A001826 -id:A001826 - cp's OEIS frontend

A188171 The number of divisors d of n of the form d == 5 (mod 8).

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 2

Offset: 1

R. J. Mathar, Mar 23 2011

a(5n) >= 1 as d=5 contributes to the count.

			a(13) = 1 because the divisor d=13 is 8+5 == 5 (mod 8).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001826, A002325, A188169, A188170, A188172.

Cf. A001620, A016631, A354634 (psi(5/8)).

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A188171 := proc(n) sigmamr(n,8,5) ; end proc:

a[n_] := DivisorSum[n, 1 &, Mod[#, 8] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A188171(n) = sumdiv(n, d, (5==(d%8)));  \\ Antti Karttunen, Jul 09 2017

A188169(n)+a(n) = A001826(n).
A188169(n)+A188170(n)-a(n)-A188172(n) = A002325(n).
G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,8) - (1 - gamma)/8 = -0.131189..., gamma(5,8) = -(psi(5/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A293451 Number of proper divisors of n of the form 4k+1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 1, 2, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Oct 19 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001620, A001826, A091954, A121262, A256778, A293513, A293901.

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

PARI
```
A293451(n) = sumdiv(n,d,(d
				
```

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

a(n) = A091954(n) - A293513(n).

a(n) = A001826(n) - A121262(n-1).

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

Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,4) - (2 - gamma)/4 = A256778 - (2 - A001620)/4 = 0.354593... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A359227 Number of divisors of 4n-3 of form 4k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 22 2022

Cf. A001826, A078703.

a[n_] := DivisorSum[4*n-3, 1 &, Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

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

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(4*k-3))))

a(n) = A001826(4*n-3).

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

A364358 Number of divisors of n of the form 4*k+1 that are at most sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2

Offset: 1

Ilya Gutkovskiy, Jul 21 2023

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001826, A038548, A364387.

f:= proc(n) nops(select(t -> t mod 4 = 1 and t^2 <= n, numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Dec 29 2024

Table[Count[Divisors[n], _?(# <= Sqrt[n] && MemberQ[{1}, Mod[#, 4]] &)], {n, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(4 k + 1)^2/(1 - x^(4 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=0} x^(4*k+1)^2 / (1 - x^(4*k+1)).

A113414 Expansion of Sum_{k>0} x^k/(1-(-x^2)^k).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 0, 1, 1, 2, 0, 2, 2, 2, 0, 1, 2, 3, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 2, 4, 0, 1, 0, 2, 0, 3, 2, 2, 0, 2, 2, 4, 0, 2, 2, 2, 0, 2, 1, 3, 0, 2, 2, 4, 0, 2, 0, 2, 0, 4, 2, 2, 0, 1, 4, 4, 0, 2, 0, 4, 0, 3, 2, 2, 0, 2, 0, 4, 0, 2, 1, 2, 0, 4, 4, 2, 0, 2, 2, 6, 0, 2, 0, 2, 0, 2, 2, 3, 0, 3, 2, 4, 0, 2, 0

Offset: 1

Michael Somos, Oct 29 2005

nonn

A001227(n) = a(2*n), A008441(n) = a(4*n+1), A099774(n) = a(4*n+2).

a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-4, d)+2*(n%2==0)*(d%4==3)))

{a(n)=if(n<1, 0, if(n%4==3, 0, if(n%4==2, numdiv(n/2), if(n%4==0, sumdiv(n,d,d%2), sumdiv(n,d,(-1)^(d\2))))))}

{a(n)=if(n<1, 0, polcoeff( sum(k=1,sqrtint(8*n+1)\2, (-1)^(k%4==2)*x^((k^2+k)/2)/(1-(-1)^(k\2)*x^k), x*O(x^n)), n))}

{a(n)=if(n<1, 0, polcoeff( sum(k=1,n, x^k/(1-(-x^2)^k), x*O(x^n)), n))}

Moebius transform is period 8 sequence [1, 0, -1, 0, 1, 2, -1, 0, ...].
G.f.: Sum_{k>0} x^k/(1-(-x^2)^k) = Sum_{k>0} x^k/(1+x^(2k))+2x^(6k)/(1-x^(8k)) = Sum_{k>0} -(-1)^k x^(2k-1)/(1+(-1)^k*x^(2k-1)).
a(4n+3) = 0.
a(n) = A001826(n) + (-1)^n * A001842(n). - David Spies, Sep 26 2012

A364584 a(n) is the least number with exactly n divisors of the form 4*k+1.

Original entry on oeis.org

1, 5, 25, 45, 441, 225, 5103, 585, 1575, 2205, 35721, 2925, 194481, 25515, 11025, 9945, 2893401, 17325, 2711943423, 28665, 127575, 178605, 480249, 45045, 275625, 972405, 121275, 266805, 18983603961, 143325, 1441237924662543, 135135, 893025, 14467005, 3189375, 225225

Offset: 1

Ilya Gutkovskiy, Jul 28 2023

nonn

Bert Dobbelaere, Table of n, a(n) for n = 1..200

Cf. A001826, A005179, A038547, A364585.

