A050449 -id:A050449 - cp's OEIS frontend

A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

1, 2, 3, 4, 6, 6, 7, 8, 10, 12, 11, 12, 14, 14, 18, 16, 18, 20, 19, 24, 22, 22, 23, 24, 31, 28, 30, 28, 30, 36, 31, 32, 34, 36, 42, 40, 38, 38, 42, 48, 42, 44, 43, 44, 60, 46, 47, 48, 50, 62, 54, 56, 54, 60, 66, 56, 58, 60, 59, 72, 62, 62, 73, 64, 84, 68

Offset: 1

N. J. A. Sloane, Dec 23 1999

Not multiplicative: a(3)*a(7) <> a(21), for example.

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

Cf. A001826, A002131, A050449, A050464, A050469, A222183.

Cf. A050461, A050462, A050463.

A050460 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if (n/d) mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050460(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

a[n_] := DivisorSum[n, Boole[Mod[n/#, 4] == 1]*#&]; Array[a, 70] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=sumdiv(n,d,if(n/d%4==1,d)) \\ Charles R Greathouse IV, Dec 04 2013

G.f.: Sum_{n>0} n*x^n/(1-x^(4*n)). - Vladeta Jovovic, Nov 14 2002
G.f.: Sum_{k>0} x^(4*k-3) / (1 - x^(4*k-3))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050464(n).
a(n) = A050469(n) + A050464(n).
a(n) = (A002131(n) + A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A222183. (End)

A284098 a(n) = Sum_{d|n, d == 1 (mod 6)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 14, 8, 1, 1, 1, 1, 20, 1, 8, 1, 1, 1, 26, 14, 1, 8, 1, 1, 32, 1, 1, 1, 8, 1, 38, 20, 14, 1, 1, 8, 44, 1, 1, 1, 1, 1, 57, 26, 1, 14, 1, 1, 56, 8, 20, 1, 1, 1, 62, 32, 8, 1, 14, 1, 68, 1, 1, 8, 1, 1, 74, 38, 26, 20, 8, 14, 80, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A086729, A109701.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), this sequence (k=6), A284099 (k=7), A284100 (k=8).

Table[Sum[If[Mod[d, 6] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 6)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%6==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (6*k + 1)*x^(6*k+1)/(1 - x^(6*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 5*x^(6*n))/(1 - x^(6*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/72 = 0.137077... (A086729). - Amiram Eldar, Nov 26 2023

A284100 a(n) = Sum_{d|n, d == 1 (mod 8)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 18, 10, 1, 1, 1, 1, 1, 1, 26, 1, 10, 1, 1, 1, 1, 1, 34, 18, 1, 10, 1, 1, 1, 1, 42, 1, 1, 1, 10, 1, 1, 1, 50, 26, 18, 1, 1, 10, 1, 1, 58, 1, 1, 1, 1, 1, 10, 1, 66, 34, 1, 18, 1, 1, 1, 10, 74, 1, 26, 1, 1, 1, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A277090.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), this sequence (k=8).

Table[Sum[If[Mod[d, 8] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,8]==1&]],{n,80}] (* or *) Table[DivisorSum[n,#&,Mod[#,8]==1&],{n,80}] (* Harvey P. Dale, Mar 28 2020 *)

for(n=1, 80, print1(sumdiv(n, d, if(Mod(d, 8)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%8==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (8*k + 1)*x^(8*k+1)/(1 - x^(8*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 7*x^(8*n))/(1 - x^(8*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/96 = 0.102808... . - Amiram Eldar, Nov 26 2023

A284313 Expansion of Product_{k>=0} (1 - x^(4*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 3, -2, 0, -1, 3, -3, 1, -1, 4, -4, 1, -1, 4, -5, 2, -1, 5, -7, 3, -1, 5, -8, 5, -2, 6, -10, 6, -2, 6, -12, 9, -3, 7, -14, 11, -4, 7, -16, 15, -6, 8, -19, 18, -8, 9, -21, 23, -11, 10, -24

Offset: 0

Seiichi Manyama, Mar 24 2017

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), this sequence (m=4), A284314 (m=5).

