Wesley Ivan Hurt - cp's OEIS frontend

A386624 a(n) = Sum_{d|n} sigma(d) * phi(d) * mu(n/d).

1, 2, 7, 11, 23, 14, 47, 46, 70, 46, 119, 77, 167, 94, 161, 188, 287, 140, 359, 253, 329, 238, 527, 322, 596, 334, 642, 517, 839, 322, 959, 760, 833, 574, 1081, 770, 1367, 718, 1169, 1058, 1679, 658, 1847, 1309, 1610, 1054, 2207, 1316, 2346, 1192, 2009, 1837, 2807, 1284, 2737, 2162, 2513, 1678, 3479, 1771, 3719, 1918, 3290, 3056, 3841, 1666, 4487, 3157, 3689

Offset: 1

Wesley Ivan Hurt, Jul 27 2025

Möbius transform of sigma(n) * phi(n) = A062354(n).

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

Cf. A000010 (phi), A000203 (sigma), A062354, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 100}]
f[p_, e_] := p^(2*e) - p^(e-1) - If[e > 1, p^(2*e-2) - p^(e-2), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 27 2025 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i, 2];  p^(2*e) - p^(e - 1) - if(e > 1, p^(2*e - 2) - p^(e - 2), 1));} \\ Amiram Eldar, Jul 27 2025

a(n) = Sum_{d|n} A062354(d) * mu(n/d).
From Amiram Eldar, Jul 27 2025: (Start)
Multiplicative with a(p) = p^2 - 2, and a(p^e) = p^(2*e) - p^(2*e-2) - p^(e-1) + p^(e-2) for e >= 2.
Dirichlet g.f.: (zeta(s-2) * zeta(s-1) / zeta(s)) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s + 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/(3*zeta(3))) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523*zeta(2)/(3*zeta(3)) = 0.24444595409976589792... . (End)

A386622 a(n) = Sum_{d|n} n^c(d), where c = A351114.

Original entry on oeis.org

1, 4, 6, 9, 6, 24, 8, 18, 19, 22, 12, 61, 14, 43, 46, 35, 18, 74, 20, 44, 44, 46, 24, 123, 27, 54, 56, 87, 30, 182, 32, 68, 68, 70, 72, 184, 38, 78, 80, 86, 42, 254, 44, 92, 138, 94, 48, 245, 51, 104, 104, 108, 54, 220, 58, 228, 116, 118, 60, 425, 62, 126, 130, 133, 68, 268, 68, 140, 140, 353, 72, 367, 74, 150, 228, 156, 80, 393, 80, 168, 165, 166, 84, 593

Offset: 1

Wesley Ivan Hurt, Jul 27 2025

nonn

Cf. A000005 (tau), A020492 (balanced numbers), A351112, A351114, A386591.

Table[Sum[n^(1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

a(n) = tau(n) + (n - 1) * Sum_{d|n} c(d), where c = A351114.

a(n) = n*A351112(n) + A386591(n).

A386596 a(n) = Sum_{d|n} d * pi(n/d).

Original entry on oeis.org

0, 1, 2, 4, 3, 10, 4, 12, 10, 15, 5, 31, 6, 21, 25, 30, 7, 45, 8, 48, 34, 29, 9, 83, 24, 34, 39, 65, 10, 102, 11, 71, 48, 42, 52, 134, 12, 47, 56, 128, 13, 141, 14, 94, 109, 55, 15, 199, 43, 108, 70, 109, 16, 169, 74, 174, 78, 65, 17, 300, 18, 71, 148, 160, 87, 204, 19, 137, 92, 216, 20, 351, 21, 82, 173, 153, 100, 238, 22, 308, 139, 89, 23, 409, 109, 94, 111

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

Dirichlet convolution of n and pi(n).

Inverse Möbius transform of A333699(n).

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

Cf. A000720 (pi), A333699.

f:= proc(n) local d;
      add(d*numtheory:-pi(n/d),d=numtheory:-divisors(n))
end proc:
map(f, [$1..100]); # Robert Israel, Aug 26 2025

Table[Sum[d*PrimePi[n/d], {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} A333699(d).

A386595 a(n) = Sum_{d|n} sigma(d)/phi(d) * c(d), where c = A351114.

