A146076 -id:A146076 - cp's OEIS frontend

A193350 Sum of even divisors of tau(n).

0, 2, 2, 0, 2, 6, 2, 6, 0, 6, 2, 8, 2, 6, 6, 0, 2, 8, 2, 8, 6, 6, 2, 14, 0, 6, 6, 8, 2, 14, 2, 8, 6, 6, 6, 0, 2, 6, 6, 14, 2, 14, 2, 8, 8, 6, 2, 12, 0, 8, 6, 8, 2, 14, 6, 14, 6, 6, 2, 24, 2, 6, 8, 0, 6, 14, 2, 8, 6, 14, 2, 24, 2, 6, 8, 8, 6, 14, 2, 12, 0, 6, 2, 24, 6, 6, 6, 14, 2, 24, 6, 8, 6, 6, 6, 24, 2, 8, 8, 0

Offset: 1

Michel Lagneau, Jul 23 2011

nonn

			a(24) = 14 because tau(24) = 8 and the sum of the 3 even divisors {2, 4, 8} is 14.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A062069, A146076, A193347, A193349.

Cf. A000290 (the positions of zeros).

Table[Total[Select[Divisors[DivisorSigma[0,n]], EvenQ[ # ]&]], {n, 74}]

PARI
```
a(n)=sumdiv(sigma(n,0),d,(1-d%2)*d);
```

a(n) = A146076(A000005(n)). - Antti Karttunen, May 28 2017

a(n) = A062069(n) - A193349(n). - Amiram Eldar, Jan 27 2025

Data section extended to 100 terms by Antti Karttunen, May 28 2017

A266537 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with 2*k-1 zeros, and the first positive element of column k is in the row A002378(k), with T(1,1) = 0.

Original entry on oeis.org

0, 2, 0, 6, 0, 10, 2, 0, 0, 14, 0, 0, 0, 18, 6, 0, 0, 22, 0, 2, 0, 0, 0, 26, 10, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 34, 14, 6, 0, 0, 0, 38, 0, 0, 2, 0, 0, 0, 0, 42, 18, 0, 0, 0, 0, 0, 0, 46, 0, 10, 0, 0, 0, 0, 0, 50, 22, 0, 0, 0, 0, 0, 0, 54, 0, 0, 6, 0, 0, 0, 0, 58, 26, 14, 0, 2

Offset: 1

Omar E. Pol, Apr 05 2016

Gives an identity for A146076. Alternating sum in row n equals the sum of even divisors of n.
Even-indexed rows of the triangle give A236106.
If T(n,k) = 6 then T(n+2,k+1) = 2, the first element of the column k+1.

			Triangle begins:
0;
2;
0;
6;
0;
10,  2;
0,   0;
14,  0;
0,   0;
18,  6;
0,   0;
22,  0,  2;
0,   0,  0;
26, 10,  0;
0,   0,  0;
30,  0,  0;
0,   0,  0;
34, 14,  6;
0,   0,  0;
38,  0,  0,  2;
0,   0,  0,  0;
42, 18,  0,  0;
0,   0,  0,  0;
46,  0, 10,  0;
0,   0,  0,  0;
50, 22,  0,  0;
0,   0,  0,  0;
54,  0,  0,  6;
0,   0,  0,  0;
58, 26, 14,  0,  2;
...
For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12 and the sum of even divisors of 12 is 2 + 4 + 6 + 12 = 24. On the other hand, the 12th row of the triangle is 22, 0, 2, so the alternating row sum is 22 - 0 + 2 = 24, equaling the sum of even divisors of 12.

Cf. A002378, A016825, A146076, A196020, A236106, A237593, A271343.

T(n,k) = 0, if n is odd.

T(n,k) = 2*A196020(n/2,k) = A236106(n/2,k), if n is even.

A281707 Even integers k such that phi(sum of even divisors of k) = sum of odd divisors of k.

