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Hardy–Ramanujan number
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.
Contents
Other properties of 1729
Some properties of 1729:
- Third Carmichael number (absolute Fermat pseudoprime) (A002997).
- First Chernick's Carmichael number (A033502). (1729 = 7 × 13 × 19 is the smallest Carmichael number of the form )
- First extended Chernick's Carmichael number.
- First absolute Euler pseudoprime (A033181).
- Third Zeisel number (A051015).
- 1729 is some figurate number, e.g. the 19th dodecagonal number (A051624) since 1729 = 19 × 91. It happens that the first and last digits of 1729 give 19, and the middle two digits 72 = 6 × 12 = (7 − 1) × (13 − 1), 7 × 13 = 91 being the reversal of 19.
- Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 1 (trivial case), 81 and 1458) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number: 1 + 7 + 2 + 9 = 19; 19 × 91 = 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 1729 n, 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
- A011541 Taxi-cab (taxicab) or Hardy–Ramanujan numbers: the smallest number that is the sum of 2 cubes in ways (an infinite sequence).
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 |