A230293 -id:A230293 - cp's OEIS frontend

A230290 a(n) = Sum_{i=1..n} d(24i+1) - Sum_{i=1..n} d(6i+1), where d(n) = A000005(n).

1, 2, 2, 1, 2, 4, 5, 4, 4, 4, 6, 7, 7, 5, 4, 10, 10, 10, 8, 9, 11, 10, 12, 10, 10, 13, 15, 14, 12, 14, 14, 14, 16, 16, 17, 17, 19, 21, 19, 20, 20, 18, 16, 16, 18, 24, 24, 23, 23, 21, 28, 28, 28, 24, 24, 26, 25, 27, 27, 28, 30, 30, 32, 28, 28, 30, 28, 30, 28, 29, 33, 39, 39, 37, 35, 39, 40, 38, 36, 36, 38

Offset: 1

N. J. A. Sloane, Oct 17 2013

nonn

Every inequality in number theory of the form f(n) >= g(n) is a potential source of a sequence floor(f(n))-ceiling(g(n)).

That sequence can be negative (e.g., floor(2/3)-ceiling(1/3)=-1), but the other 3 differences floor(f(n))-floor(g(n)), ceiling(f(n))-ceiling(g(n)), and ceiling(f(n))-floor(g(n)) are nonnegative. - Jonathan Sondow, Oct 20 2013

Robert Israel, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. (Shows that a(n) >= 0.)

Cf. A000005, A230291, A230292, A230293, A230294, A230295, A230296, A230476.

[&+[#Divisors(24*i+1):i in [1..n]] - &+[#Divisors(6*i+1):i in [1..n]]:n in [1..85]]; // Marius A. Burtea, Jan 03 2020

with(numtheory);
f:=proc(n,a,b,c,d) local i; add(tau(a*i+b),i=1..n) - add(tau(c*i+d),i=1..n); end;
[seq(f(n,24,1,6,1),n=1..120)];
# More efficient:
ListTools:-PartialSums([seq(numtheory:-tau(24*i+1)-numtheory:-tau(6*i+1),i=1..120)]); # Robert Israel, Jan 03 2020

R = Range[100];
Accumulate[DivisorSigma[0, 24R+1] - DivisorSigma[0, 6R+1]] (* Jean-François Alcover, Jan 31 2023 *)

vector(100, n, sum(i=1, n, numdiv(24*i+1)) - sum(i=1, n, numdiv(6*i+1))) \\ Michel Marcus, Oct 09 2014

a(n) = (2*log(2)/3) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

A230291 a(n) = Sum_{i=1..n} d(24i+17) - Sum_{i=1..n} d(6i+5), where d(n) = A000005(n).

Original entry on oeis.org

0, 2, 2, 2, 0, 2, 4, 6, 6, 4, 4, 4, 6, 6, 6, 6, 10, 10, 10, 10, 10, 12, 10, 10, 8, 6, 12, 14, 16, 16, 16, 18, 16, 18, 16, 14, 16, 16, 16, 12, 18, 22, 22, 24, 20, 22, 22, 24, 22, 20, 22, 28, 26, 26, 26, 24, 26, 26, 26, 26, 24, 28, 30, 30, 30, 30, 34, 34, 36, 32, 32, 32, 34, 36, 32, 34, 36, 34, 34, 34, 36, 36, 40, 42

Offset: 1

N. J. A. Sloane, Oct 17 2013

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. (Shows that a(n) >= 0.)

Cf. A000005, A230290, A230292, A230293, A230294, A230295, A230296, A230476.

[&+[#Divisors(24*i+17):i in [1..n]]- &+[#Divisors(6*i+5):i in [1..n]]:n in [1..90]]; // Marius A. Burtea, Jan 03 2020

Maple
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See A230290.
```

Accumulate[Table[DivisorSigma[0, 24*n + 17] - DivisorSigma[0, 6*n + 5], {n, 1, 100}]] (* Amiram Eldar, Apr 12 2024 *)

vector(100, n, sum(i=1, n, numdiv(24*i+17)) - sum(i=1, n, numdiv(6*i+5))) \\ Michel Marcus, Oct 09 2014

a(n) = (2*log(2)/3) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

A230292 a(n) = Sum_{i=1..n} d(24i+9) - Sum_{i=1..n} d(6i+3), where d(n) = A000005(n).

Original entry on oeis.org

1, 1, 2, 6, 6, 8, 6, 6, 11, 9, 13, 15, 14, 18, 20, 18, 14, 19, 21, 21, 25, 21, 25, 31, 33, 33, 31, 29, 33, 36, 32, 32, 34, 40, 40, 42, 41, 37, 49, 51, 51, 49, 47, 51, 52, 54, 50, 54, 54, 58, 60, 52, 56, 64, 62, 66, 68, 64, 68, 74, 72, 68, 73, 75, 75, 75, 77, 77, 85, 83, 79, 77, 72, 84, 88, 88, 84, 92, 94, 90, 98, 90

Offset: 1

N. J. A. Sloane, Oct 17 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. (Shows that a(n) >= 0.)

Cf. A000005, A230290, A230291, A230293, A230294, A230295, A230296, A230476.

Maple
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See A230290.
```

