR NOW.

As was mentioned above, 1729 is a composite number in . It is also composite in all quadratic integer rings, but its factorization differs, and it has multiple factorizations in some rings that are not unique factorization domains.

7 × 13 × 19

OR
OR
Note that is not a distinct factorization because is a divisor of . The same goes for .

A side note: some Hilbert numbers have multiple factorizations, but 1729 is not one of them, being uniquely factorable into Hilbert numbers as 13 × 133.

See also

Some integers
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
1729