A001842 -id:A001842 - cp's OEIS frontend

A293513 Number of proper divisors of n of the form 4k+3.

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

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. A001842, A091954, A121262, A293451, A293903.

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

PARI
```
A293513(n) = sumdiv(n,d,(d
				
```

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

a(n) = A091954(n) - A293451(n).

a(n) = A001842(n) - A121262(n+1).

G.f.: Sum_{k>=1} x^(8*k-2) / (1 - x^(4*k-1)). - 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(3,4) - (2 - gamma)/4 = A256846 - (2 - A001620)/4 = -0.430804... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A359290 Number of divisors of 4n-2 of form 4k+3.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 24 2022

nonn

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

Cf. A001842, A078703, A359240.

Table[Count[Divisors[4 n-2],?(IntegerQ[(#-3)/4]&)],{n,100}] (* _Harvey P. Dale, May 09 2023 *)
a[n_] := DivisorSum[4*n-2, 1 &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(4*n-2, d, d%4==3);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^(4*k-1)))))

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(3*k-1)/(1-x^(4*k-2)))))

a(n) = A001842(4*n-2).
G.f.: Sum_{k>0} x^(2*k)/(1 - x^(4*k-1)).
G.f.: Sum_{k>0} x^(3*k-1)/(1 - x^(4*k-2)).

A363904 Expansion of Sum_{k>0} x^(3k) / (1 - x^(4k))^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 0, 2, 5, 0, 0, 1, 5, 0, 3, 3, 6, 1, 0, 0, 8, 2, 0, 5, 8, 0, 4, 0, 11, 1, 0, 5, 11, 0, 0, 3, 11, 3, 5, 6, 12, 1, 2, 0, 14, 0, 0, 8, 17, 2, 6, 0, 15, 5, 0, 8, 19, 0, 0, 4, 17, 0, 7, 11, 18, 1, 0, 0, 24, 5, 5, 11, 20, 0, 8, 0, 21, 3, 0, 11, 23, 3, 0, 5, 25, 6

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A001842, A050452, A363903.

Cf. A113415, A363902.

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

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

a(n) = (1/4) * Sum_{d|n, d==3 mod 4} (d+1) = (A001842(n) + A050452(n))/4.

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

A364387 Number of divisors of n of the form 4*k+3 that are at most sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 21 2023

nonn

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

Cf. A001842, A038548, A364358.

f:= proc(n) nops(select(t -> t mod 4 = 3 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[{3}, Mod[#, 4]] &)], {n, 99}]
nmax = 99; CoefficientList[Series[Sum[x^(4 k + 3)^2/(1 - x^(4 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest

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

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

A364585 a(n) is the least number with exactly n divisors of the form 4*k+3.

Original entry on oeis.org

1, 3, 15, 63, 105, 567, 315, 3969, 945, 1575, 2835, 413343, 3465, 53361, 19845, 14175, 10395, 1750329, 17325, 26040609, 31185, 99225, 480249, 219667417263, 45045, 354375, 266805, 121275, 257985, 160137547184727, 155925, 4322241, 135135, 10333575, 8751645, 2480625

Offset: 0

Ilya Gutkovskiy, Jul 28 2023

nonn

Bert Dobbelaere, Table of n, a(n) for n = 0..200

Cf. A001842, A005179, A038547, A364584.

More terms from Bert Dobbelaere, Aug 01 2023

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.

A363973 Expansion of Sum_{k>0} k^2 * x^(4k-1) / (1 - x^(4k-1)).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 4, 0, 1, 0, 9, 1, 0, 4, 17, 0, 0, 1, 25, 0, 5, 9, 36, 1, 0, 0, 50, 4, 0, 17, 64, 0, 10, 0, 85, 1, 0, 25, 101, 0, 0, 5, 121, 9, 17, 36, 144, 1, 4, 0, 170, 0, 0, 50, 205, 4, 26, 0, 225, 17, 0, 64, 261, 0, 0, 10, 289, 0, 37, 85, 324, 1, 0, 0, 378, 25, 13, 101, 400, 0, 50, 0

Offset: 1

Seiichi Manyama, Jun 30 2023

Cf. A001842, A363904.

Cf. A050453, A363978.

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

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

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

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cp's OEIS Frontend

A359240 Number of divisors of 4*n-3 of form 4*k+3.

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A293513 Number of proper divisors of n of the form 4k+3.

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A359290 Number of divisors of 4*n-2 of form 4*k+3.

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A363904 Expansion of Sum_{k>0} x^(3*k) / (1 - x^(4*k))^2.

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A364387 Number of divisors of n of the form 4*k+3 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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A364585 a(n) is the least number with exactly n divisors of the form 4*k+3.

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

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A359240 Number of divisors of 4n-3 of form 4k+3.

A359290 Number of divisors of 4n-2 of form 4k+3.

A363904 Expansion of Sum_{k>0} x^(3k) / (1 - x^(4k))^2.

A363973 Expansion of Sum_{k>0} k^2 * x^(4k-1) / (1 - x^(4k-1)).