A146076 -id:A146076 - cp's OEIS frontend

A206436 Total sum of even parts in the last section of the set of partitions of n.

0, 2, 0, 8, 2, 18, 10, 42, 28, 80, 70, 162, 148, 290, 300, 530, 562, 918, 1020, 1570, 1780, 2602, 3022, 4286, 4992, 6858, 8110, 10872, 12888, 16962, 20178, 26134, 31138, 39728, 47412, 59848, 71312, 89072, 106176, 131440, 156400, 192164, 228330, 278616, 330502

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

Also total sum of even parts in the partitions of n that do not contain 1 as a part.
From Omar E. Pol, Apr 09 2023: (Start)
Convolution of A002865 and A146076.
a(n) is also the total sum of even divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the sum of even terms in the n-th row of the triangle A207378.
a(n) is also the sum of even terms in the n-th row of the triangle A336812. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Partial sums give A066966.

Cf. A002865, A135010, A138121, A138879, A146076, A206433, A206434, A206435, A207378, A336811, A336812.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2] +((i+1) mod 2)*h[1]*i]
      fi
    end:
a:= n-> b(n, n)[2] -`if`(n=1, 0, b(n-1, n-1)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 16 2012

b[n_, i_] := b[n, i] = Module[{g, h}, Which[n == 0, {1, 0}, i < 1, {0, 0}, True, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + Mod[i+1, 2]*h[[1]]*i}]]; a[n_] := b[n, n][[2]] - If[n == 1, 0, b[n-1, n-1][[2]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

G.f.: (Sum_{i>0} 2*i*x^(2*i)*(1-x)/(1-x^(2*i))) / Product_{i>0} (1-x^i). - Alois P. Heinz, Mar 16 2012

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2*n)). - Vaclav Kotesovec, May 29 2018

More terms from Alois P. Heinz, Mar 16 2012

A271342 Sum of all even divisors of all positive integers <= n.

Original entry on oeis.org

0, 2, 2, 8, 8, 16, 16, 30, 30, 42, 42, 66, 66, 82, 82, 112, 112, 138, 138, 174, 174, 198, 198, 254, 254, 282, 282, 330, 330, 378, 378, 440, 440, 476, 476, 554, 554, 594, 594, 678, 678, 742, 742, 814, 814, 862, 862, 982, 982, 1044, 1044, 1128, 1128, 1208, 1208, 1320, 1320, 1380, 1380, 1524, 1524, 1588, 1588, 1714, 1714

Offset: 1

Omar E. Pol, Apr 08 2016

a(n) is also the sum of all even divisors of all even positive integers <= n.
a(n) is also the total number of parts in all partitions of all positive integers <= n into an even number of equal parts. - Omar E. Pol, Jun 04 2017
The bisection of this sequence equals twice A024916 (see formulas). - Michel Marcus, Dec 14 2017

			For n = 6 the divisors of all positive integers <= 6 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6] and the even divisors of all positive integers <= 6 are [2], [2, 4], [2, 6], so a(6) = 2 + 2 + 4 + 2 + 6 = 16. On the other hand the sum of all the divisors of all positive integers <= 6/2 are [1] + [1 + 2] + [1 + 3] = A024916(3) = 8, so a(6) = 2*8 = 16.
For n = 10, (floor(10/2) = 5) numbers have divisor 2, (floor(10/4) = 2) numbers have divisor 4, ..., (floor(10/10) = 1) numbers have divisor 10. Therefore, a(10) = 5 * 2 + 2 * 4 + 1 * 6 + 1 * 8 + 1 * 10 = 42. - _David A. Corneth_, Jun 06 2017

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000203, A006128, A024916, A078471, A146076, A222171, A271343.

Partial sums of A146076.

Accumulate@ Array[DivisorSum[#, # &, EvenQ] &, 65] (* Michael De Vlieger, Jun 06 2017 *)

a(n) = sum(k=1, n, sumdiv(k, d, (1-d%2)*d)); \\ Michel Marcus, Jun 05 2017

a(n) = 2 * sum(k=1, n\2, k*(n\(k<<1))) \\ David A. Corneth, Jun 06 2017

def A271342(n): return sum(k*((n>>1)//k) for k in range(1, (n>>1)+1))<<1 # Chai Wah Wu, Apr 26 2023

from math import isqrt
def A271342(n): return -(s:=isqrt(m:=n>>1))**2*(s+1) + sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

a(1) = 0.
a(n) = 2*A024916((n-1)/2), if n is odd and n > 1.
a(n) = 2*A024916(n/2), if n is even.
a(n) = A024916(n) - A078471(n).
For n > 1, a(2*n + 1) = a(2*n). - David A. Corneth, Jun 06 2017
a(n) = c * n^2 + O(n*log(n)), where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 27 2023

A319998 a(n) = Sum_{d|n, d is even} mu(n/d)*d, where mu(n) is Moebius function A008683.

