A074400 -id:A074400 - cp's OEIS frontend

A263695 Even numbers such that the sum of the even divisors and the sum of the odd divisors are a square or a cube.

6, 14, 434, 636, 748, 762, 4620, 5964, 6204, 6324, 6580, 6820, 7084, 7660, 8404, 8636, 8804, 9010, 9710, 11342, 11920, 23622, 29820, 31020, 31620, 32844, 35420, 36204, 38964, 39804, 40044, 42020, 43180, 44020, 45724, 46004, 47564, 48484, 49146, 50644, 53444

Offset: 1

Michel Lagneau, May 28 2016

nonn

It seems that the two sums are never both a square or a cube.
Conjecture [False!]: All squares belonging to a pair are associated with a unique cube. Conversely, all cubes are associated with a unique square.
The corresponding pairs (sum of even divisors, sum of odd divisors) are (2^3, 2^2), (4^2, 2^3), (8^3, 16^2), (36^2, 6^3), (36^2, 6^3), (32^2, 8^3), 11 times the pair (24^3, 48^2), 3 times the pair (108^2, 18^3), (30^3, 30^2), (32^3, 128^2), 16 times the pair (288^2, 24^3),...
We observe several classes of numbers that generate identical pairs, for example:
{636, 748} => pair (36^2, 6^3);
{4620, 5964, 6204, 6324,... } => pair (24^3, 48^2);
{9010, 9710, 11342} => pair (108^2, 18^3);
{29820, 31020, 31620, 32844, 35420,... } => pair (288^2, 24^3);
{69576, 72168, 87752, 98552,...} => pair (56^3, 112^2);
The conjecture above is false. Consider for example the triples of numbers {69576, 938184, 7505472} or {958528, 952520, 12382760}. For the first one the (even, odd) sum of divisors pairs are (56^3, 112^2), (1568^2, 56^3), and (4704^2, 56^3). - Giovanni Resta, May 28 2016

			434 is in the sequence because the divisors are {1, 2, 7, 14, 31, 62, 217, 434} => sum of even divisors = 2+14+62+434 = 512 = 8^3 and sum of odd divisors = 1+7+31+217 = 256 = 16^2.
636 is in the sequence because the divisors are {1, 2, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636} => sum of even divisors = 2+4+6+12+106+212+318+636 = 1296 = 36^2 and sum of odd divisors = 1+3+53+159 = 216 = 6^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000593, A074400, A146076.

with(numtheory):
for n from 2 by 2  to 500000 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     ii:=0:
        for a from 1 to 1000 while(ii=0)do:
        for i from 2 to 3 do:
         if s0=a^i
          then
           for b from 1 to 1000 while(ii=0) do:
             if s1=b^(5-i)
              then
              ii:=1:printf(`%d, `,n):
              else
             fi:
           od:
          fi:
        od:
      od:
     od:

es[n_] := 2 DivisorSigma[1, n/2]; os[n_] := DivisorSigma[1, n] - es[n]; powQ[n_] := Or @@ IntegerQ /@ (n^(1/{2, 3})); Select[2 Range[10^4], powQ@ es@ # && powQ@ os@ # &] (* Giovanni Resta, May 28 2016 *)

isA002760(n)=issquare(n) || ispower(n,3)
is(n)=n%2==0 && isA002760(2*sigma(n/2)) && isA002760(sigma(n>>valuation(n,2))) \\ Charles R Greathouse IV, Jun 08 2016

A279812 Let s(k) denote the sum of the even proper divisors of k. The sequence lists the pairs of numbers (x, y) such that s(x) = y and s(y) = x.

