A065608 -id:A065608 - cp's OEIS frontend

A236632 Sum of all divisors of all positive integers <= n minus the total number of divisors of all positive integers <= n.

0, 1, 3, 7, 11, 19, 25, 36, 46, 60, 70, 92, 104, 124, 144, 170, 186, 219, 237, 273, 301, 333, 355, 407, 435, 473, 509, 559, 587, 651, 681, 738, 782, 832, 876, 958, 994, 1050, 1102, 1184, 1224, 1312, 1354, 1432, 1504, 1572, 1618, 1732, 1786, 1873, 1941

Offset: 1

Omar E. Pol, Jan 31 2014

			For n = 6 the sets of divisors of the positive integers <= 6 are {1}, {1, 2}, {1, 3}, {1, 2, 4}, {1, 5}, {1, 2, 3, 6}. There are 14 total divisors and their sum is 1 + 3 + 4 + 7 + 6 + 12 = 33, so a(6) = 33 - 14 = 19.

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

Partial sums of A065608.

Cf. A000005, A000203, A006218, A024916, A222548.

[(&+[DivisorSigma(1, k) - DivisorSigma(0, k) : k in [1..n]]): n in [1..60]]; // Vincenzo Librandi, Aug 02 2019

A236632:=n->(1/2)*add(floor(n/i)*floor((n-i)/i), i=1..n): seq(A236632(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2016
N:= 1000: # to get a(1) to a(N)
A065608:= Vector(N):
for a from 1 to floor(sqrt(N)) do for b from a to N/a do
   if b = a then
     A065608[a*b] := A065608[a*b] + a - 1
   else
     A065608[a*b] := A065608[a*b] + a + b - 2;
   fi
od od:
ListTools:-PartialSums(convert(A065608,list)); # Robert Israel, May 16 2016

Table[Sum[Floor[n/i]*Floor[(n - i)/i], {i, n}]/2, {n, 50}] (* Wesley Ivan Hurt, Jan 30 2016 *)
Table[Sum[Binomial[Floor[n/i], 2], {i, n}], {n, 51}] (* Michael De Vlieger, May 15 2016 *)
Accumulate@ Table[DivisorSum[n, # - 1 &], {n, 51}] (* or *)
Table[Sum [(k - 1) Floor[n/k], {k, n}], {n, 51}] (* Michael De Vlieger, Apr 03 2017 *)

a(n) = sum(i=1, n, sigma(i)) - sum(i=1, n, numdiv(i)); \\ Michel Marcus, Feb 01 2014

from math import isqrt
def A236632(n): return (s:=isqrt(n))**2*(1-s)+sum((q:=n//k)*((k<<1)+q-3) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

a(n) = A024916(n) - A006218(n).
a(n) = (1/2) * Sum_{i=1..n} floor(n/i) * floor((n-i)/i). - Wesley Ivan Hurt, Jan 30 2016
a(n) = Sum_{i=1..n} binomial(floor(n/i),2). - Wesley Ivan Hurt, May 08 2016
a(n) = Sum_{k=1..n} (k-1) * floor(n/k). - Wesley Ivan Hurt, Apr 02 2017
a(n) = (1/2)*(A222548(n) - A006218(n)). - Ridouane Oudra, Aug 01 2019

A296081 a(n) = gcd(tau(n)-1, sigma(n)-1), where tau = A000005 and sigma = A000203.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 7, 1, 7, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 5, 8, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A032741, A039653, A065608, A284288, A296082, A296083.

Cf. also A009205.

Array[GCD[DivisorSigma[0,#]-1,DivisorSigma[1,#]-1]&,120] (* Harvey P. Dale, Oct 18 2022 *)

(define (A296081 n) (gcd (A039653 n) (A032741 n)))

a(n) = gcd(A032741(n), A039653(n)).

a(n) = gcd(A039653(n), A065608(n)).

A152992 a(n) = sigma(n) - d(n) - pi(n).

