A140826 -id:A140826 - cp's OEIS frontend

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

Clark Kimberling

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

			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

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.

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.

List([1..80],n->n-Tau(n)); # Muniru A Asiru, Sep 28 2018

a049820 n = n - a000005 n  -- Reinhard Zumkeller, Feb 06 2012

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

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)
```

(define (A049820 n) (- n (A000005 n))) ;; Antti Karttunen, Nov 27 2015

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

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

A231167 a(1) = a(2) = 0, for n>=3: (sum of non-divisors of n) modulo (number of non-divisors of n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 8, 6, 0, 0, 0, 0, 12, 1, 0, 0, 8, 1, 16, 20, 0, 19, 0, 23, 20, 1, 24, 8, 0, 1, 24, 26, 0, 25, 0, 32, 21, 1, 0, 26, 18, 38, 32, 38, 0, 31, 40, 36, 36, 1, 0, 30, 0, 1, 31, 15, 48, 37, 0, 50, 44, 47, 0, 33, 0, 1, 35, 56

Offset: 1

Jaroslav Krizek, Nov 07 2013

nonn

a(n) = 0 for n = 1, 2 and numbers from A140826.

a(n) = 1 for numbers of form 2*p (p=prime) from A100484 and other numbers, e.g. 8 and 13456 are only numbers n < 10^5 which are not of form 2*p with a(n) = 1.

			For n=6, a(6) = A024816(6) mod A049820(6) = 9 mod 2 = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A054025 (sigma(n) mod tau(n)), A024816, A049820, A024816, A049820, A065091, A230605, A161344.

ndn[n_]:=Module[{nd=Complement[Range[n],Divisors[n]]},Mod[Total[ nd],Length[ nd]]]; Join[{0,0},Array[ndn,80,3]] (* Harvey P. Dale, Apr 11 2022 *)

a(n) = A024816(n) mod A049820(n).

A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

Original entry on oeis.org

1, 3, 24, 48, 72, 96, 120, 144, 192, 216, 240, 288, 336, 360, 384, 406, 432, 480, 576, 600, 648, 672, 720, 768, 864, 936, 960, 1008, 1080, 1104, 1152, 1200, 1224, 1296, 1320, 1344, 1368, 1440, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2160, 2208, 2304

Offset: 1

Ctibor O. Zizka, May 22 2009

nonn

Numbers k such that Sum_{i=0..k} A000032(i)/(k+1) is an integer. - Robert G. Wilson v, May 25 2009

Why do the terms in A141767 so closely correspond to A160665? Except for k = 1, 3, 406, 44758, 341446, 1413286, 3170242, 4861698, 7912534, ..., k == 0 (mod 24). - Robert G. Wilson v, May 25 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000032, A001610, A111058, A140826, A140971, A140972.

A000032 := proc(n) option remember ; if n <= 1 then 2-n; else procname(n-1)+procname(n-2) ; fi; end: A001610 := proc(n) add(A000032(i),i=0..n-1) ; end: for n from 1 to 3000 do if A001610(n) mod n = 0 then printf("%d,",n) ; fi; od: # R. J. Mathar, May 25 2009

lst = {}; a = 2; b = 1; s = 3; n = 3; While[n < 2447, c = a + b; s = s + c; If[Mod[c, n] == 0, AppendTo[lst, n]]; a = b; b = c; n++ ]; lst (* Robert G. Wilson v, May 25 2009 *)

{k: k | A001610(k)}. - R. J. Mathar, May 25 2009

More terms from R. J. Mathar and Robert G. Wilson v, May 25 2009

A230605 Composite numbers n such that number of non-divisors of n divides sum of non-divisors of n.

Original entry on oeis.org

4, 18, 20, 24, 432, 588, 692945344

Offset: 1

Jaroslav Krizek, Nov 07 2013

nonn

a(7) > 10^5.
Composite numbers n such that A231167(n) mod A049820(n) = 0. Complement of primes (A000040) with respect to A140826. Intersection of A002808 and A140826.
The terms 20 and 432 are the only composite numbers < 10^5 that are also terms of A023883.
