A050464 -id:A050464

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A050460

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

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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)

MAPLE

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:

end proc:

seq(A050460(n), n=1..40) ; # R. J. Mathar, Dec 20 2011

MATHEMATICA

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

PROG

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

CROSSREFS

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

Cf. A050461, A050462, A050463.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 23 1999

STATUS

approved

A050465

a(n) = Sum_{d|n, n/d=3 mod 4} d^2.

+10
7

0, 0, 1, 0, 0, 4, 1, 0, 9, 0, 1, 16, 0, 4, 26, 0, 0, 36, 1, 0, 58, 4, 1, 64, 0, 0, 82, 16, 0, 104, 1, 0, 130, 0, 26, 144, 0, 4, 170, 0, 0, 232, 1, 16, 234, 4, 1, 256, 49, 0, 290, 0, 0, 328, 26, 64, 370, 0, 1, 416, 0, 4, 523, 0, 0, 520, 1, 0, 538, 104, 1, 576, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A050461(n) - A050470(n). - Reinhard Zumkeller, Mar 06 2012

From Amiram Eldar, Nov 05 2023: (Start)

a(n) = A076577(n) - A050461(n).

a(n) = (A076577(n) - A050470(n))/2.

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 7*zeta(3)/16 - Pi^3/64 = 0.041426822002... . (End)

MATHEMATICA

a[n_] := DivisorSum[n, #^2 &, Mod[n/#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 05 2023 *)

PROG

(Haskell)

a050465 n = sum [d ^ 2 | d <- a027750_row n, mod (div n d) 4 == 3]

-- Reinhard Zumkeller, Mar 06 2012

(PARI) a(n) = sumdiv(n, d, (n/d % 4 == 3) * d^2); \\ Amiram Eldar, Nov 05 2023

CROSSREFS

Cf. A002117, A027750, A050461, A050470, A076577.

Cf. A050464, A050466, A050467.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 23 1999

EXTENSIONS

Offset fixed by Reinhard Zumkeller, Mar 06 2012

STATUS

approved

A285895

Sum of divisors d of n such that n/d is not congruent to 0 mod 4.

+10
6

1, 3, 4, 6, 6, 12, 8, 12, 13, 18, 12, 24, 14, 24, 24, 24, 18, 39, 20, 36, 32, 36, 24, 48, 31, 42, 40, 48, 30, 72, 32, 48, 48, 54, 48, 78, 38, 60, 56, 72, 42, 96, 44, 72, 78, 72, 48, 96, 57, 93, 72, 84, 54, 120, 72, 96, 80, 90, 60, 144, 62, 96, 104, 96, 84, 144, 68

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

a(n) = A050460(n) + A002131(n/2) + A050464(n), where A002131(.)=0 for non-integer argument. - R. J. Mathar, May 25 2020

From Amiram Eldar, Oct 30 2022: (Start)

Multiplicative with a(2^e) = 3*2^(e-1) and a(p^e) = (p^(e+1)-1)/(p-1) if p > 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 5*Pi^2/64 = 0.7710628... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/4^s). - Amiram Eldar, Dec 30 2022

EXAMPLE

The divisors of 8 are 1, 2, 4, and 8. 8/1 == 0 (mod 4) and 8/2 == 0 (mod 4). Hence, a(8) = 4 + 8 = 12.

MATHEMATICA

f[p_, e_] := If[p == 2, 3*2^(e-1), (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

PROG

(PARI) a(n)=sumdiv(n, d, if(n/d%4, d, 0)); \\ Andrew Howroyd, Jul 20 2018

CROSSREFS

Cf. A002131 (k=2), A078708 (k=3), this sequence (k=4), A285896 (k=5).

Cf. A002131, A050460, A050464.

KEYWORD

nonn,mult

AUTHOR

Seiichi Manyama, Apr 28 2017

STATUS

approved

A326400

Expansion of Sum_{k>=1} k * x^(2*k) / (1 - x^(3*k)).