Cf. A050449, A169975, A284316.

V:= Vector(100):
V[1]:= 1:
for k from 0 to 24 do
  V[4*k+2..100]:= V[4*k+2..100] - V[1..99-4*k]
od:
convert(V,list); # Robert Israel, May 03 2017

CoefficientList[Series[Product[1 - x^(4k + 1), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(4*k + 1)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A050449(k)*a(n-k), a(0) = 1.

O.g.f.: Sum_{n >= 0} (-1)^n*x^(n*(2*n-1)) / Product_{k = 1..n} ( 1 - x^(4*k) ). Cf. A284316. - Peter Bala, Nov 28 2020

A363903 Expansion of Sum_{k>0} x^k / (1 - x^(4*k))^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 5, 1, 3, 1, 6, 4, 1, 3, 7, 1, 1, 1, 10, 5, 4, 1, 9, 3, 1, 1, 10, 6, 3, 4, 11, 1, 5, 3, 12, 7, 1, 1, 18, 1, 1, 1, 14, 10, 6, 5, 15, 4, 3, 1, 16, 9, 1, 3, 17, 1, 10, 1, 24, 10, 1, 6, 19, 3, 1, 4, 20, 11, 10, 1, 21, 5, 1, 3, 25, 12, 1, 7, 30, 1, 9, 1, 24, 18, 5, 1, 25, 1, 3, 1

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363901, A363925.

Cf. A001826, A050449.

a[n_] := DivisorSum[n, # + 3 &, Mod[#, 4] == 1 &]/4; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%4==1)*(d+3))/4;
```

a(n) = (1/4) * Sum_{d|n, d==1 mod 4} (d+3) = (3 * A001826(n) + A050449(n))/4.

G.f.: Sum_{k>0} k * x^(4*k-3) / (1 - x^(4*k-3)).

A082052 Sum of divisors of n that are not of the form 4k+1.

Original entry on oeis.org

0, 2, 3, 6, 0, 11, 7, 14, 3, 12, 11, 27, 0, 23, 18, 30, 0, 29, 19, 36, 10, 35, 23, 59, 0, 28, 30, 55, 0, 66, 31, 62, 14, 36, 42, 81, 0, 59, 42, 84, 0, 74, 43, 83, 18, 71, 47, 123, 7, 62, 54, 84, 0, 110, 66, 119, 22, 60, 59, 162, 0, 95, 73, 126, 0, 110, 67, 108, 26, 138, 71

Offset: 1

Ralf Stephan, Apr 02 2003

a(A004613(n))=0.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000203, A050449, A050452, A050460, A078181, A078182, A082053.

sd[n_]:= Total[Select[Divisors[n], !IntegerQ[(# - 1) / 4]&]]; Array[sd, 100] (* Vincenzo Librandi, May 17 2013 *)
Table[DivisorSum[n,#&,(!IntegerQ[(#-1)/4]&)],{n,80}] (* Harvey P. Dale, Nov 30 2019 *)

for(n=1,100,print1(sumdiv(n,d,if(d%4!=1,d))","))

G.f.: Sum_{k>=1} x^(2*k)*(2 + 3*x^k + 4*x^(2*k) + 2*x^(4*k) + x^(5*k))/(1 - x^(4*k))^2. - Ilya Gutkovskiy, Sep 12 2019

A293901 Sum of proper divisors of n of the form 4k+1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 10, 1, 6, 1, 1, 1, 1, 6, 14, 10, 1, 1, 6, 1, 1, 1, 18, 6, 10, 1, 1, 14, 6, 1, 22, 1, 1, 15, 1, 1, 1, 1, 31, 18, 14, 1, 10, 6, 1, 1, 30, 1, 6, 1, 1, 31, 1, 19, 34, 1, 18, 1, 6, 1, 10, 1, 38, 31, 1, 1, 14, 1, 6, 10, 42, 1, 22, 23, 1, 30, 1, 1, 60, 14, 1, 1, 1, 6, 1, 1, 50, 43, 31, 1, 18, 1, 14, 27

Offset: 1

Antti Karttunen, Oct 19 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A050449, A091570, A293451, A293903.

a[n_] := DivisorSum[n, # &, # < n && Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 27 2023 *)

PARI
```
A293901(n) = sumdiv(n,d,(d
				