Original entry on oeis.org

440, 568, 2368, 2420, 5240, 5848, 10040, 11128, 12464, 12736, 21488, 21712, 24570, 29190, 34592, 36832, 126040, 133856, 133984, 134190, 139230, 142290, 152168, 159500, 175266, 177460, 200970, 244530, 244736, 246304, 248310, 279630, 283328, 284620, 306352, 337460

Offset: 1

Michel Lagneau, Dec 19 2016

nonn

"Pseudo-amicable" numbers: pairs of numbers (x, y) such that A091818(x) = y and A091818(y) = x.
A pair of numbers x and y is called "pseudo-amicable" if the sum of the proper even divisors of either one is equal to the other. The smallest pair is x = 440, y = 568.
The sequence lists the numbers in increasing order. Note that the pairs (x, y) are not adjacent to each other in the list. For instance, the pair (126040, 152168) are "pseudo-amicable" numbers.
For ordered pairs see A279950.

			The pair (a(1), a(2)) = (440, 568) is in the sequence because the even proper divisors of 440 are 2, 4, 8, 10, 20, 22, 40, 44, 88, 110, 220 with sum = 568 and the proper divisors of 568 are 2, 4, 8, 142, 284 with sum = 440.

Cf. A063990, A074400, A091818, A279950.

s[n_]:=2*(DivisorSigma[1,n/2]-n/2);P[n_]:=If[Nest[s,n,2]==n&&!s[n]==n,True,False];Select[Range[10^6],P[#]&]

A367553 a(n) = Sum_{d|n} (d+n/d)^n.

Original entry on oeis.org

2, 18, 128, 1506, 15552, 266548, 4194304, 89452674, 2010077696, 52439799700, 1486016741376, 46761291772836, 1587429546508288, 58431605635691172, 2305913377957871616, 97342665226310447618, 4371823119477393063936, 208266120619720061526886

Offset: 1

Seiichi Manyama, Nov 22 2023

nonn

a(n) is even.

Cf. A074400.

PARI
```
a(n) = sumdiv(n, d, (d+n/d)^n);
```

a(n) = Sum_{k=0..n} n^(n-k) * binomial(n,k) * (Sum_{d|n} d^(2*k-n)) = Sum_{k=0..n} binomial(n,k) * (Sum_{d|n} d^k * (n/d)^(n-k)).

A374918 Even numbers k such that lambda(sum of even divisors of k) = phi(sum of odd divisors of k) where lambda is the Carmichael function (A002322) and phi the Euler totient function (A000010).

Original entry on oeis.org

2, 6, 10, 12, 14, 18, 26, 28, 34, 36, 42, 50, 52, 62, 72, 74, 84, 100, 106, 112, 122, 124, 136, 144, 146, 148, 162, 186, 194, 200, 244, 254, 292, 296, 314, 324, 336, 372, 386, 388, 424, 434, 482, 488, 496, 508, 554, 576, 578, 584, 626, 628, 656, 674, 688, 762

Offset: 1

Michel Lagneau, Jul 23 2024

nonn

			a(18) = 100 because the divisors of 100 are {1, 2, 4, 5, 10, 20, 25, 50, 100} with lambda(2+4+10+20+50+100) = lambda(186) = 30 and phi(1+5+25) = phi(31) = 30.

Cf. A000010, A000593, A002322, A074400.

with(numtheory):nn:=800:
for n from 2 by 2 to nn do:
 d:=divisors(n):n0:=nops(d):s0:=0:s1:=0:
    for i from 1 to n0 do:
     if irem(d[i],2)=0
      then
        s0:=s0+d[i] else
s1:=s1+d[i]:
     fi:
    od:
      if lambda(s0)=phi(s1) then
printf(`%d, `,n):else fi:
   od:

Select[Range[2, 1000, 2], EulerPhi[DivisorSigma[1, #/2^IntegerExponent[#, 2]]] == CarmichaelLambda[2*DivisorSigma[1, #/2]] &] (* Amiram Eldar, Jul 23 2024 *)

A380231 Alternating row sums of triangle A237591.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 3, 4, 5, 4, 3, 6, 5, 4, 7, 8, 7, 8, 7, 10, 9, 8, 7, 10, 11, 10, 9, 12, 11, 14, 13, 14, 13, 12, 15, 16, 15, 14, 13, 16, 15, 18, 17, 16, 19, 18, 17, 20, 21, 22, 21, 20, 19, 22, 21, 24, 23, 22, 21, 24, 23, 22, 25, 26, 25, 28, 27, 26, 25, 28, 27, 32, 31, 30, 29, 28, 31, 30, 29

