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.

A352811 Table read by rows: row n gives triples (u, k, m) such that k and m are the smallest integers that respectively satisfy A352810(n) = u = A000203(k) = A024816(m).

Original entry on oeis.org

3, 2, 4, 20, 19, 7, 32, 21, 9, 54, 34, 11, 96, 42, 15, 132, 86, 18, 168, 60, 20, 217, 100, 22, 240, 114, 24, 252, 96, 23, 294, 164, 25, 338, 337, 27, 350, 349, 28, 464, 463, 31, 465, 200, 32, 582, 386, 35, 819, 288, 41, 1052, 1051, 48, 1080, 408, 47, 1182, 1181, 50
Offset: 1

Views

Author

Bernard Schott, Apr 12 2022

Keywords

Comments

A000203 is the function sigma sum of divisors, while A024816 is the antisigma function, sum of the numbers less than n that do not divide n.

Examples

			The table begins:
  ------------------------------------------------------------------
  | row |      u =        | smallest k with  |    smallest m with  |
  |  n  |   A352810(n)    |  A000203(k) = u  |     A024816(m) = u  |
  ------------------------------------------------------------------
    n=1 :         3,                   2,                   4;
    n=2 :        20,                  19,                   7;
    n=3 :        32,                  21,                   9;
    n=4 :        54,                  34,                  11;
    n=5 :        96,                  42,                  15;
    n=6 :       132,                  86,                  18;
  ...................................................................
3rd row is (32, 21, 9) because A352810(3) = 32, sigma(21) = sigma(31) = 32 and antisigma(9) = 2+4+5+6+7+8 = 32, hence 21 and 9 are respectively the smallest integers k and m such that sigma(k) = antisigma(m) = 32.
5th row is (96, 42, 15) because A352810(5) = 96 and 42 and 15 are respectively the smallest integers k and m such that sigma(k) = antisigma(m) = 96.

Crossrefs

Cf. A000203, A002191, A024816, A076617, A231365, A352810.

Programs

  • Mathematica
    m = 2000; r = Range[m]; s = DivisorSigma[1, r]; as = r*(r + 1)/2 - s; i = Select[Intersection[s, as], # <= m &]; Flatten @ Transpose @ Join[{i}, Map[Flatten[Table[FirstPosition[#, i[[k]]], {k, 1, Length[i]}]] &, {s, as}]] (* Amiram Eldar, Apr 12 2022 *)

Extensions

More terms from Amiram Eldar, Apr 13 2022

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

A353000 Quotients obtained when sigma(k) divides antisigma(k) with k = A076617(n), sigma (A000203) is the sum of divisors function and antisigma (A024816) is the sum of the non-divisors of n less than n function.

Original entry on oeis.org

0, 0, 4, 4, 4, 37, 25, 68, 49, 122, 115, 340, 544, 487, 959, 2167, 1926, 4837, 3847, 6757, 6452, 3620, 11353, 13934, 9371, 16353, 9211, 30949, 49702, 17330, 32575, 72544, 62348, 109769, 145892, 51270, 173914, 130687, 61665, 102887, 351770, 446927, 504949, 258079
Offset: 1

Views

Author

Bernard Schott, Apr 14 2022

Keywords

Comments

Note that the quotient obtained when sigma(k) divides k*(k+1)/2 with k = A076617(n) is a(n) + 1.

Examples

			A076617(6) = 95; sigma(95) = 120 and antisigma(95) = 4440, hence a(6) = 4440 / 120 = 37.

Crossrefs

Cf. A000203, A024816, A076617, A352810, A352811.

Programs

  • Mathematica
    Select[Table[(k*(k + 1)/2)/DivisorSigma[1, k] - 1, {k, 1, 10^6}], IntegerQ] (* Amiram Eldar, Apr 14 2022 *)
  • PARI
    is(n) = n*(n+1)/2%sigma(n) == 0; \\ A076617
f(n) = n*(n+1)/(2*sigma(n)) - 1;
lista(nn) = apply(f, select(is, [1..nn])); \\ Michel Marcus, Apr 15 2022

Formula

a(n) = A024816(A076617(n)) / A000203(A076617(n)).

Extensions

More terms from Amiram Eldar, Apr 14 2022
Showing 1-3 of 3 results.

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