A161664 -id:A161664 - 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

A212868 Rectangular array T(n,k) = number of nondecreasing sequences of n 1..k integers with no element dividing the sequence sum (for n, k >= 1), read by decreasing antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 5, 3, 1, 0, 0, 0, 7, 9, 5, 2, 0, 0, 0, 12, 16, 15, 6, 2, 0, 0, 0, 16, 29, 29, 22, 9, 2, 0, 0, 0, 22, 43, 59, 52, 32, 12, 3, 0, 0, 0, 28, 64, 103, 112, 82, 40, 15, 3, 0, 0, 0, 37, 92, 168, 212, 199, 122, 59, 17, 3, 0, 0, 0, 43, 127, 259, 376, 407

Offset: 1

R. H. Hardin, May 29 2012

Table starts:
0 0  0  0   0    0    0    0     0     0     0      0      0      0      0 ...
0 1  2  5   7   12   16   22    28    37    43     54     64     75     86 ...
0 1  3  9  16   29   43   64    92   127   168    219    281    355    435 ...
0 1  5 15  29   59  103  168   259   386   553    772   1043   1401   1832 ...
0 2  6 22  52  112  212  376   640  1011  1560   2293   3328   4711   6524 ...
0 2  9 32  82  199  407  796  1424  2407  3948   6166   9456  14171  20556 ...
0 2 12 40 122  319  722 1503  2872  5159  9087  15030  24441  38349  58701 ...
0 3 15 59 182  503 1214 2693  5517 10574 19715  34318  58653  96517 154975 ...
0 3 17 74 259  733 1912 4560 10052 20363 39988  73196 131054 225666 378925 ...
0 3 22 97 363 1067 2960 7533 17497 37344 77105 148113 276174 498304 873878 ...
  ...

			All solutions for n=8 and k=4:
  2   2   2   3   3   2   2   2   2   3   2   2   2   3   2
  2   3   2   4   3   2   2   2   2   3   2   2   3   3   3
  2   3   3   4   3   2   3   2   2   3   2   2   3   3   4
  2   3   3   4   4   2   3   3   2   3   3   2   3   3   4
  2   3   3   4   4   2   3   3   2   3   4   3   3   3   4
  3   3   4   4   4   2   3   3   2   3   4   4   3   3   4
  3   3   4   4   4   2   3   4   3   4   4   4   4   3   4
  3   3   4   4   4   3   4   4   4   4   4   4   4   4   4

R. H. Hardin, Table of n, a(n) for n = 1..958

Cf. A161664 (row 2, cicada cycles), A212870 (row 3), A212871 (row 4), A212872 (row 5), A212873 (row 6), A212874 (row 7).
Cf. A212864 (column 4), A212865 (column 5), A212866 (column 6), A212867 (column 7).
Cf. A212869 (superdiagonal 1).

A187489 Irregular triangle T(n,k), n>=0, 0<=k<=A068063(n), read by rows: T(n,k) is the number of k-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 5, 1, 6, 7, 1, 7, 12, 1, 1, 8, 16, 2, 1, 9, 22, 6, 1, 10, 28, 12, 1, 1, 11, 37, 22, 2, 1, 12, 43, 31, 3, 1, 13, 54, 49, 6, 1, 14, 64, 70, 10, 1, 15, 75, 99, 21, 1, 16, 86, 128, 32, 1, 17, 101, 176, 49, 1, 18, 113, 216, 65, 1, 19, 130, 284, 101

Offset: 0

Alois P. Heinz, Mar 10 2011

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

T(n,k) = 0 for k>A068063(n). The triangle contains all positive values of T.

			T(5,2) = 5, because there are 5 2-element nondividing subsets of {1,2,3,4,5}: {2,3}, {2,5}, {3,4}, {3,5}, {4,5}.  T(7,3) = 1: {4,6,7}.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2;
  1, 3, 1;
  1, 4, 2;
  1, 5, 5;
  1, 6, 7;
  1, 7, 12, 1;
  ...

Alois P. Heinz, Rows n = 0..65, flattened
Eric Weisstein's World of Mathematics, Nondividing Set

Columns k=0-8 give: A000012, A000027, A161664, A187490, A187491, A187492, A187493, A187494, A187550.
Row sums give: A051014.
Cf. A068063.

