A368742 -id:A368742 - cp's OEIS frontend

A368736 a(n) = Sum_{k = 1..n} gcd(2*k+1, n).

1, 2, 5, 4, 9, 10, 13, 8, 21, 18, 21, 20, 25, 26, 45, 16, 33, 42, 37, 36, 65, 42, 45, 40, 65, 50, 81, 52, 57, 90, 61, 32, 105, 66, 117, 84, 73, 74, 125, 72, 81, 130, 85, 84, 189, 90, 93, 80, 133, 130, 165, 100, 105, 162, 189, 104, 185, 114, 117, 180, 121, 122, 273, 64, 225, 210, 133, 132, 225, 234

Offset: 1

Peter Bala, Jan 04 2024

CF. A000010, A018804, A344372, A368737 - A368742.

seq(add(gcd(2*k+1, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(irem(d,2)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[2*k + 1, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 11 2024 *)

a(n) = sum(k = 1, n, gcd(2*k+1, n)); \\ Michel Marcus, Jan 11 2024

a(2*n) = 2*a(n); a(2*n+1) = A018804(2*n+1) = A344372(2*n+1).
a(n) = Sum_{k = 1..n} gcd(4*k+1, n) = Sum_{k = 1..n} gcd(4*k+3, n).
a(n) = Sum_{d divides n} X(d)*phi(d)*n/d, where phi(n) = A000010(n) and X(n) = 1 if n is odd, else 0, that is, X(n) is the principal Dirichlet character of the reduced residue system mod 2. See A000035.
Multiplicative: a(2^k) = 2^k and for odd prime p, a(p^e) = (e + 1)*p^e - e*p^(e-1).
Dirichlet g.f.: (1 - 2/2^s)/(1 - 1/2^s) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ n^2*(2*log(n) - 1 + 4*gamma + 4*log(2)/3 - 12*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 11 2024

A368737 a(n) = Sum_{k = 1..n} gcd(3*k, n).

Original entry on oeis.org

1, 3, 9, 8, 9, 27, 13, 20, 45, 27, 21, 72, 25, 39, 81, 48, 33, 135, 37, 72, 117, 63, 45, 180, 65, 75, 189, 104, 57, 243, 61, 112, 189, 99, 117, 360, 73, 111, 225, 180, 81, 351, 85, 168, 405, 135, 93, 432, 133, 195, 297, 200, 105, 567, 189, 260, 333, 171, 117, 648, 121, 183, 585, 256, 225, 567, 133, 264, 405, 351

Offset: 1

Peter Bala, Jan 05 2024

a(n) equals the number of solutions to the congruence 3*x*y == 0 (mod n) for 1 <= x, y <= n.

			a(6) = 27: each of the 36 pairs (x, y), 1 <= x, y <= 6, is a solution to the congruence 3*x*y == 0 (mod 6) except for the 9 pairs (x, y) with both x and y odd.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, Notes on A368737

Cf. A000010, A018804, A344372, A368736 - A368742.

seq(add(gcd(3*k, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(gcd(3,d)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[3*k, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 11 2024 *)

a(n) = sum(k = 1, n, gcd(3*k, n)); \\ Michel Marcus, Jan 11 2024

a(3*n) = 9*A018804(n); a(3*n+1) = A018804(3*n+1); a(3*n+2) = A018804(3*n+2).
a(n) = Sum_{d divides n} gcd(3, d)*phi(d)*n/d, where phi(n) = A000010(n)
Multiplicative: a(3^k) = (2*k + 1)*3^k and for prime p not equal to 3, a(p^k) = (k + 1)*p^k - k*p^(k-1).
Define D(n) = Sum_{d divides n} a(d). Then
D(3*n+1) = (3*n + 1)*tau(3*n+1) and D(3*n+2) = (3*n + 2)*tau(3*n+2), where tau(n) = A000005(n), the number of divisors of n.
The sequence {(1/9)*( D(3*n) - D(n) ) : n >= 1} begins {1, 4, 5, 12, 10, 20, 14, 32, 21, 40, 22, 60, 26, 56, 50, 80, 34, 84, 38, 120, 70, 88, ...} and appears to be multiplicative.
Dirichlet g.f.: (1 + 3/3^s)/(1 - 1/3^s) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ 9*n^2 * (log(n)/2 - 1/4 + gamma - 3*log(3)/16 - 3*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 11 2024

A368741 a(n) = Sum_{k = 1..n} gcd(5*k + 1, n).