Original entry on oeis.org

0, 0, 0, 2, 1, 5, 2, 7, 6, 10, 5, 17, 6, 14, 14, 20, 9, 26, 10, 28, 20, 24, 13, 43, 19, 29, 27, 41, 18, 54, 19, 46, 33, 39, 33, 71, 24, 44, 40, 70, 27, 75, 28, 64, 58, 54, 31, 99, 39, 72, 53, 77, 36, 96, 52, 96, 60, 70, 41, 139, 42, 74, 80, 102, 62, 118, 47, 101, 73, 117, 50

Offset: 1

Omar E. Pol, Dec 19 2008, Dec 31 2008

			a(15) = 24 - 4 - 6 = 14 because the sum of divisors of 15 is 1 + 3 + 5 + 15 = 24, the number of divisors of 15 is 4 (1,3,5,15) and the number of primes not exceeding 15 is 6 (2,3,5,7,11,13). - _Emeric Deutsch_, Dec 30 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A000720, A065608, A152770, A152991.

with(numtheory): seq(sigma(n)-tau(n)-pi(n), n = 1 .. 75); # Emeric Deutsch, Dec 30 2008

Table[DivisorSigma[1,n]-DivisorSigma[0,n]-PrimePi[n],{n,75}] (* Harvey P. Dale, Sep 19 2011 *)

a(n) = A000203(n) - A000005(n) - A000720(n) = A065608(n) - A000720(n) = A152991(n) - A000005(n).

Corrected and extended by Emeric Deutsch, Dec 30 2008

A158901 A051731 * (1, 1, 2, 3, 4, 5, ...).

Original entry on oeis.org

1, 2, 3, 5, 5, 9, 7, 12, 11, 15, 11, 23, 13, 21, 21, 27, 17, 34, 19, 37, 29, 33, 23, 53, 29, 39, 37, 51, 29, 65, 31, 58, 45, 51, 45, 83, 37, 57, 53, 83, 41, 89, 43, 79, 73, 69, 47, 115, 55, 88, 69, 93, 53, 113, 69, 113, 77, 87, 59, 157, 61, 93, 99, 121, 81, 137, 67, 121, 93, 137

Offset: 1

Gary W. Adamson, Mar 29 2009

a(n) = prime(n) if n is prime but nonprime n's can also have prime a(n).

Equals left border of triangle A158902.

			a(4) = 5 = (1, 1, 0, 1) dot (1, 1, 2, 3) = (1 + 1 + 0 + 3); where (1, 1, 0, 1) = row 4 of triangle A051731.

Cf. A158902.

L := [1,seq(n, n=1..100)] ; read("transforms"); MOBIUSi(L) ; # R. J. Mathar, Apr 02 2009

a(n) = sigma(n) - numdiv(n) + 1; \\ Michel Marcus, Sep 14 2017

A051731 * [1, 1, 2, 3, 4, 5, ...] = inverse Mobius transform of [1, 1, 2, 3, 4, ...].
a(n) = sigma(n) - d(n) + 1. - Juri-Stepan Gerasimov, Aug 30 2009
a(n) = 1 + A065608(n). - R. J. Mathar, Jan 08 2015

A386791 Triangle read by rows: T(n, k) = (n - k)/k if k divides n - k else 0 for k > 0, T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 3, 0, 1, 0, 0, 0, 0, 0, 8, 0, 2, 0, 0, 0, 0, 0, 0, 0, 9, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 5, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Aug 04 2025

			Triangle begins:
  [ 0] 1;
  [ 1] 0,  0;
  [ 2] 0,  1, 0;
  [ 3] 0,  2, 0, 0;
  [ 4] 0,  3, 1, 0, 0;
  [ 5] 0,  4, 0, 0, 0, 0;
  [ 6] 0,  5, 2, 1, 0, 0, 0;
  [ 7] 0,  6, 0, 0, 0, 0, 0, 0;
  [ 8] 0,  7, 3, 0, 1, 0, 0, 0, 0;
  [ 9] 0,  8, 0, 2, 0, 0, 0, 0, 0, 0;
  [10] 0,  9, 4, 0, 0, 1, 0, 0, 0, 0, 0;
  [11] 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  [12] 0, 11, 5, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0;

Cf. A000007 (column 0 and main diagonal), A065608 (row sums), A000012 (central terms), A175992 (subtriangle of sign), A386790.