A355145 Triangle read by rows: T(n,k) is the number of primitive subsets of {1,...,n} of cardinality k; n>=0, 0<=k<=ceiling(n/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 5, 2, 1, 6, 7, 3, 1, 7, 12, 10, 3, 1, 8, 16, 15, 5, 1, 9, 22, 26, 13, 2, 1, 10, 28, 38, 22, 4, 1, 11, 37, 66, 60, 26, 4, 1, 12, 43, 80, 76, 35, 6, 1, 13, 54, 123, 156, 111, 41, 6, 1, 14, 64, 161, 227, 180, 74, 12

Offset: 0

Marcel K. Goh, Jun 20 2022

A set is primitive if it does not contain distinct i and j such that i divides j.

For n >= 2, the alternating row sums equal -1.

			Triangle T(n,k) begins:
   n/k 0  1  2  3  4  5  6  7  8  9 10 11 12
    0  1
    1  1  1
    2  1  2
    3  1  3  1
    4  1  4  2
    5  1  5  5  2
    6  1  6  7  3
    7  1  7 12 10  3
    8  1  8 16 15  5
    9  1  9 22 26 13  2
   10  1 10 28 38 22  4
   11  1 11 37 66 60 26  4
   12  1 12 43 80 76 35  6
   ...
For n=6 and k=3 the T(6,3) = 3 primitive sets are {2,3,5}, {3,4,5}, and {4,5,6}.

Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.

Columns k=0..2 give: A000012, A000027, A161664.
Row sums give A051026.
T(2n,n) gives A174094.
T(2n-1,n) gives A192298 for n>=1.
Cf. A087077, A087086.

Sum_{k=1..ceiling(n/2)} k * T(n,k) = A087077(n). - Alois P. Heinz, Jun 24 2022

A299251 a(n) = ((Sum_{k=1..floor((n+1)^2/4)} d(k)) - T(n)) / 2, where d(n) = number of divisors of n (A000005) and T(n) = the n-th triangular number (A000217).

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 15, 21, 28, 37, 45, 55, 67, 80, 95, 110, 127, 146, 164, 187, 209, 235, 260, 286, 315, 346, 380, 413, 449, 485, 522, 564, 605, 651, 695, 743, 792, 844, 898, 950, 1006, 1064, 1123, 1185, 1250, 1318, 1384, 1451, 1523, 1596, 1670, 1747, 1828

Offset: 1

Luc Rousseau, Feb 06 2018

nonn

Twice this sequence is an attempt to find a counterpart to A161664: both compare triangular numbers T(n) and partial sums of numbers of divisors S(n). A161664 computes the excess of T(n) compared to S(n), whereas 2*a(n) computes the excess of S(n') compared to T(n), where n' is chosen equal to floor((n+1)^2/4). This choice appears structurally natural and economical when illustrated in a diagram. (See provided link.)

Luc Rousseau, Accompanying diagram

Cf. A000005, A000217, A002620, A006218, A161664.

F[n_] := Floor[(1/4)*n^2]
A[n_] := (Sum[DivisorSigma[0, k], {k, 1, F[n + 1]}] - n*(n + 1)/2)/2
Table[A[n], {n, 1, 100}]

f(n)=floor(n^2/4)
a(n)=(sum(k=1,f(n+1),numdiv(k))-n*(n+1)/2)/2
for(n=1,100,print1(a(n),", "))

from math import isqrt
def A299251(n): return (-(s:=isqrt(m:=(n+1)**2>>2))**2-(n*(n+1)>>1)>>1)+sum(m//k for k in range(1,s+1)) # Chai Wah Wu, Oct 23 2023

a(n) = (A006218(A002620(n + 1)) - A000217(n)) / 2.

A362864 Numbers k that divide Sum_{i=1..k} (i - d(i)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 5, 8, 15, 24, 26, 47, 121, 204, 347, 562, 4204, 6937, 6947, 31108, 379097, 379131, 379133, 2801205, 12554202, 20698345, 56264197, 13767391064, 37423648626, 37423648726, 61701166395, 276525443156, 276525443176, 455913379395, 455913379831, 751674084802

Offset: 1

Ctibor O. Zizka, May 06 2023

nonn

Numbers k such that the mean number of nondivisors in the range 1..k is an integer.
Numbers k such that A161664(k) is divisible by k.
Numbers k such that (A000217(k) - A006218(k)) is divisible by k.
The subsequence of odd terms k equals the intersection of A050226 and this sequence.

			k = 5: Sum_{i=1..5} (i - d(i))/k = 5/5 = 1, so k = 5 is a term.

