Mian–Chowla sequence
Encyclopedia
In mathematics
, the Mian–Chowla sequence is an integer sequence
defined
recursively
in the following way. The sequence starts with
Then for
, 
is the smallest integer such that the pairwise sum
is distinct, for all
and 
less than or equal to 
.
Initially, with
, there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence, 
, is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then, 
can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that 
, with the pairwise sums being 2, 3, 4, 5, 6 and 8. The sequence thus begins
If we define
, the resulting sequence is the same except each term is one less (that is, 0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, ... ).
The sequence was invented by Abdul Majid Mian and Sarvadaman Chowla.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Mian–Chowla sequence is an integer sequence
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
defined
recursively
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
in the following way. The sequence starts with
Then for
, is the smallest integer such that the pairwise sumis distinct, for all
and less than or equal to .Initially, with
, there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence, , is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then, can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that , with the pairwise sums being 2, 3, 4, 5, 6 and 8. The sequence thus begins
- 1, 2, 4, 8, 1313 (number)13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
, 2121 (number)21 is the natural number following 20 and preceding 22.-In mathematics:Twenty-one is the fifth discrete Semiprime and the second in the family. With 22 it forms the second discrete Semiprime pair...
, 3131 (number)31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...
, 4545 (number)45 is the natural number following 44 and followed by 46.- In mathematics :Forty-five is a triangular number, a hexagonal and 16-gonal number, a Kaprekar number, and a Harshad number....
, 6666 (number)66 is the natural number following 65 and preceding 67.Usages of this number include:-Mathematics:*66 is a sphenic number, a triangular number, a hexagonal number, and a semi-meandric number...
, 8181 (number)81 is the natural number following 80 and preceding 82.-In mathematics:Eighty-one is the square of 9 and the fourth power of 3. Like all powers of three, 81 is a perfect totient number. It is a heptagonal number and a centered octagonal number. It is also a tribonacci number, and an open meandric...
, 9797 (number)97 is the natural number following 96 and preceding 98.-In mathematics:97 is the 25th prime number , following 89 and preceding 101. 97 is a Proth prime as it is 3 × 25 + 1.The numbers 97, 907, 9007, 90007 and 900007 are happy primes...
, 123123 (number)123 is the natural number following 122 and preceding 124.-In mathematics:* 123 is a Lucas number...
, 148148 (number)148 is the natural number following 147 and before 149.-In mathematics:* 148 is the second number to be both a heptagonal number and a centered heptagonal number * 148 is the twelfth member of the Mian–Chowla sequence...
, 182182 (number)182 is the natural number following 181 and preceding 183.-In mathematics:* 182 is an even number* 182 is a composite number, as it is a positive integer with a positive divisor other than one or itself* 182 is a deficient number, as 154 is less than 182...
, 204204 (number)204 is the natural number following 203 and preceding 205. It may be written as "two hundred four" or "two hundred and four".-In mathematics:...
, 252, 290290 (number)290 is the natural number after 289 and before 291.-In mathematics:The product of three primes, 290 is a sphenic number, and the sum of four consecutive primes . The sum of the squares of the divisors of 17 is 290...
, 361, 401, 475, ... .
If we define
, the resulting sequence is the same except each term is one less (that is, 0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, ... ).The sequence was invented by Abdul Majid Mian and Sarvadaman Chowla.