Offset: 1

Omar E. Pol, Jan 17 2025

Consider the symmetric Dyck path in the first quadrant of the square grid described in the n-th row of A237593. Let C = (A240542(n), A240542(n)) be the middle point of the Dyck path.
a(n) is also the coordinate on the x axis of the point (a(n),n) and also the coordinate on the y axis of the point (n,a(n)) such that the middle point of the line segment [(a(n),n),(n,a(n))] coincides with the middle point C of the symmetric Dyck path.
The three line segments [(a(n),n),C], [(n,a(n)),C] and [(n,n),C] have the same length.
For n > 2 the points (n,n), C and (a(n),n) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (n,n), C and (n,a(n)) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (a(n),n), (n,n) and (n,a(n)) are the vertices of a virtual isosceles right triangle.

			For n = 14 the 14th row of A237591 is [8, 3, 1, 2] hence the alternating row sum is 8 - 3 + 1 - 2 = 4, so a(14) = 4.
On the other hand the 14th row of A237593 is the 14th row of A237591 together with the 14 th row of A237591 in reverse order as follows: [8, 3, 1, 2, 2, 1, 3, 8].
Then with the terms of the 14th row of A237593 we can draw a Dyck path in the first quadrant of the square grid as shown below:
.
         (y axis)
          .
          .
          .    (4,14)              (14,14)
          ._ _ _ . _ _ _ _            .
          .               |
          .               |
          .               |_
          .                 |
          .                 |_ _
          .                C    |_ _ _
          .                           |
          .                           |
          .                           |
          .                           |
          .                           . (14,4)
          .                           |
          .                           |
          . . . . . . . . . . . . . . | . . . (x axis)
        (0,0)
.
In the example the point C is the point (9,9).
The three line segments [(4,14),(9,9)], [(14,4),(9,9)] and [(14,14),(9,9)] have the same length.
The points (14,14), (9,9) and (4,14) are the vertices of a virtual isosceles right triangle.
The points (14,14), (9,9) and (14,4) are the vertices of a virtual isosceles right triangle.
The points (4,14), (14,14) and (14,4) are the vertices of a virtual isosceles right triangle.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Other alternating row sums (ARS) related to the Dyck paths of A237593 and the stepped pyramid described in A245092 are as follows:
ARS of A237593 give A000004.
ARS of A196020 give A000203.
ARS of A252117 give A000203.
ARS of A271343 give A000593.
ARS of A231347 give A001065.
ARS of A236112 give A004125.
ARS of A236104 give A024916.
ARS of A249120 give A024916.
ARS of A271344 give A033879.
ARS of A231345 give A033880.
ARS of A239313 give A048050.
ARS of A237048 give A067742.
ARS of A236106 give A074400.
ARS of A235794 give A120444.
ARS of A266537 give A146076.
ARS of A236540 give A153485.
ARS of A262612 give A175254.
ARS of A353690 give A175254.
ARS of A239446 give A235796.
ARS of A239662 give A239050.
ARS of A235791 give A240542.
ARS of A272026 give A272027.
ARS of A211343 give A336305.
Cf. A221529, A237270, A237271, A237591, A262626, A322141.

A380231[n_] := 2*Sum[(-1)^(k + 1)*Ceiling[(n + 1)/k - (k + 1)/2], {k,  Quotient[Sqrt[8*n + 1] - 1, 2]}] - n;
Array[A380231 , 100] (* Paolo Xausa, Sep 06 2025 *)

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
a(n) = my(orow = concat(row235791(n), 0)); vecsum(vector(#orow-1, i, (-1)^(i+1)*(orow[i] - orow[i+1]))); \\ Michel Marcus, Apr 13 2025

a(n) = 2*A240542(n) - n.
a(n) = n - 2*A322141(n).
a(n) = A240542(n) - A322141(n).