Original entry on oeis.org

2, 6, 14, 42, 62, 186, 254, 434, 762, 1302, 1778, 5334, 7874, 16382, 23622, 49146, 55118, 114674, 165354, 262142, 344022, 507842, 786426, 1048574, 1523526, 1834994, 2080514, 3145722, 3554894, 5504982, 6241542, 7340018, 8126402, 10664682, 14563598, 22020054

Offset: 1

Michel Lagneau, Jan 28 2017

nonn

The number of divisors of a(n) is a power of 2, and sum of even divisors = 2^(m+1), sum of odd divisors = 2^m for some m.
a(n) == 2, 6 (mod 8) or a(n) == 2, 6 (mod 12).
a(n) is of the form 2*p1*p2*...pk where p1, p2, ..., pk are Mersenne primes = 3, 7, 31, 127, 8191, ... (see A000668).

			62 is a term because its divisors are 1, 2, 31 and 62, the sum of the even divisors of 62 = 62 + 2 = 2^6, the sum of odd divisors = 1 + 31 = 2^5, and phi(2^6) = 2^5.

Cf. A000010, A000593, A000668, A146076.

with(numtheory):
for n from 2 by 2 to 10^6 do:
x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
   for k from 1 to n1 do:
    if irem(x[k],2)=0
     then
     s0:=s0+ x[k]:
     else
     s1:=s1+ x[k]:
    fi:
  od:
    if s1=phi(s0)
     then
     print(n):
     else
    fi:
od:

Select[2 * Range[10^6], (sodd = (s = DivisorSigma[1, #])/(2^(IntegerExponent[#, 2]+1) - 1)) == EulerPhi[s - sodd] &] (* Amiram Eldar, Aug 12 2023 *)

isok(n) = eulerphi(sumdiv(n, d, d*((d % 2)==0))) == sumdiv(n, d, d*(d%2)); \\ Michel Marcus, Jan 28 2017

a(1) inserted by Amiram Eldar, Aug 12 2023

A360156 a(n) is the sum of the even unitary divisors of 2*n.

Original entry on oeis.org

2, 4, 8, 8, 12, 16, 16, 16, 20, 24, 24, 32, 28, 32, 48, 32, 36, 40, 40, 48, 64, 48, 48, 64, 52, 56, 56, 64, 60, 96, 64, 64, 96, 72, 96, 80, 76, 80, 112, 96, 84, 128, 88, 96, 120, 96, 96, 128, 100, 104, 144, 112, 108, 112, 144, 128, 160, 120, 120, 192, 124, 128

Offset: 1

Amiram Eldar, Jan 28 2023

a(n) is the unitary analog of A146076(2*n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Wang-Sun Formula in GL(Z/2kZ), INTEGERS, Vol. 23 (2023), #A37; arXiv preprint, arXiv:2207.14495 [math.NT], 2022.

Cf. A002117, A034448, A100008, A146076, A171977, A192066, A328258.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; a[n_] := Module[{e = IntegerExponent[n, 2]}, 2^(e + 1) * usigma[n/2^e]]; Array[a, 100]

usigma(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + 1)} ;
a(n) = {my(e = valuation(n, 2)); (1 << (e+1)) * usigma(n >> e); }

a(n) = Sum_{even d|(2*n), gcd(d, 2*n/d)=1} d.
a(n) = A034448(2*n) - A192066(2*n).
a(n) = A192066(2*n) - A328258(2*n).
a(n) = A171977(n) * A192066(n).
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (7*zeta(3)).
Dirichlet g.f. of b(n): (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(s+1)-2)/(2^(2*s)-2), where b(n) is the sum of the even unitary divisors of n: b(n) = a(n/2) if n is even and 0 otherwise.

A361879 Sum of even middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 4, 0, 6, 0, 4, 0, 0, 0, 6, 0, 0, 0, 8, 0, 6, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 6, 0, 8, 0, 0, 0, 16, 0, 0, 0, 8, 0, 6, 0, 0, 0, 10, 0, 14, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 12, 0, 0, 0, 8, 0, 10, 0, 0, 0, 0, 0, 20

Offset: 1

Omar E. Pol, Mar 27 2023

Sum of even divisors of n in the half-open interval [sqrt(n/2), sqrt(n*2)).

Also sum of even numbers in the n-th row of A299761.

			For n = 18 the middle divisor of 18 is [3]. There are no even middle divisors of 18 so a(18) = 0.
For n = 20 the middle divisors of 20 are [4, 5]. There is only one even middle divisor of 20 so a(20) = 4.
For n = 24 the middle divisors of 24 are [4, 6]. There are two even middle divisors of 24 so a(24) = 4 + 6 = 10.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^16 (ignoring zeros).

