[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a050464 -id:a050464
Displaying 1-6 of 6 results found.
page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A050460
a(n) = Sum_{d|n, n/d=1 mod 4} d.
+0
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:
a;
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.
+0
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
A050466
a(n) = Sum_{d|n, n/d=3 mod 4} d^3.
+0
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.
+0
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
A285895
Sum of divisors d of n such that n/d is not congruent to 0 mod 4.
+0
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)).
+0
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
page 1

Search completed in 0.004 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified November 30 14:52 EST 2024. Contains 378264 sequences.
License Agreements, Terms of Use, Privacy Policy