A340896 Irregular triangle in which the n-th row consists of all multiples of n that have fewer than twice as many divisors as n.

1, 2, 4, 3, 9, 4, 8, 16, 5, 25, 6, 12, 18, 7, 49, 8, 16, 32, 64, 9, 27, 81, 10, 20, 50, 11, 121, 12, 24, 36, 48, 13, 169, 14, 28, 98, 15, 45, 75, 16, 32, 64, 128, 256, 17, 289, 18, 36, 54, 162, 19, 361, 20, 40, 80, 100, 21, 63, 147, 22, 44, 242, 23, 529

Offset: 1

Author

J. Lowell, Jan 25 2021

Keywords

• nonn
• tabf

Comments

If n has d divisors, n has an infinite number of multiples with exactly 2d divisors, but only a finite number of multiples with fewer than 2d divisors.

Conjecture: row n includes n^2 if and only if n is a power of a prime number (A000961).

Examples

			Triangle begins:
   1;
   2,   4;
   3,   9;
   4,   8, 16;
   5,  25;
   6,  12, 18;
   7,  49;
   8,  16, 32,  64;
   9,  27, 81;
  10,  20, 50;
  11, 121;
  12,  24, 36,  48;
  13, 169;
  14,  28, 98;
  15,  45, 75;
  16,  32, 64, 128, 256;
  ...
		

Crossrefs

Columns k=1..2 give: A000027, A285109 (for n>=2).

Last elements of rows give A225004.

Cf. A000005, A000961.

Programs

PARI

row(n) = select(x->((numdiv(x)<2*numdiv(n)) && !(x % n)), [1..n^2]); \\ Michel Marcus, Jan 26 2021