Original entry on oeis.org

1, 4, 3, 4, 1, 12, 1, 4, 3, 4, 1, 19, 1, 8, 6, 4, 1, 12, 1, 4, 3, 4, 1, 19, 1, 4, 3, 8, 1, 24, 1, 4, 3, 4, 3, 19, 1, 4, 3, 4, 1, 24, 1, 4, 6, 4, 1, 19, 1, 4, 3, 4, 1, 12, 1, 13, 3, 4, 1, 31, 1, 4, 3, 4, 1, 12, 1, 4, 3, 16, 1, 19, 1, 4, 6, 4, 1, 19, 1, 4, 3, 4, 1, 31, 1, 4, 3, 4, 1, 24, 1, 4, 3, 4, 1, 19, 1, 8, 3, 4

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

nonn

For each divisor d of n, add sigma(d)/phi(d) if phi(d) | sigma(d), else add 0.

Cf. A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A351114.

Table[Sum[(DivisorSigma[1, d]/EulerPhi[d])*(1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

A386594 a(n) = Sum_{d|n} d * c(n/d), where c = A351114.

Original entry on oeis.org

1, 3, 4, 6, 5, 12, 7, 12, 12, 15, 11, 25, 13, 22, 21, 24, 17, 36, 19, 30, 28, 33, 23, 50, 25, 39, 36, 44, 29, 63, 31, 48, 44, 51, 36, 75, 37, 57, 52, 60, 41, 88, 43, 66, 63, 69, 47, 100, 49, 75, 68, 78, 53, 108, 55, 89, 76, 87, 59, 131, 61, 93, 84, 96, 65, 132, 67, 102, 92, 113, 71, 150, 73, 111, 105, 114, 77, 157, 79, 120, 108, 123, 83, 183, 85, 129, 116

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

nonn

Dirichlet convolution of n and c(n), where c = A351114.

For each divisor d of n, add n/d if phi(d) | sigma(d), else add 0.

Cf. A020492 (balanced numbers), A351114.

Table[Sum[d*(1 - Ceiling[DivisorSigma[1, n/d]/EulerPhi[n/d]] + Floor[DivisorSigma[1, n/d]/EulerPhi[n/d]]), {d, Divisors[n]}], {n, 100}]

A386592 Sum of the divisors of n that are not balanced numbers.

Original entry on oeis.org

0, 0, 0, 4, 5, 0, 7, 12, 9, 15, 11, 4, 13, 7, 5, 28, 17, 27, 19, 39, 28, 33, 23, 36, 30, 39, 36, 39, 29, 15, 31, 60, 44, 51, 12, 67, 37, 57, 52, 87, 41, 28, 43, 81, 59, 69, 47, 100, 56, 90, 68, 95, 53, 108, 71, 47, 76, 87, 59, 99, 61, 93, 100, 124, 83, 132, 67, 123, 92, 22, 71, 171, 73, 111, 105, 137, 95, 78, 79, 183, 117, 123, 83, 144, 107, 129, 116, 177

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

Sum of the divisors d of n such that phi(d) does not divide sigma(d).

Inverse Möbius transform of n * (1 - c(n)), where c = A351114.

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

Cf. A000010 (phi), A000203 (sigma), A351113, A351114, A387333.

g:= proc(n) option remember; numtheory:-sigma(n) mod numtheory:-phi(n) <> 0 end proc:
f:= n -> convert(select(g,numtheory:-divisors(n)),`+`):
map(f, [$1..100]); # Robert Israel, Aug 26 2025

Table[Sum[d (Ceiling[DivisorSigma[1, d]/EulerPhi[d]] - Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} d * (1 - c(d)), where c = A351114.

a(n) = A000203(n) - A351113(n).

A386591 Number of divisors of n that are not balanced numbers.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 4, 2, 2, 2, 4, 1, 2, 2, 6, 1, 2, 1, 4, 3, 2, 1, 5, 2, 4, 2, 4, 1, 4, 3, 4, 2, 2, 1, 5, 1, 2, 4, 5, 3, 4, 1, 4, 2, 3, 1, 7, 1, 2, 3, 4, 3, 3, 1, 8, 3, 2, 1, 5, 3, 2, 2, 6, 1, 6, 3, 4, 2, 2, 3, 7, 1, 3, 4, 7

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

Number of divisors d of n such that phi(d) does not divide sigma(d).

Inverse Möbius transform of 1 - c(n), where c = A351114.