+10
6

0, 1, 0, 2, 1, 3, 0, 5, 0, 7, 1, 6, 0, 8, 3, 10, 1, 9, 0, 15, 0, 13, 1, 15, 5, 14, 0, 16, 1, 21, 0, 21, 3, 19, 8, 18, 0, 20, 0, 35, 1, 24, 0, 27, 9, 25, 1, 30, 0, 36, 3, 28, 1, 27, 16, 40, 0, 31, 1, 45, 0, 32, 0, 42, 14, 39, 0, 39, 3, 56, 1, 45, 0, 38, 15, 40, 8, 42, 0, 71

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

FORMULA

a(n) = Sum_{d|n, n/d==2 (mod 3)} d.

G.f.: Sum_{k>0} x^(3*k-1) / (1 - x^(3*k-1))^2. - Seiichi Manyama, Jun 29 2023

MATHEMATICA

nmax = 80; CoefficientList[Series[Sum[k x^(2 k)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Table[DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 80}]

CROSSREFS

Cf. A001822, A002131, A016789, A078182, A078708, A326399, A326401.

Cf. A050464, A363900.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Sep 11 2019

STATUS

approved

A050466

a(n) = Sum_{d|n, n/d=3 mod 4} d^3.

+10
5

0, 0, 1, 0, 0, 8, 1, 0, 27, 0, 1, 64, 0, 8, 126, 0, 0, 216, 1, 0, 370, 8, 1, 512, 0, 0, 730, 64, 0, 1008, 1, 0, 1358, 0, 126, 1728, 0, 8, 2198, 0, 0, 2960, 1, 64, 3402, 8, 1, 4096, 343, 0, 4914, 0, 0, 5840, 126, 512, 6886, 0, 1, 8064, 0, 8, 9991, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

From Robert G. Wilson v, Mar 26 2015: (Start)

a(n) = 0 for n = 1, 2, 4, 5, 8, 10, 13, 16, 17, 20, 25, ... (A072437).

a(n) = 1 for n = 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, ... (A002145). (End)

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

FORMULA

From Amiram Eldar, Nov 05 2023: (Start)

a(n) = A007331(n) - A050462(n).

a(n) = A050462(n) - A050471(n).

a(n) = (A007331(n) - A050471(n))/2.

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = Pi^4/192 - A175572/2 = 0.0128667399315... . (End)

MATHEMATICA

a[n_] := Total[(n/Select[Divisors@ n, Mod[#, 4] == 3 &])^3]; Array[a, 64] (* Robert G. Wilson v, Mar 26 2015 *)

a[n_] := DivisorSum[n, #^3 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)

PROG

(PARI) a(n) = sumdiv(n, d, ((n/d % 4)== 3)* d^3); \\ Michel Marcus, Mar 26 2015

CROSSREFS

Cf. A007331, A050462, A050471, A175572.

Cf. A050464, A050465, A050467.

Cf. A002145, A072437.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 23 1999

EXTENSIONS

Offset changed from 0 to 1 by Robert G. Wilson v, Mar 27 2015

STATUS

approved

A050467

a(n) = Sum_{d|n, n/d=3 mod 4} d^4.

+10
5

0, 0, 1, 0, 0, 16, 1, 0, 81, 0, 1, 256, 0, 16, 626, 0, 0, 1296, 1, 0, 2482, 16, 1, 4096, 0, 0, 6562, 256, 0, 10016, 1, 0, 14722, 0, 626, 20736, 0, 16, 28562, 0, 0, 39712, 1, 256, 50706, 16, 1, 65536, 2401, 0, 83522, 0, 0, 104992, 626, 4096, 130402

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

FORMULA

From Amiram Eldar, Nov 05 2023: (Start)

a(n) = A285989(n) - A050463(n).

a(n) = A050463(n) - A050468(n).

a(n) = (A285989(n) - A050468(n))/2.

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 31*zeta(5)/64 - 5*Pi^5/3072 = 0.00418296735902... . (End)

MATHEMATICA

Table[Total[Select[Divisors[n], Mod[n/#, 4]==3&]^4], {n, 60}] (* Harvey P. Dale, Jun 10 2023 *)

a[n_] := DivisorSum[n, #^4 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)

PROG

(PARI) a(n) = sumdiv(n, d, (n/d % 4 == 3) * d^4); \\ Amiram Eldar, Nov 05 2023

CROSSREFS

Cf. A013663, A050463, A050468, A285989.

Cf. A050464, A050465, A050466.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 23 1999

EXTENSIONS

Offset corrected by Amiram Eldar, Nov 05 2023

STATUS

approved

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