A069733 -id:A069733 - cp's OEIS frontend

A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 0

Jaume Oliver Lafont, Oct 15 2009

			G.f. = x + x^2 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^13 + x^14 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Characteristic function of A042968, whose complement A008586 gives the positions of zeros (after its initial term).
Absolute values of A046978, A075553, A131729, A358839, and for n >= 1, also of A112299 and of A257196.
Sequence A152822 shifted by two terms.
Parity of A327860, A331733, A342002, A351083 and A345000.
Row 3 of A225145, Column 2 of A229940 (after the initial term).
First differences of A057353. Sum of A359370 and A359372.
Cf. A000035, A011655, A011558, A097325, A109720, A168181, A168182, A168184, A145568, A168185 (characteristic functions for numbers that are not multiples of k = 2, 3 and 5..12).
Cf. A016628, A164985, A165132.
Cf. A010873, A033436, A069733 (inverse Möbius transform), A121262 (one's complement), A190621 [= n*a(n)], A355689 (Dirichlet inverse).

[Ceiling(n/4)-Floor(n/4) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

seq(1/2*((n^3+n) mod 4), n=0..50); # Gary Detlefs, Mar 20 2010

PadRight[{},120,{0,1,1,1}] (* Harvey P. Dale, Jul 04 2013 *)
Table[Ceiling[n/4] - Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)
a[ n_] := Sign[ Mod[n, 4]]; (* Michael Somos, May 05 2015 *)

PARI
```
{a(n) = !!(n%4)};
```

def A166486(n): return (0,1,1,1)[n&3] # Chai Wah Wu, Jan 03 2023

G.f.: (x + x^2 + x^3) / (1 - x^4) = x * (1 + x + x^2) / ((1 - x) * (1 + x) * (1 + x^2)) = x * (1 - x^3) / ((1 - x) * (1 - x^4)).
a(n) = (3 - i^n - (-i)^n - (-1)^n) / 4, where i=sqrt(-1).
Sum_{k>0} a(k)/(k*3^k) = log(5)/4.
From Reinhard Zumkeller, Nov 30 2009: (Start)
Multiplicative with a(p^e) = (if p=2 then 0^(e-1) else 1), p prime and e>0.
a(n) = 1-A121262(n).
a(A042968(n))=1; a(A008586(n))=0.
A033436(n) = Sum{k=0..n} a(k)*(n-k). (End)
a(n) = 1/2*((n^3+n) mod 4). - Gary Detlefs, Mar 20 2010
a(n) = (Fibonacci(n)*Fibonacci(3n) mod 3)/2. - Gary Detlefs Dec 21 2010
Euler transform of length 4 sequence [ 1, 0, -1, 1]. - Michael Somos, Feb 12 2011
Dirichlet g.f. (1-1/4^s)*zeta(s). - R. J. Mathar, Feb 19 2011
a(n) = Fibonacci(n)^2 mod 3. - Gary Detlefs, May 16 2011
a(n) = -1/4*cos(Pi*n)-1/2*cos(1/2*Pi*n)+3/4. - Leonid Bedratyuk, May 13 2012
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = ceiling(n/4) - floor(n/4). - Wesley Ivan Hurt, Jun 20 2014
a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015
For n >= 1, a(n) = A053866(A225546(n)) = A000035(A331733(n)). - Antti Karttunen, Jul 07 2020
a(n) = signum(n mod 4). - Alois P. Heinz, May 12 2021
From Antti Karttunen, Dec 28 2022: (Start)
a(n) = [A010873(n) > 0], where [ ] is the Iverson bracket.
a(n) = abs(A046978(n)) = abs(A075553(n)) = abs(A131729(n)) = abs(A358839(n)).
For all n >= 1, a(n) = abs(A112299(n)) = abs(A257196(n))
a(n) = A152822(2+n).
a(n) = A359370(n) + A359372(n). (End)
E.g.f.: (cosh(x) - cos(x))/2 + sinh(x). - Stefano Spezia, Aug 04 2025

Secondary definition (from Reinhard Zumkeller's Nov 30 2009 comment) added to the name by Antti Karttunen, Dec 20 2022

A046897 Sum of divisors of n that are not divisible by 4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The o.g.f. is (theta_3(0,x)^4 - 1)/8, see the Hardy reference, eqs. 9.2.1, 9.2.3 and 9.2.4 on p. 133 for Sum' m*u_m. Also Hardy-Wright, p. 314. See also the Somos, Jan 25 2008 formula below. - Wolfdieter Lang, Dec 11 2016

			G.f. = q + 3*q^2 + 4*q^3 + 3*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 3*q^8 + 13*q^9 + ...

