A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

0, 0, 1, 1, 3, 2, 5, 4, 6, 6, 9, 6, 11, 10, 11, 11, 15, 12, 17, 14, 17, 18, 21, 16, 22, 22, 23, 22, 27, 22, 29, 26, 29, 30, 31, 27, 35, 34, 35, 32, 39, 34, 41, 38, 39, 42, 45, 38, 46, 44, 47, 46, 51, 46, 51, 48, 53, 54, 57, 48, 59, 58, 57, 57, 61, 58, 65

Offset

1,5

Comments

a(n) is the number of non-divisors of n in 1..n. - Jaroslav Krizek, Nov 14 2009

Also equal to the number of partitions p of n such that max(p)-min(p) = 1. The number of partitions of n with max(p)-min(p) <= 1 is n; there is one with k parts for each 1 <= k <= n. max(p)-min(p) = 0 iff k divides n, leaving n-d(n) with a difference of 1. It is easiest to see this by looking at fixed k with increasing n: for k=3, starting with n=3 the partitions are [1,1,1], [2,1,1], [2,2,1], [2,2,2], [3,2,2], etc. - Giovanni Resta, Feb 06 2006 and Franklin T. Adams-Watters, Jan 30 2011

Number of positive numbers in n-th row of array T given by A049816.

Number of proper non-divisors of n. - Omar E. Pol, May 25 2010

a(n+2) is the sum of the n-th antidiagonal of A225145. - Richard R. Forberg, May 02 2013

For n > 2, number of nonzero terms in n-th row of triangle A051778. - Reinhard Zumkeller, Dec 03 2014

Number of partitions of n of the form [j,j,...,j,i] (j > i). Example: a(7)=5 because we have [6,1], [5,2], [4,3], [3,3,1], and [2,2,2,1]. - Emeric Deutsch, Sep 22 2016

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

G. E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014-2015.

Formula

a(n) = Sum_{k=1..n} ceiling(n/k)-floor(n/k). - Benoit Cloitre, May 11 2003

G.f.: Sum_{k>0} x^(2*k+1)/(1-x^k)/(1-x^(k+1)). - Emeric Deutsch, Mar 01 2006

a(n) = A006590(n) - A006218(n) = A161886(n) - A000005(n) - A006218(n) + 1 for n >= 1. - Jaroslav Krizek, Nov 14 2009

a(n) = Sum_{k=1..n} A000007(A051731(n,k)). - Reinhard Zumkeller, Mar 09 2010

a(n) = A076627(n) / A000005(n). - Reinhard Zumkeller, Feb 06 2012

For n >= 2, a(n) = A094181(n) / A051953(n). - Antti Karttunen, Nov 27 2015

a(n) = Sum_{k=1..n} ((n mod k) + (-n mod k))/k. - Wesley Ivan Hurt, Dec 28 2015

G.f.: Sum_{j>=2} (x^(j+1)*(1-x^(j-1))/(1-x^j))/(1-x). - Emeric Deutsch, Sep 22 2016

Dirichlet g.f.: zeta(s-1)- zeta(s)^2. - Ilya Gutkovskiy, Apr 12 2017

a(n) = Sum_{i=1..n-1} sign(i mod n-i). - Wesley Ivan Hurt, Sep 27 2018

Example

a(7) = 5; the 5 non-divisors of 7 in 1..7 are 2, 3, 4, 5, and 6.
The 5 partitions of 7 with max(p) - min(p) = 1 are [4,3], [3,2,2], [2,2,2,1], [2,2,1,1,1] and [2,1,1,1,1,1]. - Emeric Deutsch, Mar 01 2006

Maple

A049820 := n->n-numtheory[tau](n):
seq(A049820(n), n=1..100);

Mathematica

Table[n - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 19 2014 *)
Array[(# - DivisorSigma[0, #])&, 70] (* Vincenzo Librandi, Dec 29 2015 *)

Prog

(PARI) a(n)=n-numdiv(n)
(Haskell)
a049820 n = n - a000005 n  -- Reinhard Zumkeller, Feb 06 2012
(Scheme) (define (A049820 n) (- n (A000005 n))) ;; Antti Karttunen, Nov 27 2015
(GAP) List([1..80], n->n-Tau(n)); # Muniru A Asiru, Sep 28 2018

Crossrefs

Cf. A000005.

One less than A062968, two less than A059292.

Cf. A161664 (partial sums).

Cf. A060990 (number of solutions to a(x) = n).

Cf. A045765 (numbers not occurring in this sequence).

Cf. A236561 (same sequence sorted into ascending order), A236562 (with also duplicates removed), A236565, A262901 and A262903.

Cf. A262511 (numbers that occur only once).

Cf. A055927 (positions of repeated terms).

Cf. A245388 (positions of squares).

Cf. A155043 (number of steps needed to reach zero when iterating a(n)), A262680 (number of nonzero squares encountered).

Cf. A259934 (an infinite trunk of the tree defined by edge-relation a(child) = parent, conjectured to be unique).

Cf. tables and arrays A047916, A051731, A051778, A173540, A173541.

Cf. also arrays A225145, A262898, A263255 and tables A263265, A263267.

Other related sequences: A006218, A006590, A051953, A070824, A094181, A062249, A067391, A076627, A128508, A131187, A134156, A140826, A161886, A177235, A177236, A227874, A228453, A230653, A230654, A231167, A245197, A253473.

Sequence in context: A075491 A326730 A089279 * A109712 A095049 A118209

Adjacent sequences: A049817 A049818 A049819 * A049821 A049822 A049823

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Edited by Franklin T. Adams-Watters, Jan 30 2012

Status

approved