A061017 List in which n appears d(n) times, where d(n) [A000005] is the number of divisors of n.

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24

Offset: 1

Author

Jont Allen (jba(AT)research.att.com), May 25 2001

Keywords

• nonn
• easy

Comments

The union of N, 2N, 3N, ..., where N = {1, 2, 3, 4, 5, 6, ...}. In other words, the numbers {m*n, m >= 1, n >= 1} sorted into nondecreasing order.

Considering the maximal rectangle in each of the Ferrers graphs of partitions of n, a(n) is the smallest such maximal rectangle; a(n) is also an inverse of A006218. - Henry Bottomley, Mar 11 2002

The numbers in A003991 arranged in numerical order. - Matthew Vandermast, Feb 28 2003

Least k such that tau(1) + tau(2) + tau(3) + ... + tau(k) >= n. - Michel Lagneau, Jan 04 2012

The number 1 appears only once, primes appear twice, squares of primes appear thrice. All other positive integers appear at least four times. - Alonso del Arte, Nov 24 2013

Examples

			Array begins:
   1
   2  2
   3  3
   4  4  4
   5  5
   6  6  6  6
   7  7
   8  8  8  8
   9  9  9
  10 10 10 10
  11 11
  12 12 12 12 12 12
  13 13
  14 14 14 14
  15 15 15 15
  16 16 16 16 16
  17 17
  18 18 18 18 18 18
  19 19
  20 20 20 20 20 20
  21 21 21 21
  22 22 22 22
  23 23
  24 24 24 24 24 24 24 24
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..7069
• Hayato Kobayashi, Perplexity on Reduced Corpora, in: Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics, Baltimore, Maryland, USA, June 23-25 2014, Association for Computational Linguistics, 2014, pp. 797-806.

Crossrefs

Cf. A000005. An inverse to A006218.

Cf. A027750, A056538, A072691.

Programs

Maple

with(numtheory); t1:=[]; for i from 1 to 1000 do for j from 1 to tau(i) do t1:=[op(t1),i]; od: od: t1:=sort(t1);

Mathematica

Flatten[Table[Table[n, {Length[Divisors[n]]}], {n, 30}]]

PARI

a(n)=if(n<0,0,t=1;while(sum(k=1,t,floor(t/k))Benoit Cloitre, Nov 08 2009

Formula

a(n) >= pi(n+1) for all n; a(n) >= pi(n) + 1 for all n >= 24 (cf. A098357, A088526, A006218, A052511). - N. J. A. Sloane, Oct 22 2008

a(n) = A027750(n) * A056538(n). - Charles Kusniec, Jan 21 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Jan 14 2024

Extensions

More terms from Erich Friedman, Jun 01 2001