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

%I A061017 #61 Jan 14 2024 02:22:05
%S A061017 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,
%T A061017 12,12,12,12,12,13,13,14,14,14,14,15,15,15,15,16,16,16,16,16,17,17,18,
%U A061017 18,18,18,18,18,19,19,20,20,20,20,20,20,21,21,21,21,22,22,22,22,23,23,24
%N A061017 List in which n appears d(n) times, where d(n) [A000005] is the number of divisors of n.
%C A061017 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.
%C A061017 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
%C A061017 The numbers in A003991 arranged in numerical order. - _Matthew Vandermast_, Feb 28 2003
%C A061017 Least k such that tau(1) + tau(2) + tau(3) + ... + tau(k) >= n. - _Michel Lagneau_, Jan 04 2012
%C A061017 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
%H A061017 N. J. A. Sloane, <a href="/A061017/b061017.txt">Table of n, a(n) for n = 1..7069</a>
%H A061017 Hayato Kobayashi, <a href="http://hayatokobayashi.com/paper/ACL2014_Kobayashi.pdf">Perplexity on Reduced Corpora</a>, 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.
%F A061017 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
%F A061017 a(n) = A027750(n) * A056538(n). - _Charles Kusniec_, Jan 21 2021
%F A061017 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). - _Amiram Eldar_, Jan 14 2024
%e A061017 Array begins:
%e A061017    1
%e A061017    2  2
%e A061017    3  3
%e A061017    4  4  4
%e A061017    5  5
%e A061017    6  6  6  6
%e A061017    7  7
%e A061017    8  8  8  8
%e A061017    9  9  9
%e A061017   10 10 10 10
%e A061017   11 11
%e A061017   12 12 12 12 12 12
%e A061017   13 13
%e A061017   14 14 14 14
%e A061017   15 15 15 15
%e A061017   16 16 16 16 16
%e A061017   17 17
%e A061017   18 18 18 18 18 18
%e A061017   19 19
%e A061017   20 20 20 20 20 20
%e A061017   21 21 21 21
%e A061017   22 22 22 22
%e A061017   23 23
%e A061017   24 24 24 24 24 24 24 24
%p A061017 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);
%t A061017 Flatten[Table[Table[n, {Length[Divisors[n]]}], {n, 30}]]
%o A061017 (PARI) a(n)=if(n<0,0,t=1;while(sum(k=1,t,floor(t/k))<n,t++);t) \\ _Benoit Cloitre_, Nov 08 2009
%Y A061017 Cf. A000005. An inverse to A006218.
%Y A061017 Cf. A027750, A056538, A072691.
%K A061017 nonn,easy
%O A061017 1,2
%A A061017 Jont Allen (jba(AT)research.att.com), May 25 2001
%E A061017 More terms from _Erich Friedman_, Jun 01 2001