cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

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

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

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.

Programs

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

Extensions

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

A024816 Antisigma(n): Sum of the numbers less than n that do not divide n.

Original entry on oeis.org

0, 0, 2, 3, 9, 9, 20, 21, 32, 37, 54, 50, 77, 81, 96, 105, 135, 132, 170, 168, 199, 217, 252, 240, 294, 309, 338, 350, 405, 393, 464, 465, 513, 541, 582, 575, 665, 681, 724, 730, 819, 807, 902, 906, 957, 1009, 1080, 1052, 1168, 1182, 1254, 1280, 1377, 1365
Offset: 1

Views

Author

Paul Jobling (paul.jobling(AT)whitecross.com)

Keywords

Comments

a(n) is the sum of proper non-divisors of n, the row sum in triangle A173541. - Omar E. Pol, May 25 2010
a(n) is divisible by A000203(n) iff n is in A076617. - Bernard Schott, Apr 12 2022

Examples

			a(12)=50 as 5+7+8+9+10+11 = 50 (1,2,3,4,6 not included as they divide 12).

Links

Crossrefs

Cf. A000203 (sigma), A000217, A004125, A023896, A024916, A066760, A076617, A153485, A173539, A173540, A173541, A244048, A352810, A352811.
Cf. A342344 (for a symmetric representation).

Programs

  • Haskell
    a024816 = sum . a173541_row  -- Reinhard Zumkeller, Feb 19 2014
  • Magma
    [n*(n+1) div 2- SumOfDivisors(n): n in [1..60]]; // Vincenzo Librandi, Dec 29 2015
  • Maple
    A024816 := proc(n)
    n*(n+1)/2-numtheory[sigma](n) ;
end proc: # R. J. Mathar, Aug 03 2013
  • Mathematica
    Table[n(n + 1)/2 - DivisorSigma[1, n], {n, 55}] (* Robert G. Wilson v *)
Table[Total[Complement[Range[n],Divisors[n]]],{n,60}] (* Harvey P. Dale, Sep 23 2012 *)
With[{nn=60},#[[1]]-#[[2]]&/@Thread[{Accumulate[Range[nn]],DivisorSigma[ 1,Range[nn]]}]] (* Harvey P. Dale, Nov 22 2014 *)
  • PARI
    a(n)=n*(n+1)/2-sigma(n) \\ Charles R Greathouse IV, Mar 19 2012
  • Python
    from sympy import divisor_sigma
def A024816(n): return (n*(n+1)>>1)-divisor_sigma(n) # Chai Wah Wu, Apr 28 2023
  • SageMath
    def A024816(n): return sum(k for k in (0..n-1) if not k.divides(n))
print([A024816(n) for n in srange(1, 55)])  # Peter Luschny, Nov 14 2023

Formula

a(n) = n*(n+1)/2 - sigma(n) = A000217(n) - A000203(n).
a(n) = A024916(n-1) - A153485(n), n > 1. - Omar E. Pol, Jun 24 2014
From Wesley Ivan Hurt, Jul 16 2014, Dec 28 2015: (Start)
a(n) = Sum_{i=1..n} i * ( ceiling(n/i) - floor(n/i) ).
a(n) = Sum_{k=1..n} (n mod k) + (-n mod k). (End)
G.f.: x/(1 - x)^3 - Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017
From Omar E. Pol, Mar 21 2021: (Start)
a(n) = A244048(n) + A004125(n).
a(n) = A153485(n-1) + A004125(n), n >= 2. (End)
a(p) = (p-2)*(p+1)/2 for p prime. - Bernard Schott, Apr 12 2022

A173539 Square array read by antidiagonals: T(n,k)=0 if k is a divisor of n, otherwise T(n,k)=k.

Original entry on oeis.org

0, 0, 2, 0, 0, 3, 0, 2, 3, 4, 0, 0, 0, 4, 5, 0, 2, 3, 4, 5, 6, 0, 0, 3, 0, 5, 6, 7, 0, 2, 0, 4, 5, 6, 7, 8, 0, 0, 3, 4, 0, 6, 7, 8, 9, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, 0, 0, 5, 0, 7, 8, 9, 10, 11, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 0, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 0, 2, 0, 4, 0, 6, 7, 8, 9
Offset: 1

Views

Author

Omar E. Pol, May 25 2010

Keywords

Comments

Observation: Column k is the sequence defined as: Period k: repeat (k-1 numbers k together with zero).
Note that the positive terms in row n are the non-divisors of n (See also A173540 and A173541).

Examples

			Array begins:
0,2,3,4,5,6,7,8,9,10,11,12;
0,0,3,4,5,6,7,8,9,10,11;
0,2,0,4,5,6,7,8,9,10;
0,0,3,0,5,6,7,8,9;
0,2,3,4,0,6,7,8;
0,0,0,4,5,0,7;
0,2,3,4,5,6;
0,0,3,0,5;
0,2,0,4;
0,0,3;
0,2;
0;

Crossrefs

Cf. A007978, A173540, A173541.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.