Cf. A000203, A067742, A071090, A071562, A146076, A299761, A299777, A303297, A358434, A361561, A361824.

f:= proc(n) local D;
     if n::odd then return 0 fi;
     D:= select(proc(d) local s; if d::odd then return false fi; s:= d^2; s >= n/2 and s < 2*n end proc, numtheory:-divisors(n)); convert(D,`+`) end proc:
map(f, [$1..100]); # Robert Israel, Mar 18 2024

Table[DivisorSum[n, # &, And[EvenQ[#], Sqrt[n/2] <= # < Sqrt[2 n]] &], {n, 120}] (* Michael De Vlieger, Mar 28 2023 *)

a(n) = vecsum(select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2)) && !(x%2)), divisors(n))); \\ Michel Marcus, Mar 31 2023

a(n) = A071090(n) - A361824(n).

A193388 Sum of even divisors of phi(n).

Original entry on oeis.org

0, 0, 2, 2, 6, 2, 8, 6, 8, 6, 12, 6, 24, 8, 14, 14, 30, 8, 26, 14, 24, 12, 24, 14, 36, 24, 26, 24, 48, 14, 48, 30, 36, 30, 56, 24, 78, 26, 56, 30, 84, 24, 64, 36, 56, 24, 48, 30, 64, 36, 62, 56, 84, 26, 84, 56, 78, 48, 60, 30, 144, 48, 78, 62, 120, 36, 96

Offset: 1

Michel Lagneau, Jul 25 2011

nonn

			a(13) = 24 because phi(13) = 12 and the sum of the 4 even divisors { 2, 4, 6, 12} is 24.

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

Cf. A000010, A062402, A146076, A193454.

Cf. also A193322, A193386.

Table[Total[Select[Divisors[EulerPhi[n]], EvenQ[ # ]&]], {n, 58}]

A193388(n) = { my(s=eulerphi(n)); sumdiv(s,d,(!(d%2))*d); }; \\ Antti Karttunen, Dec 05 2017

a(n) = A146076(A000010(n)) = A062402(n) - A193454(n). - Antti Karttunen, Dec 05 2017

A193526 Sum of even divisors of sopf(n).

Original entry on oeis.org

0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 14, 2, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 2, 16, 0, 24, 0, 0, 0, 30, 0, 0, 24, 0, 0, 14, 0, 0, 0, 0, 0, 36, 0, 0, 0, 30, 0, 24, 0, 0, 12, 0, 0, 12, 2, 26, 30, 0, 0, 28, 16, 0, 0, 0, 0, 14, 0, 26, 26, 0, 0

Offset: 1

Michel Lagneau, Jul 29 2011

Sopf(n) is the sum of the distinct primes dividing n (A008472).

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008472, A146076, A193525.

Table[Total[Select[Divisors[Plus @@ First[Transpose[FactorInteger[n]]]], EvenQ[#] &]], {n, 100}]

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472
A146076(n) = if (n%2, 0, 2*sigma(n/2)); \\ From A146076
A193526(n) = if(1==n,0,A146076(A008472(n))); \\ Antti Karttunen, Dec 23 2018

a(1) = 0; for n > 1, a(n) = A146076(A008472(n)). - Antti Karttunen, Dec 23 2018

A255891 Numbers n such that the sum of the even divisors of n is equal to m! and the sum of the odd divisors of n is equal to k! for some integers m and k.

Original entry on oeis.org

2, 4, 240, 348, 368, 380, 19364665320, 20210069880, 20328267960, 20673770040, 20681420760, 20735165880, 20940748920, 20959618680, 21135474360, 21196014840, 21256222680, 21302746920, 21380630040, 21405023640, 21426252120, 21465896760, 21522002040, 21544621560

Offset: 1

Michel Lagneau, Mar 09 2015

nonn

Numbers n such that A000593(n) = m! and A146076(n) = k! for some m and k.
Is this sequence finite? No further terms less than 10^6.
No further terms less than 10^9. - Michel Marcus, Mar 10 2015
sigma(a(25711)) >= 29! + 30!. - Hiroaki Yamanouchi, Mar 26 2015

			240 is in the sequence because A000593(240)= 24 = 4! and A146076(240)= 720 = 6!

