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

1,2

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).

Links

Table of n, a(n) for n=1..64.

Example

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;
  ...

Prog

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

Crossrefs

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

Last elements of rows give A225004.

Cf. A000005, A000961.

Sequence in context: A155749 A198931 A344425 * A293779 A063379 A000463

Adjacent sequences: A340893 A340894 A340895 * A340897 A340898 A340899

Keyword

nonn,tabf

Author

J. Lowell, Jan 25 2021

Status

approved