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 194.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island 2002, p. 133.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 314.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 31, Article 273.
C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall, 2006.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000203, A000118, A051731, A069733, A027748, A124010, A190621, A000593 (not divis. by 2), A046913 (not divis. by 3), A116073 (not divis. by 5).

a046897 1 = 1
a046897 n = product $ zipWith
            (\p e -> if p == 2 then 3 else div (p ^ (e + 1) - 1) (p - 1))
            (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Aug 12 2015

A := Basis( ModularForms( Gamma0(4), 2), 72); B := (A[1] - 1)/8 + A[2]; B; /* Michael Somos, Dec 30 2014 */

A046897 := proc(n) if n mod 4 = 0 then numtheory[sigma](n)-4*numtheory[sigma](n/4) ; else numtheory[sigma](n) ; end if; end proc: # R. J. Mathar, Mar 23 2011

a[n_] := Sum[ Boole[ !Divisible[d, 4]]*d, {d, Divisors[n]}]; Table[ a[n], {n, 1, 71}] (* Jean-François Alcover, Dec 12 2011 *)
DivisorSum[#1, # &, Mod[#, 4] != 0 &] & /@ Range[71] (* Jayanta Basu, Jun 30 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - 1) / 8, {q, 0, n}]; (* Michael Somos, Dec 30 2014 *)
f[2, e_] := 3; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

{a(n) = if( n<1, 0, sumdiv(n, d, if(d%4, d)))};

a(n) = (-1)^(n+1)*Sum_{d divides n} (-1)^(n/d+d)*d. Multiplicative with a(2^e) = 3, a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p. - Vladeta Jovovic, Sep 10 2002 [For a proof of the multiplicative property, see for example Moreno and Wagstaff, p. 33. - N. J. A. Sloane, Nov 09 2016]
G.f.: Sum_{k>0} x^k/(1+(-x)^k)^2, or Sum_{k>0} k*x^k/(1+(-x)^k). - Vladeta Jovovic, Dec 16 2002
Expansion of (1 - phi(q)^4) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jan 25 2008
Equals inverse Mobius transform of A190621. - Gary W. Adamson, Jul 03 2008
A000118(n) = 8*a(n) for all n>0.
Dirichlet g.f.: (1 - 4^(1-s)) * zeta(s) * zeta(s-1). - Michael Somos, Oct 21 2015
L.g.f.: log(Product_{k>=1} (1 - x^(4*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
From Peter Bala, Dec 19 2021: (Start)
Logarithmic g.f.: Sum_{n >= 1} a(n)*x^n/n = Sum_{n >= 1} x^n*(1 + x^n + x^(2*n))/( n*(1 - x^(4*n)) )
G.f.: Sum_{n >= 1} x^n*(x^(6*n) + 2*x^(5*n) + 3*x^(4*n) + 3*x^(2*n) + 2*x^n + 1)/(1 - x^(4*n))^2. (End)
Sum_{k=1..n} a(k) ~ (Pi^2/16) * n^2. - Amiram Eldar, Oct 04 2022

A069734 Number of pairs (p,q), 0<=p<=q, such that p+q divides n.

Original entry on oeis.org

1, 3, 3, 6, 4, 9, 5, 11, 8, 12, 7, 19, 8, 15, 14, 20, 10, 24, 11, 26, 18, 21, 13, 37, 17, 24, 22, 33, 16, 42, 17, 37, 26, 30, 26, 53, 20, 33, 30, 52, 22, 54, 23, 47, 42, 39, 25, 71, 30, 51, 38, 54, 28, 66, 38, 67, 42, 48, 31, 94, 32, 51, 55, 70, 44, 78, 35, 68, 50, 78, 37, 108

Offset: 1

Valery A. Liskovets, Apr 07 2002

Also number of orientable coverings of the Klein bottle with 2n lists (orientable m-list coverings exist only for even m).
Equals row sums of triangle A178650. - Gary W. Adamson, May 31 2010
Also number of inequivalent sublattices of index n of the rectangular lattice, that has the p2mm (pmm) symmetry group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - Andrey Zabolotskiy, Mar 12 2018

			There are 9 pairs (p,q), 0<=p<=q, such that p+q divides 6: (0,1), (0,2), (0,3), (0,6), (1,1), (1, 2), (1, 5), (2, 4), (3, 3); thus a(6) = 9.
x + 3*x^2 + 3*x^3 + 6*x^4 + 4*x^5 + 9*x^6 + 5*x^7 + 11*x^8 + 8*x^9 + ...

