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
Examples
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
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.
Crossrefs
Cf. A000005.
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).
Programs
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GAP
List([1..80],n->n-Tau(n)); # Muniru A Asiru, Sep 28 2018
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Haskell
a049820 n = n - a000005 n -- Reinhard Zumkeller, Feb 06 2012
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Maple
A049820 := n->n-numtheory[tau](n): seq(A049820(n), n=1..100);
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Mathematica
Table[n - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 19 2014 *) Array[(# - DivisorSigma[0, #])&, 70] (* Vincenzo Librandi, Dec 29 2015 *) -
PARI
a(n)=n-numdiv(n)
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Scheme
(define (A049820 n) (- n (A000005 n))) ;; Antti Karttunen, Nov 27 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} ((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
Extensions
Edited by Franklin T. Adams-Watters, Jan 30 2012
Comments