A383963 Irregular triangle read by rows: T(n,k) is the sum of the k-th pair of conjugate divisors of n. If n is a square then the central term in the row n is equal to 2*sqrt(n), with n >= 1, 1 <= k <= A000005(n).

Original entry on oeis.org

2, 3, 3, 4, 4, 5, 4, 5, 6, 6, 7, 5, 5, 7, 8, 8, 9, 6, 6, 9, 10, 6, 10, 11, 7, 7, 11, 12, 12, 13, 8, 7, 7, 8, 13, 14, 14, 15, 9, 9, 15, 16, 8, 8, 16, 17, 10, 8, 10, 17, 18, 18, 19, 11, 9, 9, 11, 19, 20, 20, 21, 12, 9, 9, 12, 21, 22, 10, 10, 22, 23, 13, 13, 23, 24, 24, 25, 14, 11, 10, 10, 11, 14, 25

Offset: 1

Omar E. Pol, Jun 17 2025

Row n is a palindromic composition of A074400(n) = 2*A000203(n).

Shares infinitely many rows with the virtual sequence 2*A237270.

			Triangle begins:
   n |   Row n
   1 |   2;
   2 |   3,  3;
   3 |   4,  4;
   4 |   5,  4,  5;
   5 |   6,  6;
   6 |   7,  5,  5,  7;
   7 |   8,  8;
   8 |   9,  6,  6,  9;
   9 |  10,  6, 10;
  10 |  11,  7,  7, 11;
  11 |  12, 12;
  12 |  13,  8,  7,  7,  8, 13;
  13 |  14, 14;
  14 |  15,  9,  9, 15;
  15 |  16,  8,  8, 16;
  16 |  17, 10,  8, 10, 17;
  ...
For n = 8 the divisors of 8 are [1, 2, 4, 8] and the sums of the conjugate divisors are respectively [1 + 8 = 9], [2 + 4 = 6], [4 + 2 = 6], [8 + 1 = 9], so the 8th row is [9, 6, 6, 9].
For n = 9 the divisors of 9 are [1, 3, 9] and the sums of the conjugate divisors are respectively [1 + 9 = 10], [3 + 3 = 6], [9 + 1 = 10], so the 9th row is [10, 6, 10]. Since 9 is a square then the central term in the row is equal to 2*sqrt(9) = 2*3 = 6. Also in this case the 9th row is the same as the 9th row of the virtual sequence 2*A237270 because the 9th row of A237270 is [5, 3, 5].

Row lengths give A000005.
Row sums give A074400 = 2*A000203.
Column 1 gives A000027 except the 1, the same for the right border.
Cf. A027750, A056538, A237270, A272025.

row[n_] := Module[{d = Divisors[n]}, d + Reverse[d]]; Array[row, 24] // Flatten (* Amiram Eldar, Jun 18 2025 *)

row(n) = my(d=divisors(n)); vector(#d, k, d[k]+n/d[k]); \\ Michel Marcus, Jun 18 2025

T(n,k) = A027750(n,k) + A056538(n,k).

cp's OEIS Frontend

A263695 Even numbers such that the sum of the even divisors and the sum of the odd divisors are a square or a cube.

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A279812 Let s(k) denote the sum of the even proper divisors of k. The sequence lists the pairs of numbers (x, y) such that s(x) = y and s(y) = x.

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Mathematica

A367553 a(n) = Sum_{d|n} (d+n/d)^n.

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A374918 Even numbers k such that lambda(sum of even divisors of k) = phi(sum of odd divisors of k) where lambda is the Carmichael function (A002322) and phi the Euler totient function (A000010).

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Maple

Mathematica

A380231 Alternating row sums of triangle A237591.

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Programs

Mathematica

PARI

Formula

A383963 Irregular triangle read by rows: T(n,k) is the sum of the k-th pair of conjugate divisors of n. If n is a square then the central term in the row n is equal to 2*sqrt(n), with n >= 1, 1 <= k <= A000005(n).

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