Original entry on oeis.org

0, 2, 0, 2, 0, 4, 0, 4, 0, 8, 0, 4, 0, 12, 0, 8, 0, 12, 0, 8, 0, 20, 0, 8, 0, 24, 0, 12, 0, 16, 0, 16, 0, 32, 0, 12, 0, 36, 0, 16, 0, 24, 0, 20, 0, 44, 0, 16, 0, 40, 0, 24, 0, 36, 0, 24, 0, 56, 0, 16, 0, 60, 0, 32, 0, 40, 0, 32, 0, 48, 0, 24, 0, 72, 0, 36, 0, 48, 0, 32, 0, 80, 0, 24, 0, 84, 0, 40, 0, 48, 0, 44, 0, 92, 0, 32, 0, 84, 0, 40, 0, 64, 0, 48, 0

Offset: 1

Antti Karttunen, Oct 31 2018

nonn

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

Cf. A000010, A008683, A059841, A140434, A146076, A319997.

Rest[CoefficientList[Series[Sum[2*MoebiusMu[k]*x^(2*k)/(1 - x^(2*k))^2, {k, 1, 100}], {x, 0, 100}], x]] (* Vaclav Kotesovec, Nov 03 2018 *)

A319998(n) = sumdiv(n,d,(!(d%2))*moebius(n/d)*d);

A319998(n) = if(n%2, 0, 2*eulerphi(n/2));

a(n) = Sum_{d|n} A059841(d)*A008683(n/d)*d.
a(n) = A000010(n) - A319997(n).
a(2n) = 2*A000010(n), a(2n+1) = 0.
G.f.: Sum_{k>=1} 2*mu(k)*x^(2*k)/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Nov 02 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/(2*Pi^2) = 0.151981... . - Amiram Eldar, Nov 12 2022

A323238 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291750(n) for all n, except for odd numbers n > 1, f(n) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 23, 3, 24, 3, 17, 3, 25, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 23, 3, 32, 3, 33, 3, 34, 3, 33, 3, 35, 3, 36, 3, 37, 3, 38, 3, 39, 3, 40, 3, 41, 3, 42, 3, 43, 3, 44, 3, 31, 3, 33, 3, 45, 3, 46, 3, 47, 3, 48, 3, 49, 3

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  a(i) = a(j) => A146076(i) = A146076(j),
  a(i) = a(j) => A319697(i) = A319697(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A048250, A146076, A291750, A291751, A319698, A319697, A319701, A322588, A323237, A323241.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
Aux323238(n) = if((n>1)&&(n%2),0,(1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n)));
v323238 = rgs_transform(vector(up_to, n, Aux323238(n)));
A323238(n) = v323238[n];

A271343 Triangle read by rows: T(n,k) = A196020(n,k) - A266537(n,k), n>=1, k>=1.

Original entry on oeis.org

1, 1, 5, 1, 1, 0, 9, 3, 1, -2, 1, 13, 5, 0, 1, 0, 0, 17, 7, 3, 1, -6, 0, 1, 21, 9, 0, 0, 1, 0, 3, 0, 25, 11, 0, 0, 1, -10, 0, 3, 29, 13, 7, 0, 1, 1, 0, 0, 0, 0, 33, 15, 0, 0, 0, 1, -14, 3, 5, 0, 37, 17, 0, 0, 0, 1, 0, 0, -2, 3, 41, 19, 11, 0, 0, 1, 1, -18, 0, 7, 0, 0, 45, 21, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0

Offset: 1

Omar E. Pol, Apr 06 2016

Gives an identity for A000593. Alternating sum of row n equals the sum of odd divisors of n, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A000593(n).
Row n has length A003056(n) hence the column k starts in row A000217(k).
Since the odd-indexed rows of the triangle A266537 contain all zeros then odd-indexed rows of this triangle are the same as the odd-indexed rows of the triangle A196020.
If T(n,k) is the second odd number in the column k then T(n+1,k+1) = 1 is the first element in the column k+1.
Alternating row sums of A196020 give A000203.
Alternating row sums of A266537 give A146076.