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
V. A. Liskovets and A. Mednykh, Number of non-orientable coverings of the Klein bottle
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2]. [From _N. J. A. Sloane_, Feb 23 2009]
Index entries for sequences related to sublattices

Cf. A069733, A069735, A046524, A178650.
Cf. A000203, A145391-A145394, A003051.
Cf. A304182, A304183, A008619, A007503, A183063.

with(numtheory): a := n -> (sigma(n) + tau(n) + `if`(irem(n,2) = 1, 0, tau(n/2)))/2: seq(a(n), n=1..72); # Peter Luschny, Jul 20 2019

a[n_] := (DivisorSigma[1, n] + DivisorSigma[0, n] + If[OddQ[n], 0, DivisorSigma[0, n/2]])/2;
Array[a, 72] (* Jean-François Alcover, Aug 27 2019, from Maple *)

{a(n) = if( n<1, 0, sum( k=1, n, sum( j=0, k, n%(j+k) == 0)))} /* Michael Somos, Mar 24 2012 */

a(n) = A046524(2n) - A069733(2n).
Inverse Moebius transform of: 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... G.f.: Sum_{n>0} x^n*(1+x^n-x^(2*n))/(1-x^(2*n))/(1-x^n). - Vladeta Jovovic, Feb 03 2003
a(n) = (A000203(n) + A069735(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A304182(n/m^2) + A304183(n/m^2) = A069735(n) + Sum_{ m: m^2|n } A304183(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A008619(d) = Sum_{ d|n } (1 + floor(d/2)). - Andrey Zabolotskiy, Jul 20 2019
a(n) = (A007503(n) + A183063(n))/2. - Peter Luschny, Jul 20 2019

New description from Vladeta Jovovic, Feb 03 2003

A069184 Sum of divisors d of n such that d or n/d is odd.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 13, 18, 12, 20, 14, 24, 24, 17, 18, 39, 20, 30, 32, 36, 24, 36, 31, 42, 40, 40, 30, 72, 32, 33, 48, 54, 48, 65, 38, 60, 56, 54, 42, 96, 44, 60, 78, 72, 48, 68, 57, 93, 72, 70, 54, 120, 72, 72, 80, 90, 60, 120, 62, 96, 104, 65, 84, 144, 68, 90, 96

Offset: 1

Vladeta Jovovic, Apr 10 2002

Might be called UnitaryOrdinarySigma(n): If n=Product p_i^r_i then UOSigma(n)=UnitarySigma(2^r_1)*Sigma(n/2^r_1)=(2^r_1+1)*Product (p_i^(r_i+1)-1)/(p_i-1), p_i is not 2. - Yasutoshi Kohmoto, Jun 11 2005

			UOSigma(2^4*7^2) = UnitarySigma(2^4)*sigma(7^2) = 17*57 = 969.

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

Cf. A069733, A107749, A092356.

A069184 := proc(n) local a,f,p,e; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; if p = 2 then a := a*(2^e+1) ; else a := a*(p^(e+1)-1)/(p-1) ; end if; end do; a ; end proc: # R. J. Mathar, Jun 02 2011

Table[ Sum[ d*Boole[ OddQ[d] || OddQ[n/d] ], {d, Divisors[n]}], {n, 1, 69}] (* Jean-François Alcover, Mar 26 2013 *)
f[2, e_] := 2^e+1; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n) = sumdiv(n, d, d*((d % 2) || ((n/d) % 2))); \\ Michel Marcus, Apr 10 2014

a(n)=my(e=valuation(n,2)); sigma(n>>e) * if(e,2^e+1,1) \\ Charles R Greathouse IV, Apr 10 2014

Multiplicative with a(2^e) = 2^e+1 and a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p.
G.f.: Sum_{m>0} m*x^m*(1+x^m+x^(2*m)-x^(3*m))/(1-x^(4*m)).
Dirichlet g.f.: zeta(s) *zeta(s-1) *(1-2^(1-2s)). - R. J. Mathar, Jun 02 2011 [simplified by Michael Shamos, May 14 2025]
Sum_{k=1..n} a(k) ~ 7*Pi^2*n^2 / 96. - Vaclav Kotesovec, Feb 08 2019

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A035207 Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = 25.