No more terms up to 10^8. - Michel Marcus, Nov 09 2013
a(8) > 10^11. - Donovan Johnson, Nov 12 2013
a(8) > 4*10^12. - Giovanni Resta, Feb 11 2014

			20 is in sequence because A024816(20) mod A049820(20) = 168 mod 14 = 0.

Cf. A231167, A000040, A049820, A024816.

isok(n) = (nnd = n - numdiv(n)) && !isprime(n) && !((n*(n+1)/2-sigma(n)) % nnd); \\ Michel Marcus, Nov 09 2013

a(7) from Donovan Johnson, Nov 10 2013

A279815 Numbers n such that the average of the squares of the numbers less than n that do not divide n is an integer.

Original entry on oeis.org

3, 4, 7, 13, 16, 19, 20, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 188, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619, 631, 643, 661, 673, 691, 709, 727, 733, 739, 751, 757, 769, 787

Offset: 1

Ilya Gutkovskiy, Dec 19 2016

Numbers n such that A049820(n) divides A276984(n).

			7 is in the sequence because 7 has 2 divisors {1,7} therefore 5 non-divisors {2,3,4,5,6}, 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 90 and 5 divides 90.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A020486, A049820, A140826, A276984.

Superset of A045375(m) (m > 1) ?

Select[Range[800], Mod[#1 (#1 + 1) ((2 #1 + 1)/6) - DivisorSigma[2, #1], #1 - DivisorSigma[0, #1]] == 0 & ]

is(n)=my(f=factor(n)); n>2 && ((2*n^3+3*n^2+n)/6-sigma(f,2))%(n-numdiv(f))==0 \\ Charles R Greathouse IV, Dec 19 2016

A351398 Numbers k >= 3 such that the arithmetic mean of the divisors of k AND the arithmetic mean of the nondivisors of k are integers.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Ctibor O. Zizka, Feb 09 2022

nonn

This sequence includes all primes >= 3 because A000203(p)/A000005(p) = (p + 1)/2 AND (A000217(p) - A000203(p))/A049820(p) = (p + 1)/2.

Up to 2 * 10^8 the only nonprime terms are 20 and 432. - Robert Israel, May 06 2024

			k = 13, A000203(13)/A000005(13) = 14/2 = 7, A024816(13)/A049820(13) = 77/11 = 7, so 13 is a term.
k = 20, A000203(20)/A000005(20) = 42/6 = 7, A024816(20)/A049820(20) = 168/14 = 12, so 20 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A000217, A024816, A049820.

Intersection of A003601 and A140826.

filter:= proc(n) local s,t;
  s:= numtheory:-sigma(n);
  t:= numtheory:-tau(n);
  (s/t)::integer and ((n*(n+1)/2 - s)/(n-t))::integer;
end proc:
select(filter, [$2..1000]); # Robert Israel, May 06 2024

Select[Range[2, 350], Divisible[(s = DivisorSigma[1, #]), (d = DivisorSigma[0, #])] && Divisible[#*(# + 1)/2 - s, # - d] &] (* Amiram Eldar, Feb 09 2022 *)

isok(k) = if (k>=3, my(sk=sigma(k), nk=numdiv(k), tk=k*(k+1)/2); !(sk % nk) && !((tk - sk) % (k - nk))); \\ Michel Marcus, Feb 10 2022

cp's OEIS Frontend

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

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A231167 a(1) = a(2) = 0, for n>=3: (sum of non-divisors of n) modulo (number of non-divisors of n).

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A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

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A230605 Composite numbers n such that number of non-divisors of n divides sum of non-divisors of n.

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A279815 Numbers n such that the average of the squares of the numbers less than n that do not divide n is an integer.

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A351398 Numbers k >= 3 such that the arithmetic mean of the divisors of k AND the arithmetic mean of the nondivisors of k are integers.

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