```

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

a(n) = A091570(n) - A293903(n).

G.f.: Sum_{k>=1} (4*k-3) * x^(8*k-6) / (1 - x^(4*k-3)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 - 1/8 = 0.0806167... . - Amiram Eldar, Nov 27 2023

A050448 a(n) = Sum_{d|n, d==1 (mod 4)} d^4.

Original entry on oeis.org

1, 1, 1, 1, 626, 1, 1, 1, 6562, 626, 1, 1, 28562, 1, 626, 1, 83522, 6562, 1, 626, 194482, 1, 1, 1, 391251, 28562, 6562, 1, 707282, 626, 1, 1, 1185922, 83522, 626, 6562, 1874162, 1, 28562, 626, 2825762, 194482, 1, 1, 4107812, 1, 1, 1

Offset: 1

N. J. A. Sloane, Dec 23 1999

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001159, A050449, A050450, A050451.

a[n_] := DivisorSum[n, #^4 &, Mod[#, 4] == 1 &]; Array[a, 50] (* Amiram Eldar, Jul 08 2023 *)

a(n) = sumdiv(n, d, if ((d%4)==1, d^4)); \\ Michel Marcus, Aug 16 2021

Offset corrected by Sean A. Irvine, Aug 15 2021

A082053 Sum of divisors of n that are not of the form 4k+3.

Original entry on oeis.org

1, 3, 1, 7, 6, 9, 1, 15, 10, 18, 1, 25, 14, 17, 6, 31, 18, 36, 1, 42, 22, 25, 1, 57, 31, 42, 10, 49, 30, 54, 1, 63, 34, 54, 6, 88, 38, 41, 14, 90, 42, 86, 1, 73, 60, 49, 1, 121, 50, 93, 18, 98, 54, 90, 6, 113, 58, 90, 1, 150, 62, 65, 31, 127, 84, 130, 1, 126, 70, 102, 1, 192

Offset: 1

Ralf Stephan, Apr 02 2003

a(A002145(n))=1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000203, A050449, A050452, A050460, A078181, A078182, A082052.

sd[n_]:= Total[Select[Divisors[n], !IntegerQ[(# - 3) / 4]&]]; Array[sd, 100] (* Vincenzo Librandi, May 17 2013 *)

for(n=1,100,print1(sumdiv(n,d,if(d%4!=3,d))","))

G.f.: Sum_{k>=1} x^k*(1 + 2*x^k + 4*x^(3*k) + 3*x^(4*k) + 2*x^(5*k))/(1 - x^(4*k))^2. - Ilya Gutkovskiy, Sep 12 2019

A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 35, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 46, 24, 1, 13, 1, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 68, 35, 24, 1, 1, 13, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 13, 1

Offset: 1

Seiichi Manyama, Jan 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8), this sequence (k=11).

Cf. A357911.

a[n_] := DivisorSum[n, # &, Mod[#, 11] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)

a(n) = sumdiv(n, d, (Mod(d, 11)==1)*d);

my(N=100, x='x+O('x^N)); Vec(sum(k=0, N, (11*k+1)*x^(11*k+1)/(1-x^(11*k+1))))

G.f.: Sum_{k>=0} (11*k+1) * x^(11*k+1)/(1 - x^(11*k+1)).

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cp's OEIS Frontend

A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

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A284098 a(n) = Sum_{d|n, d == 1 (mod 6)} d.

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A284100 a(n) = Sum_{d|n, d == 1 (mod 8)} d.

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A284313 Expansion of Product_{k>=0} (1 - x^(4*k+1)) in powers of x.

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A363903 Expansion of Sum_{k>0} x^k / (1 - x^(4*k))^2.

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A082052 Sum of divisors of n that are not of the form 4k+1.

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A293901 Sum of proper divisors of n of the form 4k+1.

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A050448 a(n) = Sum_{d|n, d==1 (mod 4)} d^4.

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