Original entry on oeis.org

1, 3, 5, 8, 5, 15, 13, 20, 21, 15, 21, 40, 25, 39, 25, 48, 33, 63, 37, 40, 65, 63, 45, 100, 25, 75, 81, 104, 57, 75, 61, 112, 105, 99, 65, 168, 73, 111, 125, 100, 81, 195, 85, 168, 105, 135, 93, 240, 133, 75, 165, 200, 105, 243, 105, 260, 185, 171, 117, 200, 121, 183, 273, 256, 125, 315, 133, 264, 225, 195

Offset: 1

Peter Bala, Jan 08 2024

Cf. A000010, A011558, A018804, A368736 - A368742.

with(numtheory): seq(add(gcd(5*k+1, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(irem(d^4,5)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[5*k+1, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 12 2024 *)

a(n) = Sum_{k = 1..n} gcd(5*k + r, n) for 1 <= r <= 4.
a(5*n) = 5*a(n); a(5*n+r) = A018804(5*n+r) for 1 <= r <= 4.
a(n) = Sum_{d divides n} X(d)*phi(d)*n/d, where phi(n) = A000010(n) and X(n) = A011558(n) is the principal Dirichlet character of the reduced residue system mod 5.
Multiplicative: a(5^k) = 5^k and for prime p not equal to 5, a(p^k) = (k + 1)*p^k - k*p^(k-1).
Dirichlet g.f.: (1 - 5/5^s)/(1 - 1/5^s) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ 5*n^2 * (log(n)/2 - 1/4 + gamma + 5*log(5)/48 - 3*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 12 2024

A368738 a(n) = Sum_{k = 1..n} gcd(3*k + 1, n).

Original entry on oeis.org

1, 3, 3, 8, 9, 9, 13, 20, 9, 27, 21, 24, 25, 39, 27, 48, 33, 27, 37, 72, 39, 63, 45, 60, 65, 75, 27, 104, 57, 81, 61, 112, 63, 99, 117, 72, 73, 111, 75, 180, 81, 117, 85, 168, 81, 135, 93, 144, 133, 195, 99, 200, 105, 81, 189, 260, 111, 171, 117, 216, 121, 183, 117, 256, 225, 189, 133, 264, 135, 351

Offset: 1

Peter Bala, Jan 05 2024

Cf. A000010, A011655, A018804, A368736 - A368742.

seq(add(gcd(3*k+1, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(irem(d^2,3)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[3*k+1, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 12 2024 *)

a(n) = Sum_{k = 1..n} gcd(3*k + 2, n).
a(n) = Sum_{k = 1..n} gcd(9*k + r) for r = 1, 2, 4, 5, 7 and 8.
a(3*n) = 3*a(n); a(3*n+1) = A018804(3*n+1); a(3*n+2) = A018804(3*n+2).
a(n) = Sum_{d divides n} X(d)*phi(d)*n/d, where phi(n) = A000010(n) and X(n) = A011655(n) is the principal Dirichlet character of the reduced residue system mod 3.
Multiplicative: a(3^k) = 3^k and for prime p not equal to 3, a(p^k) = (k + 1)*p^k - k*p^(k-1).
Dirichlet g.f.: (1 - 3/3^s)/(1 - 1/3^s) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ 9*n^2 * (log(n)/2 - 1/4 + gamma + 3*log(3)/16 - 3*zeta'(2)/Pi^2) / (2*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 12 2024

A368739 a(n) = Sum_{k = 1..n} gcd(4*k, n).