A386791 := (n, k) -> ifelse(k = 0, 0^n, ifelse(modp(n-k, k) = 0, iquo(n-k, k), 0)):
seq(seq(A386791(n, k), k = 0..n), n = 0..12);

A386791[n_, k_] := If[k == 0, Boole[k == n], If[Divisible[n-k, k], Quotient[n-k, k], 0]];
Table[A386791[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

def A386791(n, k):
    if k == 0: return int(k == n)
    return (n - k) // k if k.divides(n - k) else 0
for n in (0..12):
    print([A386791(n, k) for k in (0..n)])

Sum_{k=0..n} T(n, k) = A000203(n) - A000005(n) = A065608(n) for n > 0.
sign(T(n, k)) = A175992(n, k) for n, k >= 1.
T(2*n, n) = A000012(n).

A165797 a(n) = n^( sigma(n) - tau(n) ).

Original entry on oeis.org

1, 2, 9, 256, 625, 1679616, 117649, 8589934592, 3486784401, 100000000000000, 25937424601, 552061438912436417593344, 23298085122481, 83668255425284801560576, 332525673007965087890625

Offset: 1

Jaroslav Krizek, Sep 27 2009

nonn

The power of n with exponent given by the difference between its sum of divisors and its count of divisors.

			a(4) = 4^(sigma(4)-tau(4)) = 4^(7-3) = 4^4 = 256.

Jaroslav Krizek, Table of n, a(n) for n = 1..50

Table[n^[ DivisorSigma[1, n] - DivisorSigma[0, n]], {n, 50}]

a(n) = n^(A000203(n)-A000005(n)) = n^A000203(n) / n^A000005(n) = n^A065608(n).
a(n) = A100879(n) / A062758(n).
a(p) = p^(p-1) for p = prime.

Slightly edited by R. J. Mathar, Sep 29 2009

A176919 Triangle by columns: (1, 2, 3, ...) in each column interleaved with (0, 1, 2, ...) zeros. Columns > 1 shifted down twice.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson & Mats Granvik, Apr 29 2010

Row sums of n-th row = A065608(n+1).

			First few rows of the triangle:
   1;
   2, 0;
   3, 1, 0;
   4, 0, 0, 0;
   5, 2, 1, 0, 0;
   6, 0, 0, 0, 0, 0;
   7, 3, 0, 1, 0, 0, 0;
   8, 0, 2, 0, 0, 0, 0, 0;
   9, 4, 0, 0, 1, 0, 0, 0, 0;
  10, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  11, 5, 3, 2, 0, 1, 0, 0, 0, 0, 0;
  12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  13, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0;
  14, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  15, 7, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0;
  16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  17, 8, 5, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0;
  18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  19, 9, 0, 4, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  ...

Cf. A065608.

Triangle, left border = (1, 2, 3, ...). Each successive column is shifted down twice from previous column and interleaved with k zeros.

A329104 Numbers m such that sigma(m) - tau(m) = 2m.

Original entry on oeis.org

56, 7192, 7232, 7912, 10792, 17272, 30592, 114256, 2154584, 3428368, 89245784

Offset: 1

Jaroslav Krizek, Nov 04 2019

Abundant numbers m with abundance A(m) = tau(m).
Corresponding values of A(m) = tau(m): 8, 16, 14, 16, 16, 16, 16, 20, 32, 30, 32, ...
a(12) > 10^13, if it exists. - Giovanni Resta, Nov 07 2019

			Number 56 is in the sequence because sigma(56) - tau(56) = 2*56; 120 - 8 = 112.

Cf. A000005, A000203, A033880, A065608.

Cf. A083874 (deficient numbers m with deficiency D(m) = tau(m)).

[m: m in [1..10^5] | SumOfDivisors(m) - NumberOfDivisors(m) eq 2*m];

Select[Range[4*10^6], DivisorSigma[1, #] - DivisorSigma[0, #] == 2 # &] (* Amiram Eldar, Nov 04 2019 *)

isok(m) = my(f=factor(m)); sigma(f) - numdiv(f) == 2*m; \\ Michel Marcus, Nov 05 2019

a(11) from Amiram Eldar, Nov 04 2019

A330866 a(n) = Sum_{d|n, d

Original entry on oeis.org

0, 2, 6, 16, 20, 48, 42, 88, 90, 140, 110, 264, 156, 280, 300, 416, 272, 594, 342, 720, 588, 704, 506, 1248, 700, 988, 972, 1400, 812, 1920, 930, 1824, 1452, 1700, 1540, 2952, 1332, 2128, 2028, 3280, 1640, 3696, 1806, 3432, 3240, 3128, 2162, 5472, 2646, 4350, 3468