Original entry on oeis.org

1, 2, 2, 3, 1, 4, 2, 4, 3, 2, 2, 6, 2, 4, 2, 5, 2, 6, 2, 3, 4, 4, 2, 8, 1, 4, 4, 6, 2, 4, 2, 6, 4, 4, 2, 9, 2, 4, 4, 4, 2, 8, 2, 6, 3, 4, 2, 10, 3, 2, 4, 6, 2, 8, 2, 8, 4, 4, 2, 6, 2, 4, 6, 7, 2, 8, 2, 6, 4, 4, 2, 12, 2, 4, 2, 6, 4, 8, 2, 5, 5, 4, 2, 12, 2, 4, 4, 8, 2, 6, 4, 6, 4, 4, 2, 12, 2, 6, 6, 3, 2, 8, 2

Offset: 1

N. J. A. Sloane

Number of divisors of n not congruent to 0 mod 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A001620, A035191, A069733.

Cf. A116073 (sum of divisors of n not congruent to 0 mod 5).

[NumberOfDivisors(n)/Valuation(5*n, 5): n in [1..100]]; // Vincenzo Librandi, Jun 03 2019

for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=5 then b := 1:else b := e+1:fi:s := s*b:od:printf(`%d,`,s); od:

Table[Count[Divisors[n],?(!Divisible[#,5]&)],{n,110}] (* _Harvey P. Dale, Apr 08 2015 *)
f[5, e_] := 1; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

{a(n)=if(n<1, 0, sumdiv(n, d, d%5>0))} /* Michael Somos, Oct 31 2006 */

{a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/if(p==5, 1, 1-X))[n])} /* Michael Somos, Oct 31 2006 */

Multiplicative with a(5^e)=1 and a(p^e)=e+1 for p<>5.
Moebius transform is period 5 sequence A011558. - Michael Somos, Oct 31 2006
G.f.: Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k))/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 11 2019
a(n) = tau(5*n) - tau(n). - Ridouane Oudra, Sep 05 2020
From Amiram Eldar, Nov 27 2022: (Start)
Dirichlet g.f.: zeta(s)^2 * (1 - 1/5^s).
Sum_{k=1..n} a(k) ~ (4*n*log(n) + (8*gamma + log(5) - 4)*n)/5, where gamma is Euler's constant (A001620). (End)

Additional comments from Vladeta Jovovic, Oct 26 2001

A046524 Number of coverings of Klein bottle with n lists.

Original entry on oeis.org

1, 3, 2, 5, 2, 7, 2, 8, 3, 8, 2, 13, 2, 9, 4, 13, 2, 14, 2, 16, 4, 11, 2, 23, 3, 12, 4, 19, 2, 22, 2, 22, 4, 14, 4, 30, 2, 15, 4, 30, 2, 26, 2, 25, 6, 17, 2, 41, 3, 23, 4, 28, 2, 30, 4, 37, 4, 20, 2, 50, 2, 21, 6, 39, 4, 34, 2, 34, 4, 34, 2, 59, 2, 24, 6, 37, 4, 38, 2, 56, 5, 26, 2, 62, 4, 27, 4

Offset: 1

Valery A. Liskovets

T. D. Noe, Table of n, a(n) for n=1..1000
V. A. Liskovets and A. Mednykh, Number of non-orientable coverings of the Klein bottle
A. D. Mednykh, On the number of subgroups in the fundamental group of a closed surface, Commun. in Algebra, 16, No 10 (1988), 2137-2148.