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

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A351112, A351114.

g:= proc(n) option remember; numtheory:-sigma(n) mod numtheory:-phi(n) <> 0 end proc:
f:= n -> nops(select(g,numtheory:-divisors(n))):
map(f, [$1..100]); # Robert Israel, Aug 26 2025

Table[Sum[Ceiling[DivisorSigma[1, d]/EulerPhi[d]] - Floor[DivisorSigma[1, d]/EulerPhi[d]], {d, Divisors[n]}], {n, 100}]

a(n) = sumdiv(n, d, sigma(d)%eulerphi(d) != 0); \\ Michel Marcus, Aug 26 2025

a(n) = Sum_{d|n} (1 - c(d)), where c = A351114.

a(n) = A000005(n) - A351112(n).

A386589 a(n) = Sum_{d|n} d^c(d), where c = A351114.

Original entry on oeis.org

1, 3, 4, 4, 2, 12, 2, 5, 5, 5, 2, 25, 2, 18, 20, 6, 2, 14, 2, 7, 6, 5, 2, 27, 3, 5, 6, 20, 2, 59, 2, 7, 6, 5, 38, 28, 2, 5, 6, 9, 2, 70, 2, 7, 22, 5, 2, 29, 3, 7, 6, 7, 2, 16, 4, 77, 6, 5, 2, 74, 2, 5, 8, 8, 4, 16, 2, 7, 6, 125, 2, 31, 2, 5, 22, 7, 4, 93, 2, 11, 7, 5, 2, 85, 4, 5, 6, 9, 2, 63, 4, 7, 6, 5, 4, 31, 2, 20, 8, 10

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

nonn

Inverse Möbius transform of n^c(n), where c = A351114.

For each divisor d of n, add d if d is a balanced number (A020492), else add 1.

Cf. A000005 (tau), A020492 (balanced numbers), A351112, A351113, A351114.

Table[Sum[d^(1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} 1 + (d - 1)*c(d), where c = A351114.

a(n) = A000005(n) + A351113(n) - A351112(n).

A386574 Number of squarefree balanced divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 3, 3, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 6, 1, 2, 2, 2, 2, 4, 1, 2, 2, 2, 1, 6, 1, 2, 3, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 2, 3, 2, 1, 5, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

nonn

Inverse Möbius transform of c(n) * mu(n)^2, where c = A351114.

Cf. A005117 (squarefree numbers), A020492 (balanced numbers), A351114, A386573.

Table[Sum[(1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]) MoebiusMu[d]^2, {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} c(d) * mu(d)^2, where c = A351114.

A386573 Sum of the squarefree balanced divisors of n.

Original entry on oeis.org

1, 3, 4, 3, 1, 12, 1, 3, 4, 3, 1, 12, 1, 17, 19, 3, 1, 12, 1, 3, 4, 3, 1, 12, 1, 3, 4, 17, 1, 57, 1, 3, 4, 3, 36, 12, 1, 3, 4, 3, 1, 68, 1, 3, 19, 3, 1, 12, 1, 3, 4, 3, 1, 12, 1, 17, 4, 3, 1, 57, 1, 3, 4, 3, 1, 12, 1, 3, 4, 122, 1, 12, 1, 3, 19, 3, 1, 90, 1, 3, 4, 3, 1, 68, 1, 3, 4, 3, 1, 57, 1, 3, 4, 3, 1, 12, 1, 17, 4, 3

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

nonn

Inverse Möbius transform of n * c(n) * mu(n)^2, where c = A351114.

Cf. A005117 (squarefree numbers), A020492 (balanced numbers), A078557, A351114, A386574.

Table[Sum[d (1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]) MoebiusMu[d]^2, {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} d * c(d) * mu(d)^2, where c = A351114.

cp's OEIS Frontend

User: Wesley Ivan Hurt

A386624 a(n) = Sum_{d|n} sigma(d) * phi(d) * mu(n/d).

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A386622 a(n) = Sum_{d|n} n^c(d), where c = A351114.

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A386596 a(n) = Sum_{d|n} d * pi(n/d).

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A386595 a(n) = Sum_{d|n} sigma(d)/phi(d) * c(d), where c = A351114.

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A386594 a(n) = Sum_{d|n} d * c(n/d), where c = A351114.

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A386592 Sum of the divisors of n that are not balanced numbers.

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A386591 Number of divisors of n that are not balanced numbers.

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A386589 a(n) = Sum_{d|n} d^c(d), where c = A351114.

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