Offset: 1

Wesley Ivan Hurt, Apr 28 2020

Total area of all distinct L X W rectangles such that s + t = n, 1 <= s <= t, s | n, L = t and W = n/s. - Wesley Ivan Hurt, Aug 01 2025

			a(6) = 48; The proper divisors of 6 are 1, 2 and 3. We have (6/1)*(6-1) + (6/2)*(6-2) + (6/3)*(6-3) = 30 + 12 + 6 = 48.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A001065, A065608.

with(numtheory): seq(n*sigma(n) - n*tau(n), n=1..100); # Ridouane Oudra, Apr 11 2024

Table[Sum[(n/i) (n-i) (1 - Ceiling[n/i] + Floor[n/i]), {i, n-1}], {n, 80}]

a(n)={sumdiv(n, d, (n-d)*n/d)} \\ Andrew Howroyd, Apr 28 2020

a(p^k) = p^k * (p^(k+1) - p*(k+1) + k) / (p-1), where p is prime and k is a positive integer.

a(n) = n*sigma(n) - n*tau(n) = n*A065608(n). - Ridouane Oudra, Apr 11 2024

A355750 Sum of the divisors of 2n minus the number of divisors of 2n.

Original entry on oeis.org

1, 4, 8, 11, 14, 22, 20, 26, 33, 36, 32, 52, 38, 50, 64, 57, 50, 82, 56, 82, 88, 78, 68, 114, 87, 92, 112, 112, 86, 156, 92, 120, 136, 120, 136, 183, 110, 134, 160, 176, 122, 212, 128, 172, 222, 162, 140, 240, 165, 208, 208, 202, 158, 268, 208, 238, 232, 204, 176, 344, 182

Offset: 1

Wesley Ivan Hurt, Jul 15 2022

Consider the partitions of 2n into 2 parts (s,t), where s <= t. a(n) gives the sum of all the quotients t/s such that t/s is an integer. (See example.)

			a(7) = 20; the partitions of 2*7 = 14 into two parts (s,t) where s <= t are: (1,13), (2,12), (3,11), (4,10), (5,9), (6,8), and (7,7). The sum of the quotients t/s such that each t/s is an integer is then: 13/1 + 12/2 + 7/7 = 13 + 6 + 1 = 20.

Cf. A000005 (tau), A000203 (sigma), A062731, A099777.

Bisection of A065608.

Table[DivisorSigma[1, 2 n] - DivisorSigma[0, 2 n], {n, 80}]

a(n) = my(f=factor(2*n)); sigma(f) - numdiv(f); \\ Michel Marcus, Jul 16 2022

a(n) = sigma(2n) - tau(2n).
a(n) = Sum_{d|2n} (2n-d)/d.
a(n) = A065608(2n) = A000203(2n) - A000005(2n).
a(n) = A062731(n) - A099777(n).
a(n) = Sum_{k=1..n} m*c(m), where m=(2n-k)/k and c(m)=1-ceiling(m)+floor(m).

cp's OEIS Frontend

A236632 Sum of all divisors of all positive integers <= n minus the total number of divisors of all positive integers <= n.

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A296081 a(n) = gcd(tau(n)-1, sigma(n)-1), where tau = A000005 and sigma = A000203.

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A152992 a(n) = sigma(n) - d(n) - pi(n).

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A158901 A051731 * (1, 1, 2, 3, 4, 5, ...).

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A386791 Triangle read by rows: T(n, k) = (n - k)/k if k divides n - k else 0 for k > 0, T(n, 0) = 0^n.

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A165797 a(n) = n^( sigma(n) - tau(n) ).

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A176919 Triangle by columns: (1, 2, 3, ...) in each column interleaved with (0, 1, 2, ...) zeros. Columns > 1 shifted down twice.

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A329104 Numbers m such that sigma(m) - tau(m) = 2m.

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