Cf. A027842, A027844, A000005, A000203, A069733, A069734, A069735.

with(numtheory); A046524:=n->`if`(type(n/2, integer), (3*tau(n) + sigma(n/2) - tau(n/2))/2, tau(n)); seq(A046524(n), n=1..100); # Wesley Ivan Hurt, Feb 14 2014

kb[n_]:=If[OddQ[n],DivisorSigma[0,n],(3DivisorSigma[0,n]+ DivisorSigma[ 1,n/2]- DivisorSigma[0,n/2])/2]; Array[kb,90] (* Harvey P. Dale, Oct 08 2011 *)

def A046524(n) :
    f = lambda n : 1 if n % 2 == 1 else (n+7)//4
    return add(f(d) for d in divisors(n))
[A046524(n) for n in (1..87)] # Peter Luschny, Jul 23 2012

a(n)=d(n) (the number of divisors) for odd n.
a(n)=[3d(n)+sigma(n/2)-d(n/2)]/2 for even n where d(n) is the number and sigma(n) the sum of divisors of n (A000005 and A000203).
Inverse Moebius transform of 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 5, 1, 5, 1, 6, 1, 6, 1, 7, 1, 7, ... . G.f.: Sum_{n>1} x^n*(1+2*x^n-x^(4*n)-x^(5*n))/(1+x^(2*n))/(1-x^(2*n))^2. - Vladeta Jovovic, Feb 03 2003

More terms from Vladeta Jovovic, Feb 03 2003

A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 3, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 3, 6, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 6, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A034444 at n = 25.

The sum of these divisors is A365211(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A007310, A035218, A065330, A065331, A069733, A365211.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p <= 3 , 2, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, 2, f[i,2]+1));}

Multiplicative with a(p^e) = 2 for p = 2 and 3, and a(p^e) = e+1 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) >= A034444(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) = A000005(A065330(n)) * A034444(A065331(n)).
Dirichlet g.f.: (1-1/4^s) * (1-1/9^s) * zeta(s)^2.
Sum_{k=1..n} a(k) ~ (2*n/3) * (log(n) + 2*gamma - 1 + 2*log(2)/3 + log(3)/4), where gamma is Euler's constant (A001620).

A076598 Sum of squares of divisors d of n such that d or n/d is odd.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 50, 65, 91, 130, 122, 170, 170, 250, 260, 257, 290, 455, 362, 442, 500, 610, 530, 650, 651, 850, 820, 850, 842, 1300, 962, 1025, 1220, 1450, 1300, 1547, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 2074, 2366, 2650, 2210, 2570, 2451, 3255

Offset: 1

Vladeta Jovovic, Oct 20 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A069733, A069184, A001157, A050999, A076577.

f[2, e_] := 4^e+1 ; f[p_, e_] := (p^(2*e+2)-1)/(p^2-1) ; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Aug 01 2019 *)

a(n) = sumdiv(n, d, if ((d % 2) || (n/d % 2), d^2)); \\ Michel Marcus, Oct 30 2022

Multiplicative with a(2^e) = 4^e+1, a(p^e) = (p^(2*e+2)-1)/(p^2-1) for an odd prime p.
G.f.: Sum_{m>0} m^2*x^m*(1+2*x^m+3*x^(2*m))/(1+x^(2*m))/(1+x^m).
More generally, if b(n, k) is sum of k-th powers of divisors d of n such that d or n/d is odd then b(n, k) = sigma_k(n)-2^k*sigma_k(n/4) if n mod 4=0, otherwise b(n, k) = sigma_k(n).
G.f. for b(n, k): Sum_{m>0} m^k*x^m*(1+x^m+x^(2*m)-(2^k-1)*x^(3*m))/(1-x^(4*m)). b(n, k) is multiplicative and b(2^e, k) = 2^(k*e)+1, b(p^e, k) = (p^(k*e+k)-1)/(p^k-1) for an odd prime p.
a(n) = sigma_2(n)-4*sigma_2(n/4) if n mod 4=0, otherwise a(n) = sigma_2(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 5*zeta(3)/16 = 0.375642... . - Amiram Eldar, Oct 30 2022

cp's OEIS Frontend

A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

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A046897 Sum of divisors of n that are not divisible by 4.

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A069734 Number of pairs (p,q), 0<=p<=q, such that p+q divides n.

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A069184 Sum of divisors d of n such that d or n/d is odd.

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A035207 Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = 25.

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A046524 Number of coverings of Klein bottle with n lists.

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A035207 Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = 25.