			Triangle begins:
1;
1;
5,   1;
1,   0;
9,   3;
1,  -2,  1;
13,  5,  0;
1,   0,  0;
17,  7,  3;
1,  -6,  0,  1;
21,  9,  0,  0;
1,   0,  3,  0;
25, 11,  0,  0;
1, -10,  0,  3;
29, 13,  7,  0,  1;
1,   0,  0,  0,  0;
33, 15,  0,  0,  0;
1, -14,  3,  5,  0;
37, 17,  0,  0,  0;
1,   0,  0, -2,  3;
41, 19, 11,  0,  0,  1;
1, -18,  0,  7,  0,  0;
45, 21,  0,  0,  0,  0;
1,   0,  3,  0,  0,  0;
49, 23,  0,  0,  5,  0;
1, -22,  0,  9,  0,  0;
53, 25, 15,  0,  0,  3;
1,   0,  0, -6,  0,  0,  1;
...
For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18 and the sum of odd divisors of 18 is 1 + 3 + 9 = 13. On the other hand, the 18th row of the triangle is 1, -14, 3, 5, 0, so the alternating row sum is 1 -(-14) + 3 - 5 + 0 = 13, equaling the sum of odd divisors of 18.

Column 1 is A266072.

Cf. A000217, A000593, A001227, A003056, A146076, A196020, A236104, A237593, A261699, A266537.

A319697 Sum of even squarefree divisors of n.

Original entry on oeis.org

0, 2, 0, 2, 0, 8, 0, 2, 0, 12, 0, 8, 0, 16, 0, 2, 0, 8, 0, 12, 0, 24, 0, 8, 0, 28, 0, 16, 0, 48, 0, 2, 0, 36, 0, 8, 0, 40, 0, 12, 0, 64, 0, 24, 0, 48, 0, 8, 0, 12, 0, 28, 0, 8, 0, 16, 0, 60, 0, 48, 0, 64, 0, 2, 0, 96, 0, 36, 0, 96, 0, 8, 0, 76, 0, 40, 0, 112, 0, 12, 0, 84, 0, 64, 0, 88, 0, 24, 0, 48, 0, 48, 0, 96

Offset: 1

Antti Karttunen, Oct 31 2018

nonn

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

Cf. A048250, A146076, A183063, A206787, A319698.

Table[Total[Select[Divisors[n],EvenQ[#]&&SquareFreeQ[#]&]],{n,100}] (* Harvey P. Dale, May 18 2019 *)
f[2, e_] := 2; f[p_, e_] := p + 1; a[n_] := If[OddQ[n], 0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);

a(n) = Sum_{d|n} A059841(d)*A008966(d)*d.

a(n) = A048250(n) - A206787(n).

A193322 Sum of even divisors of lambda(n).

Original entry on oeis.org

0, 0, 2, 2, 6, 2, 8, 2, 8, 6, 12, 2, 24, 8, 6, 6, 30, 8, 26, 6, 8, 12, 24, 2, 36, 24, 26, 8, 48, 6, 48, 14, 12, 30, 24, 8, 78, 26, 24, 6, 84, 8, 64, 12, 24, 24, 48, 6, 64, 36, 30, 24, 84, 26, 36, 8, 26, 48, 60, 6, 144, 48, 8, 30, 24, 12, 96, 30, 24, 24, 96, 8, 182, 78, 36, 26, 48, 24, 112, 6, 80, 84, 84, 8, 30, 64

Offset: 1

Michel Lagneau, Jul 22 2011

nonn

Lambda is the function in A002322.

			a(17) = 30 because lambda(17) = 16 and the sum of the 4 even divisors { 2, 4, 8, 16} is 30.

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

Cf. A002322, A146076, A191613.

Table[Total[Select[Divisors[CarmichaelLambda[n]], EvenQ[ # ]&]], {n, 62}]
(* Second program: *)
Array[DivisorSum[CarmichaelLambda@ #, # &, EvenQ] &, 86] (* Michael De Vlieger, Dec 04 2017 *)

a(n) = sumdiv(lcm(znstar(n)[2]), d, d*(1-(d%2))); \\ Michel Marcus, Mar 18 2016

a(n) = A146076(A002322(n)). - Michel Marcus, Mar 18 2016

More terms from Antti Karttunen, Dec 04 2017

A193511 a(n) = Sum of even divisors of Omega(n), a(1) = 0.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 2, 6, 0, 0, 0, 0, 2, 2, 0, 6, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 6, 0, 2, 2, 6, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 6, 2, 6, 2, 2, 0, 6, 0, 2, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0

Offset: 1

Michel Lagneau, Jul 29 2011

nonn

Omega(n) = number of prime divisors of n counted with multiplicity : A001222 (also called bigomega(n)).

a(1) = 0 by convention.

			a(16) = 6 because Omega(16) = 4 and the sum of the even divisors of 4 {2, 4} is 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001222, A146076, A193510, A193512, A290080.

