A076617 -id:A076617 - cp's OEIS frontend

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.

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

Bernard Schott, Apr 14 2022

nonn

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

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

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

Select[Table[(k*(k + 1)/2)/DivisorSigma[1, k] - 1, {k, 1, 10^6}], IntegerQ] (* Amiram Eldar, Apr 14 2022 *)

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

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

More terms from Amiram Eldar, Apr 14 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

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

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

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

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

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

Cf. A342344 (for a symmetric representation).

a024816 = sum . a173541_row  -- Reinhard Zumkeller, Feb 19 2014

[n*(n+1) div 2- SumOfDivisors(n): n in [1..60]]; // Vincenzo Librandi, Dec 29 2015

A024816 := proc(n)
    n*(n+1)/2-numtheory[sigma](n) ;
end proc: # R. J. Mathar, Aug 03 2013

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 *)

a(n)=n*(n+1)/2-sigma(n) \\ Charles R Greathouse IV, Mar 19 2012

from sympy import divisor_sigma
def A024816(n): return (n*(n+1)>>1)-divisor_sigma(n) # Chai Wah Wu, Apr 28 2023

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

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

A232324 n(n+1)/2 modulo sigma(n).

Original entry on oeis.org

0, 0, 2, 3, 3, 9, 4, 6, 6, 1, 6, 22, 7, 9, 0, 12, 9, 15, 10, 0, 7, 1, 12, 0, 15, 15, 18, 14, 15, 33, 16, 24, 33, 1, 6, 29, 19, 21, 52, 10, 21, 39, 22, 66, 21, 1, 24, 60, 28, 66, 30, 6, 27, 45, 28, 36, 53, 1, 30, 150, 31, 33, 40, 48, 45, 51, 34, 78, 15, 37, 36

Offset: 1

Jaroslav Krizek, Nov 25 2013

nonn

Also antisigma(n) modulo sigma(n). Antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n, sigma(n) = A000203(n) = the sum of the divisors of n.

a(n) = 0 for numbers from A076617, a(n) = 1 for numbers from A232540, a(n) = n for numbers from A232538.

			For n=10, a(10) = antisigma(10) mod sigma(10) = 37 mod 18 = 1.

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

Cf. A000203, A076617, A024816, A232538, A232540.

Table[Mod[n (n + 1)/2, DivisorSigma[1, n]], {n, 100}] (* T. D. Noe, Nov 27 2013 *)

a(n) = n(n+1)/2 mod A000203(n).

A066860 The sum of the non-divisors of n (less than n) is a multiple of the sum of the divisors of n.

Original entry on oeis.org

15, 20, 24, 95, 104, 207, 224, 287, 464, 1023, 1199, 1952, 4095, 4607, 8036, 12095, 15872, 16895, 19359, 22932, 23519, 28799, 45440, 45695, 54144, 77375, 101567, 102024, 130304, 159599, 163295, 223199, 296207, 317184, 352799, 522752, 524160

Offset: 1

Joseph L. Pe, Jan 25 2002

nonn

			Divisors of 15 = {1, 3, 5, 15}, which sum to 24. Non-divisors of 15 less than 15 = {2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14}, which sum to 96, a multiple of 24. So 15 is a term of the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..300

Cf. A000203, A024816, A076617, A232324.

f[n_] := Module[{a, b, c}, a = Divisors[n]; b = Apply[Plus, Complement[Range[1, n], a]]; c = Apply[Plus, a]; Mod[b, c] == 0]; Do[If[f[n] == True, Print[n]], {n, 3, 23519}]
Select[Range[3, 10000], Mod[# (# + 1)/2, DivisorSigma[1, #]] == 0 &] (* T. D. Noe, Nov 27 2013 *)

a(n) = A076617(n+2). - Alex Ratushnyak, Jul 02 2013

A232324(a(n)) = A024816(a(n)) mod A000203(a(n)) = 0. - Jaroslav Krizek, Nov 25 2013

More terms from Lior Manor, Feb 10 2002

A232538 Numbers n such that (n(n+1)/2) modulo sigma(n) = n.

Original entry on oeis.org

33, 136, 145, 261, 897, 1441, 2016, 2241, 2353, 3808, 4320, 7201, 17101, 26937, 30721, 32896, 46593, 70561, 148960, 151633, 169345, 174592, 208801, 400401, 578593, 712801, 803800, 1040401, 1103233, 1596673, 2265121, 2377089, 3330001, 4357153, 5953024, 5962321

Offset: 1

Jaroslav Krizek, Nov 25 2013

nonn

Also numbers n such that antisigma(n) modulo sigma(n) = n. Antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n, sigma(n) = A000203(n) = the sum of the divisors of n.
Numbers n such that A232324(n) = n.
a(19)  > 10^5.

			136 is in sequence because antisigma(136) mod sigma(136) = 9046 mod 270 = 136.

Cf. A000203, A076617, A024816, A232324, A232540.

Select[Range[6*10^6],Mod[(#(#+1))/2,DivisorSigma[1,#]]==#&] (* Harvey P. Dale, Sep 12 2019 *)

isok(n) = (n*(n+1)/2 - sigma(n)) % sigma(n) == n; \\ Michel Marcus, Nov 25 2013

A232324(a(n)) = n.

More terms from Michel Marcus, Nov 25 2013

A232540 Numbers n such that (n(n+1)/2) modulo sigma(n) = 1.

Original entry on oeis.org

10, 22, 34, 46, 58, 82, 94, 106, 118, 142, 166, 178, 202, 214, 226, 262, 274, 298, 334, 346, 358, 382, 385, 394, 430, 454, 466, 478, 502, 514, 526, 538, 562, 586, 622, 634, 694, 706, 718, 766, 778, 802, 838, 862, 886, 898, 922, 934, 958, 982, 1006, 1018, 1042

Offset: 1

Jaroslav Krizek, Nov 25 2013

nonn

Also numbers n such that antisigma(n) modulo sigma(n) = 1. Antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n, sigma(n) = A000203(n) = the sum of the divisors of n.
Numbers n such that A232324(n) = 1.
Number 5950 is only squareful number from first 1400 terms (< 50000) of this sequence.
Conjecture: supersequence of A112774 (semiprimes of the form 6n+4).

			106 is in sequence because antisigma(106) mod sigma(106) = 5509 mod 162 = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000203, A076617, A024816, A112774, A232324, A232538.

Select[Range[1100],Mod[(#(#+1))/2,DivisorSigma[1,#]]==1&] (* Harvey P. Dale, Sep 08 2017 *)

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

Bernard Schott, Apr 12 2022

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.

			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.

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

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 *)

More terms from Amiram Eldar, Apr 13 2022

cp's OEIS Frontend

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.

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

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A232324 n(n+1)/2 modulo sigma(n).

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A066860 The sum of the non-divisors of n (less than n) is a multiple of the sum of the divisors of n.

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A232538 Numbers n such that (n(n+1)/2) modulo sigma(n) = n.

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A232540 Numbers n such that (n(n+1)/2) modulo sigma(n) = 1.

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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).

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