More terms from Bert Dobbelaere, Jul 31 2023

A359289 Number of divisors of 4n-2 of form 4k+1.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 2, 2, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 1, 3, 1, 4, 2, 2, 2, 2, 2, 2, 3, 2, 1, 4, 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 1, 2, 4, 2, 1, 2, 2, 4, 3, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 3, 3, 2, 3, 2, 1

Offset: 1

Seiichi Manyama, Dec 24 2022

Cf. A001826, A078703, A359227.

a[n_] := DivisorSum[4*n-2, 1 &, Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(4*n-2, d, d%4==1);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(4*k-2))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(4*k-3))))

a(n) = A001826(4*n-2).
G.f.: Sum_{k>0} x^k/(1 - x^(4*k-2)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(4*k-3)).

A363341 Number of positive integers k <= n such that round(n/k) is odd.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 4, 6, 7, 6, 5, 9, 8, 9, 9, 10, 10, 11, 12, 13, 12, 13, 12, 15, 16, 17, 16, 17, 16, 17, 17, 20, 21, 20, 20, 23, 22, 21, 22, 24, 23, 26, 25, 28, 27, 26, 25, 27, 29, 30, 31, 32, 31, 32, 31, 32, 33, 34, 33, 35, 34, 37, 37, 40, 39, 38, 39, 40

Offset: 1

Caleb M. Shor, May 28 2023

nonn

Here round(x) = floor(x + 1/2).

a(n) is related to the number of lattice points in a circle. Let C(x) equal the number of square lattice points in a circle of radius sqrt(x) centered at the origin. Then a(n) = (C(2n) - 4n - 1)/4. (Prop 3.5 in Dent & Shor paper)

			For n=5: round(5/1), round(5/2), round(5/3), round(5/4), round(5/5) = 5, 3, 2, 1, 1 among which 4 are odd so a(5)=4.

Robert Israel, Table of n, a(n) for n = 1..10000
Nicholas Dent and Caleb M. Shor, On residues of rounded shifted fractions with a common numerator, arXiv:2206.15452 [math.NT], 2022.

Cf. A059851 (number of k=1..n such that floor(n/k) is odd).
Cf. A330926 (number of k=1..n such that ceiling(n/k) is odd).
Cf. A057655 (number of lattice points in circle).
Cf. A001826 (d_1), A001842 (d_3), A002654 (d_1-d_3).
Cf. A077024 (n + floor(2n/3) + floor(2n/5) + floor(2n/7) + ...).

f:= proc(n) local k;
   nops(select(k -> floor(n/k + 1/2)::odd, [$1..n]))
end proc:
map(f, [$1..120]); # Robert Israel, Aug 03 2025

a(n) = sum(k=1, n, round(n/k)%2) \\ Andrew Howroyd, May 28 2023

a(n) = n - floor(2n/3) + floor(2n/5) - floor(2n/7) + ...

a(n) = -n + Sum_{k=1..2n} d_1(k) - d_3(k), where d_i(k) is the number of divisors of k that are congruent to i modulo 4.

A363972 Expansion of Sum_{k>0} k^2 * x^(4k-3) / (1 - x^(4k-3)).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 10, 5, 1, 1, 17, 1, 5, 1, 26, 10, 1, 5, 37, 1, 1, 1, 54, 17, 10, 1, 65, 5, 1, 1, 82, 26, 5, 10, 101, 1, 17, 5, 122, 37, 1, 1, 158, 1, 1, 1, 170, 54, 26, 17, 197, 10, 5, 1, 226, 65, 1, 5, 257, 1, 46, 1, 310, 82, 1, 26, 325, 5, 1, 10, 362, 101, 54, 1, 401, 17, 1, 5, 451, 122, 1

Offset: 1

Seiichi Manyama, Jun 30 2023

Cf. A001826, A363903.

Cf. A050450, A363977.

a[n_] := DivisorSum[n, ((#+3)/4)^2 &, Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 30 2023 *)

a(n) = sumdiv(n, d, (d%4==1)*((d+3)/4)^2);

a(n) = Sum_{d|n, d==1 mod 4} ((d+3)/4)^2.

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cp's OEIS Frontend

A188171 The number of divisors d of n of the form d == 5 (mod 8).

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

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A293451 Number of proper divisors of n of the form 4k+1.

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A359227 Number of divisors of 4*n-3 of form 4*k+1.

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A364358 Number of divisors of n of the form 4*k+1 that are at most sqrt(n).

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A113414 Expansion of Sum_{k>0} x^k/(1-(-x^2)^k).

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A364584 a(n) is the least number with exactly n divisors of the form 4*k+1.

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A359289 Number of divisors of 4*n-2 of form 4*k+1.

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A359227 Number of divisors of 4n-3 of form 4k+1.

A359289 Number of divisors of 4n-2 of form 4k+1.

A363972 Expansion of Sum_{k>0} k^2 * x^(4k-3) / (1 - x^(4k-3)).