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

A146076(n) = if(n%2, 0, 2*sigma(n/2)); \\ This function from Michel Marcus, Apr 01 2015
A193511(n) = if(1==n,0,A146076(bigomega(n))); \\ Antti Karttunen, Jul 23 2017

From Antti Karttunen, Jul 23 2017: (Start)
a(1) = 0, for n > 1, a(n) = A146076(A001222(n)).
a(n) + A193512(n) = A290080(n).
(End)

Description clarified by Antti Karttunen, Jul 23 2017

A194771 Even numbers that divide the sum of their even divisors.

Original entry on oeis.org

2, 12, 56, 240, 992, 1344, 16256, 60480, 65520, 1047552, 4357080, 47139840, 67100672, 91065600, 285981696, 919636480, 2758909440, 2952609792, 17179738112, 28364878080, 63996791040, 87722956800, 102002360320, 132867440640, 137438691328

Offset: 1

Michel Lagneau, Sep 02 2011

nonn

Since the sum of even divisors of an odd number is zero, every odd number divides its sum of even divisors. - Nathaniel Johnston, Sep 02 2011

			The divisors of 56 are { 1, 2, 4, 7, 8, 14, 28, 56 } and the sum of the even divisors is 2 + 4 + 8 + 14 + 28 + 56 = 112, hence 56 divides 112, so 56 is in the sequence.

Cf. A146076.

with(numtheory):for n from 1 to 10000000 do if(sigma(n) mod n = 0)then print(2*n):fi:od:

a(n) = 2*A007691(n).

a(11)-a(25) from Nathaniel Johnston, Sep 02 2011

A193336 Sum of even divisors of sigma(n).

Original entry on oeis.org

0, 0, 6, 0, 8, 24, 14, 0, 0, 26, 24, 48, 16, 56, 56, 0, 26, 0, 36, 64, 62, 78, 56, 144, 0, 64, 84, 112, 48, 182, 62, 0, 120, 80, 120, 0, 40, 144, 112, 156, 64, 248, 72, 192, 112, 182, 120, 192, 0, 0, 182, 114, 80, 336, 182, 336, 180, 156, 144, 448, 64, 248, 196, 0, 192, 390, 108, 208, 248, 390, 182, 0, 76, 160, 192, 288, 248, 448, 180, 256, 0

Offset: 1

Michel Lagneau, Jul 23 2011

nonn

sigma(n) = sum of divisors of n: A000203 (also called sigma_1(n)).

			a(14) = 56 because sigma(14) = 24 and the sum of the 6 even divisors {2, 4, 6, 8, 12, 24} is 56.

Cf. A000203, A028982, A051027, A146076, A193334, A193337.

Table[Total[Select[Divisors[DivisorSigma[1,n]], EvenQ[ # ]&]], {n, 53}]

A193336(n) = { my(s=sigma(n)); sumdiv(s,d,(!(d%2))*d); }; \\ Antti Karttunen, Nov 18 2017

a(n) + A193337(n) = A051027(n). - Antti Karttunen, Nov 18 2017
From Amiram Eldar, Mar 30 2024: (Start)
a(n) = A146076(A000203(n)).
a(n) = 0 if and only if n is in A028982. (End)

More terms from Antti Karttunen, Nov 18 2017

cp's OEIS Frontend

A206436 Total sum of even parts in the last section of the set of partitions of n.

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A271342 Sum of all even divisors of all positive integers <= n.

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A319998 a(n) = Sum_{d|n, d is even} mu(n/d)*d, where mu(n) is Moebius function A008683.

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A323238 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291750(n) for all n, except for odd numbers n > 1, f(n) = 0.

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A271343 Triangle read by rows: T(n,k) = A196020(n,k) - A266537(n,k), n>=1, k>=1.

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A319697 Sum of even squarefree divisors of n.

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A193322 Sum of even divisors of lambda(n).

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A193511 a(n) = Sum of even divisors of Omega(n), a(1) = 0.

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