* sum of 1/2/3/4/5/6/7/8/9/10 etc alternate numbers are divisible by 1/2/3/4/5/6/7/8/9/10 etc respectively.

* One can test this above.

Pernicious Numbers

Pernicious numbers are positive integers, whose binary representation contains prime no. of 1s  and the sum of binary digits is a prime. First pernicious no. is 3 ->11₂ -->1+1=2 which is a prime. Next is 5 ->101₂ -->1+1=2 which is a prime. Next is 6->110₂ -->1+1=2 which is a prime. Then are 7,9,10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 47, 48, 49, 50, 52, 55, 56, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 79, 80, 81, 82, 84, 87, 88, 91, 93, 94, 96, 97, 98, 100 etc.

* No power of 2 is a pernicious number since it is 1 followed by zeroes.

* Every even Perfect Number is a pernicious number since they are of the form 2^p-1(2^p -1 )where p is a prime no. Such no. are represented in binary as p no. of 1s and (p-1)no. of zeroes.

* A number of the form (2^p -1 )where p is a prime is a pernicious no. known as Pernicious Mersenne Number whose binary representation contains no zeroes.