Virtual Home of Paul Muljadi

Web Hosting by Netfirms | Free Domain Names by Netfirms

Home Median Numbers and Median Primes
It's my delight to share with you my discovery that some numbers and primes can be classified as median numbers and median primes. A median number is a special positive integer. The (i,j)-th median number is defined mathematically as

M(i,j) = (i^2 + j) / 2,

where i and j are alternating positive odd and even integers.

The first eight (i,1)-th median numbers, where i is odd, are

M(1,1) = (12 + 1)/2 = 1
M(3,1) = (32 + 1)/2 = 5
M(5,1) = (52 + 1)/2 = 13
M(7,1) = (72 + 1)/2 = 25
M(9,1) = (92 + 1)/2 = 41
M(11,1) = (112 + 1)/2 = 61
M(13,1) = (132 + 1)/2 = 85
M(15,1) = (152 + 1)/2 = 113
M(17,1) = (172 + 1)/2 = 145

In the above sequence, the one's digits follow the pattern 1-5-3-5-1. Hence, the (i,1)-th median number sequence is also defined as the 1-5-3-5-1 sequence.

The second case is for j = 2 where i is an even integer, which generates

M(2,2) = (22 + 2)/2 = 3
M(4,2) = (42 + 2)/2 = 9
M(6,2) = (62 + 2)/2 = 19
M(8,2) = (82 + 2)/2 = 33
M(10,2) = (102 + 2)/2 = 51
M(12,2) = (122 + 2)/2 = 73
M(14,2) = (142 + 2)/2 = 99
M(16,2) = (162 + 2)/2 = 113
M(18,2) = (182 + 2)/2 = 145

A generalization to other number systems is possible.

The alternating odd and even integer for i and j simulate a sieving process, which is normally used in generating prime sequences such as the lucky primes. A median number that is also prime is called a median prime. Five of the above eight median numbers, for a = 1, M3, M5, M9, M11, and M15 are prime numbers. The first median primes are

5 13 41 61 113 181 313 421 613 761 1013 1201 1301 1741 1861 2113 2381 2521 3121 3613 4513 5101 5941 7081 7321 ....

All median primes after 5 end with numbers 01, 13, 21, 41, 61, and 81.

If you see an error in my discovery, please let me know. Paul