A123031 Array read by antidiagonals: row i (i>=0) contains those positive integers n >= 2 for which the multiset { n mod k : k=2,3,...,n } contains exactly one copy of i.

2, 3, 3, 5, 4, 5, 7, 6, 6, 7, 9, 8, 7, 8, 11, 11, 10, 9, 9, 12, 13, 13, 12, 11, 10, 13, 14, 17, 15, 14, 13, 12, 12, 15, 18, 19, 17, 16, 15, 14, 13, 14, 19, 20, 23, 19, 18, 17, 16, 15, 15, 16, 21, 24, 29, 21, 20, 19, 18, 17, 16, 17, 20, 25, 30, 31, 23, 22, 21, 20, 19, 18, 18, 21, 22

Offset: 1

Author

Jared B. Ricks (jaredricks(AT)yahoo.com), Sep 24 2006

Keywords

• nonn
• tabl

Comments

In other words, for i >= 1, the i-th row contains all numbers n>2i such that n-i does not have divisors d with i < d < n-i. If p is the smallest prime divisor of n-i then (n-i)/p <= i.

Alternatively, the i-th row (i>=1) consists of 2i+1 and positive integers n>2i+1 such that the smallest prime divisor of n-i is greater than or equal to (n-i)/i = n/i - 1.

Examples

			For example, the 0th row obviously contains all prime numbers.
The first few rows of the array are
0) 2, 3, 5, 7, 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
1) 3, 4, 6, 8, 12,14,18,20,24,30,32,38,42,44,48,54,60,62,68,72,74,80,84,90,98,
2) 5, 6, 7, 9, 13,15,19,21,25,31,33,39,43,45,49,55,61,63,69,73,75,81,85,91,99,
3) 7, 8, 9, 10,12,14,16,20,22,26,32,34,40,44,46,50,56,62,64,70,74,76,82,86,92,
4) 9, 10,11,12,13,15,17,21,23,27,33,35,41,45,47,51,57,63,65,71,75,77,83,87,93,
5) 11,12,13,14,15,16,18,20,22,24,28,30,34,36,42,46,48,52,58,64,66,72,76,78,84,
6) 13,14,15,16,17,18,19,21,23,25,29,31,35,37,43,47,49,53,59,65,67,73,77,79,85,
...
		

Crossrefs

Rows: A000040, A008864, ...; columns: A004280, A051755, ...; diagonal starting with 2: A033627.

Extensions

Additional comments from Max Alekseyev, Sep 26 2006