Martin Ehrenstein, Table of n, a(n) for n = 1..38

Cf. A000005, A000217, A006218, A049820, A050226, A161664.

seq[kmax_] := Module[{sum = 0, s = {}}, Do[sum += k - DivisorSigma[0, k]; If[Divisible[sum, k], AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6] (* Amiram Eldar, May 06 2023 *)

isok(k) = !(sum(i=1, k, i - numdiv(i)) % k); \\ Michel Marcus, May 06 2023

from itertools import count, islice
from sympy import divisor_count
def A362864_gen(): # generator of terms
    c = 0
    for k in count(1):
        if not (c:=c+k-divisor_count(k))%k:
            yield k
A362864_list = list(islice(A362864_gen(),15)) # Chai Wah Wu, May 20 2023

More terms from Amiram Eldar, May 06 2023

a(24)-a(32) from Martin Ehrenstein, May 22 2023

A368592 a(n) = numerator of -(1/4)n!(2 + n!)(-2 + 1/(1 + floor(n/2 - 1/2))) - n!Sum_{m=1..1 + 2*floor(n/2 - 1/2)} 1/m.

Original entry on oeis.org

-1, 0, 7, 190, 5826, 214956, 11104542, 711175536, 59256152496, 5925678248160, 730285755406560, 105161159860398720, 18003044434808914560, 3528596711774282883840, 801568243461355261718400, 205201470326854119387494400, 59742508072063053997776844800

Offset: 1

Mats Granvik, Dec 31 2023

In the sum formula below, changing n! to n in the outer summation yields A161664.

			The fractions, of which a(n) is the numerator, begin: -1/4, 0, 7, 190, 5826, ...

Cf. A161664, A006218.

Numerator[Table[-1/4*n!*(2 + n!)*(-2 + 1/(1 + Floor[n/2 - 1/2])) - n!*Sum[1/m, {m, 1, 1 + 2*Floor[n/2 - 1/2]}], {n, 1, 17}]]

For n>1: a(n) = Sum_{h=1..n!} Sum_{m=1..1 + 2*floor(n/2 - 1/2)} Sum_{k=1 + floor(h/(m + 1))..floor(h/m - 1/m)} 1.

A280981 Partial products of A049820; a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 30, 120, 720, 4320, 38880, 233280, 2566080, 25660800, 282268800, 3104956800, 46574352000, 558892224000, 9501167808000, 133016349312000, 2261277938304000, 40703002889472000, 854763060678912000, 13676208970862592000, 300876597358977024000

Offset: 1

Jaroslav Krizek, Jan 11 2017

nonn

Cf. A049820(n) = number of nondivisors of n.

Cf. A049820, A161664.

[1, 1] cat [&*[#[c: c in [1..k] | k mod c ne 0]: k in [3..n]]: n in [3..100]]

FoldList[#1 #2 &, Table[Boole[n <= 2] + n - DivisorSigma[0, n], {n, 25}]] (* Michael De Vlieger, Jan 11 2017 *)

a(1) = a(2) = 1; for n>2, a(n) = Product_{i=3..n} A049820(i).

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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A212868 Rectangular array T(n,k) = number of nondecreasing sequences of n 1..k integers with no element dividing the sequence sum (for n, k >= 1), read by decreasing antidiagonals.

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A187489 Irregular triangle T(n,k), n>=0, 0<=k<=A068063(n), read by rows: T(n,k) is the number of k-element nondividing subsets of {1, 2, ..., n}.

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A355145 Triangle read by rows: T(n,k) is the number of primitive subsets of {1,...,n} of cardinality k; n>=0, 0<=k<=ceiling(n/2).

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A299251 a(n) = ((Sum_{k=1..floor((n+1)^2/4)} d(k)) - T(n)) / 2, where d(n) = number of divisors of n (A000005) and T(n) = the n-th triangular number (A000217).

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A362864 Numbers k that divide Sum_{i=1..k} (i - d(i)), where d(n) is the number of divisors of n (A000005).

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A368592 a(n) = numerator of -(1/4)*n!*(2 + n!)*(-2 + 1/(1 + floor(n/2 - 1/2))) - n!*Sum_{m=1..1 + 2*floor(n/2 - 1/2)} 1/m.

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A280981 Partial products of A049820; a(1) = a(2) = 1.

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A368592 a(n) = numerator of -(1/4)n!(2 + n!)(-2 + 1/(1 + floor(n/2 - 1/2))) - n!Sum_{m=1..1 + 2*floor(n/2 - 1/2)} 1/m.