b) 12 is the first sublime number.

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.

The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

There are only two known sublime numbers: 12 and

. The second of these has 76 decimal digits:6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.

c) We know 6 is the first perfect number : Sum of the factors whose proper factors sum to the number itself.

(1 + 2 + 3 = 6).

28 is the second perfect number.

496 is the third perfect number.

8128 is the fourth perfect number.

d) 17 is the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.

e) 25 is the first centered square number besides 1 that is also a square number.

In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.

f) 30 is the smallest sphenic number.

In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers.

Definition : A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.

Examples : The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ...

g) 142857 is the smallest base 10 cyclic number.

A cyclic number is an integer in which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are

142857 × 1 = 142857

142857 × 2 = 285714

142857 × 3 = 428571

142857 × 4 = 571428

142857 × 5 = 714285

142857 × 6 = 857142.

h) 9814072356 is the largest perfect power that contains no repeated digits in base ten.

i) Pandigital number: In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by :

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689.

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