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..25710

Cf. A000593, A146076, A245015.

for n from 2 by 2  to 20000 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 20 while(ii=0)do:
         if s0=a!
          then
           for b from 1 to 20 while(ii=0) do:
             if s1=b!
              then
              ii:=1:print(n):
              else
             fi:
           od:
          fi:
        od:
      od:

fQ[n_] := Block[{d = Divisors@ n, lst = Array[Factorial, {449}]}, MemberQ[lst, Plus @@ Select[d, EvenQ]] && MemberQ[lst, Plus @@ Select[d, OddQ]]]; Select[Range@10000, fQ] (* Michael De Vlieger, Mar 10 2015 *)

isoks(s) = {if (s==1, return (1)); f = 1; for (k=2, s, f *= k; if (f == s, return (1)); if (f > s, return (0)););}
isok(n) = my(sod = sumdiv(n, d, d*(d%2))); my(sed = sigma(n) - sod); sod && sed && isoks(sed) && isoks(sod); \\ Michel Marcus, Mar 10 2015

a(7)-a(24) from Hiroaki Yamanouchi, Mar 26 2015

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

Original entry on oeis.org

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

A293356 Even integers k such that lambda(sum of even divisors of k) = sum of odd divisors of k.

Original entry on oeis.org

2, 20, 40, 48, 68, 176, 212, 304, 328, 944, 1360, 1712, 1888, 2320, 2344, 2864, 4240, 7120, 7888, 7984, 8448, 8960, 11920, 12032, 14416, 14592, 15536, 17492, 20224, 21520, 23984, 24208, 24592, 25904, 26112, 28160, 29440, 30464, 34560, 35920, 36352, 40528, 41296

Offset: 1

Michel Lagneau, Oct 07 2017

nonn

Or even integers k such that A002322(A146076(k)) = A000593(k).
Observations:
The primes a(n)/4: {5, 17, 53, 4373, 13121, ...} are of the form 2*3^m - 1, m > 0 (A079363).
The primes a(n)/8: {5, 41, 293, 4941257, ...} are of the form 6*7^m - 1, m = 0, 1, ... (primes in A198688).
The set of the primes {a(n)/16} = {3, 11, 19, 59, 107, 179, 499, 971, 1499, 1619, ...} contains the primes of the form 4*3^(2m+1) - 1 = {11, 107, 971, ...}, m = 0, 1, ...

			68 is in the sequence because A002322(A146076(68)) = A002322(108) = 18 and A000593(68) = 18.

Robert Israel, Table of n, a(n) for n = 1..738

Cf. A000593, A000668, A002322, A146076, A281707.

with(numtheory):
for n from 2 by 2 to 10^6 do:
x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
   for k from 1 to n1 do:
    if type(x[k],even)
     then
     s0:=s0+ x[k]:
     else
     s1:=s1+ x[k]:
    fi:
  od:
    if s1=lambda(s0)
     then
     printf(`%d, `,n):
     else
    fi:
od:

fQ[n_] :=
Block[{d = Divisors@n},
  CarmichaelLambda[Plus @@ Select[d, EvenQ]] ==
Plus @@ Select[d, OddQ]]; Select[2 Range@2000, fQ] (* Robert G. Wilson v, Oct 07 2017 *)

is(n)=if(n%2, return(0)); my(s=valuation(n,2),d=sigma(n>>s)); lcm(znstar(d*(2^(s+1)-2))[2])==d \\ Charles R Greathouse IV, Dec 26 2017

Edited by Robert Israel, Dec 28 2017

cp's OEIS Frontend

A193350 Sum of even divisors of tau(n).

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A266537 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with 2*k-1 zeros, and the first positive element of column k is in the row A002378(k), with T(1,1) = 0.

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A281707 Even integers k such that phi(sum of even divisors of k) = sum of odd divisors of k.

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A360156 a(n) is the sum of the even unitary divisors of 2*n.

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A361879 Sum of even middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

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A193388 Sum of even divisors of phi(n).

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A193526 Sum of even divisors of sopf(n).

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A255891 Numbers n such that the sum of the even divisors of n is equal to m! and the sum of the odd divisors of n is equal to k! for some integers m and k.