Original entry on oeis.org

1, 4, 5, 16, 9, 20, 13, 48, 21, 36, 21, 80, 25, 52, 45, 128, 33, 84, 37, 144, 65, 84, 45, 240, 65, 100, 81, 208, 57, 180, 61, 320, 105, 132, 117, 336, 73, 148, 125, 432, 81, 260, 85, 336, 189, 180, 93, 640, 133, 260, 165, 400, 105, 324, 189, 624, 185, 228, 117, 720, 121, 244, 273, 768

Offset: 1

Peter Bala, Jan 07 2024

Cf. A000010, A018804, A344372, A368736 - A368742.

seq(add(gcd(4*k, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(gcd(4,d)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[4*k, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 12 2024 *)

a(4*n) = 16*A018804(n); a(4*n+2) = 4*A018804(2*n+1); a(4*n+r) = A018804(4*n+r) for r = 1 and 3.
a(n) = Sum_{d divides n} gcd(4, d)*phi(d)*n/d, where phi(n) = A000010(n)
Multiplicative: a(2^k) = k*2^(k+1) for k >= 1; for odd prime p, a(p^k) = (k + 1)*p^k - k*p^(k-1).
Define D(n) = Sum_{d divides n} a(d). Then
D(4*n+r) = (4*n + r)*tau(4*n+r) for r = 1 and r = 3, where tau(n) = A000005(n), the number of divisors of n.
D(4*n+2) = (5/4)*(4*n + 2)*tau(4*n+2).
The sequence defined for n >= 1 by u(n) = (1/4)*( D(4*n) - D(n) ) begins {5, 16, 30, 44, 50, 96, 70, 112, 135, 160, 110, 264, 130, 224, 300, 272, 170, 432, 190, 440, 420, 352, ...} and appears to be multiplicative: that is, u(1)*u(n*m) = u(n)*u(m) for n and m coprime.
Dirichlet g.f.: (1 + 4/4^s)/(1 - 1/2^s) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ n^2 * (5*log(n) - 5/2 + 10*gamma - 11*log(2)/3 - 30*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 12 2024

A368740 a(n) = Sum_{k = 1..n} gcd(5*k, n).

Original entry on oeis.org

1, 3, 5, 8, 25, 15, 13, 20, 21, 75, 21, 40, 25, 39, 125, 48, 33, 63, 37, 200, 65, 63, 45, 100, 225, 75, 81, 104, 57, 375, 61, 112, 105, 99, 325, 168, 73, 111, 125, 500, 81, 195, 85, 168, 525, 135, 93, 240, 133, 675, 165, 200, 105, 243, 525, 260, 185, 171, 117, 1000, 121, 183, 273, 256, 625

Offset: 1

Peter Bala, Jan 07 2024

a(n) equals the number of solutions to the congruence 5*x*y == 0 (mod n) for 1 <= x, y <= n.

			a(10) = 75: each of the 100 pairs (x, y), 1 <= x, y <= 10, is a solution to the congruence 5*x*y == 0 (mod 10) except for the 25 pairs (x, y) with both x and y odd.

Cf. A000010, A018804, A344372, A368736 - A368742.

seq(add(gcd(5*k, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(gcd(5,d)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[5*k, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 12 2024 *)

a(5*n) = 25*A018804(n); a(5*n+r) = A018804(5*n+r) for 1 <= r <= 4.
a(n) = Sum_{d divides n} gcd(5, d)*phi(d)*n/d, where phi(n) = A000010(n).
Multiplicative: a(5^k) = (4*k + 1)*5^k and for prime p not equal to 5, a(p^k) = (k + 1)*p^k - k*p^(k-1).
Define D(n) = Sum_{d divides n} a(d). Then
D(5*n+r) = (5*n + r)*tau(5*n+r) for 1 <= r <= 4, where tau(n) = A000005(n), the number of divisors of n.
The sequence {(1/25)*( D(5*n) - D(n) ) : n >= 1} begins {1, 4, 6, 12, 9, 24, 14, 32, 27, 36, 22, 72, 26, 56, 54, 80, 34, 108, 38, 108, 84, 88, ...} and appears to be multiplicative.
Dirichlet g.f.: (1 + 15/5^s)/(1 - 1/5^s) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ n^2 * (5*log(n) - 5/2 + 10*gamma - 25*log(5)/12 - 30*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 12 2024

cp's OEIS Frontend

A368736 a(n) = Sum_{k = 1..n} gcd(2*k+1, n).

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A368737 a(n) = Sum_{k = 1..n} gcd(3*k, n).

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A368741 a(n) = Sum_{k = 1..n} gcd(5*k + 1, n).

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A368738 a(n) = Sum_{k = 1..n} gcd(3*k + 1, n).

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A368739 a(n) = Sum_{k = 1..n} gcd(4*k, n).

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A368740 a(n) = Sum_{k = 1..n} gcd(5*k, n).

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