Accumulate[Table[DivisorSigma[0, 24*n + 9] - DivisorSigma[0, 6*n + 3], {n, 1, 100}]] (* Amiram Eldar, Apr 12 2024 *)

vector(100, n, sum(i=1, n, numdiv(24*i+9)) - sum(i=1, n, numdiv(6*i+3))) \\ Michel Marcus, Oct 09 2014

a(n) = (5*log(2)/3) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

A230294 a(n) = Sum_{i=1..n} d(4i+1) - Sum_{i=1..n} d(2i+1), where d(n) = A000005(n).

Original entry on oeis.org

0, 1, 1, 0, 2, 3, 1, 3, 3, 1, 5, 5, 3, 5, 5, 5, 5, 5, 5, 8, 10, 6, 8, 7, 5, 11, 9, 7, 11, 12, 10, 10, 12, 10, 12, 14, 10, 12, 12, 11, 17, 16, 14, 16, 14, 14, 18, 18, 14, 16, 18, 14, 16, 18, 18, 25, 23, 19, 19, 18, 20, 20, 22, 20, 24, 24, 18, 24, 24, 22, 26, 25, 21, 27, 29, 27, 27, 27, 25, 25, 29, 25, 29, 28, 26, 32

Offset: 1

N. J. A. Sloane, Oct 17 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. (Shows that a(n) >= 0.)

Cf. A000005, A230290, A230291, A230292, A230293, A230295, A230296, A230476.

Maple
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See A230290.
```

Accumulate[Table[DivisorSigma[0, 4*n + 1] - DivisorSigma[0, 2*n + 1], {n, 1, 100}]] (* Amiram Eldar, Apr 12 2024 *)

vector(100, n, sum(i=1, n, numdiv(4*i+1)) - sum(i=1, n, numdiv(2*i+1))) \\ Michel Marcus, Oct 09 2014

a(n) = A230476(n) - A230293(n). - Jonathan Sondow, Oct 20 2013

a(n) = (log(2)/2) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

A230295 a(n) = Sum_{i=1..n} d(8i+5) - Sum_{i=1..n} d(4i+3), where d(n) = A000005(n).

Original entry on oeis.org

0, 2, 0, 0, 4, 2, 2, 2, 2, 4, 6, 4, 2, 6, 4, 6, 8, 4, 4, 10, 8, 6, 10, 6, 8, 10, 10, 8, 8, 10, 12, 16, 10, 10, 14, 10, 12, 12, 10, 14, 18, 16, 12, 18, 18, 16, 18, 12, 12, 18, 16, 16, 20, 20, 22, 24, 18, 18, 22, 20, 20, 22, 16, 16, 26, 26, 26, 26, 22, 24, 24, 24, 24, 24, 26, 26, 32, 24, 26, 30, 28, 28, 28, 26, 26, 36

Offset: 1

N. J. A. Sloane, Oct 17 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. (Shows that a(n) >= 0.)

Cf. A000005, A230290, A230291, A230292, A230293, A230294, A230296, A230476.

Maple
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See A230290.
```

Accumulate[Table[DivisorSigma[0, 8*n + 5] - DivisorSigma[0, 4*n + 3], {n, 1, 100}]] (* Amiram Eldar, Apr 12 2024 *)

vector(100, n, sum(i=1, n, numdiv(8*i+5)) - sum(i=1, n, numdiv(4*i+3))) \\ Michel Marcus, Oct 09 2014

a(n) = (log(2)/2) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

A230296 a(n) = Sum_{i=1..n} d(72i+1) - Sum_{i=1..n} d(18i+1), where d(n) = A000005(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 2, 4, 2, 4, 6, 8, 8, 6, 4, 6, 5, 12, 8, 7, 8, 10, 12, 10, 16, 14, 12, 14, 12, 12, 12, 16, 18, 12, 14, 14, 12, 16, 20, 19, 19, 19, 26, 24, 22, 24, 24, 22, 20, 20, 20, 20, 26, 26, 24, 26, 28, 28, 22, 24, 24, 24, 30, 30, 32, 34, 36, 38, 33, 31, 30, 30, 36, 36, 35, 35, 36, 36, 36, 36, 36, 38, 38, 34, 34

Offset: 1

N. J. A. Sloane, Oct 17 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. (Shows that a(n) >= 0.)

Cf. A000005, A230290, A230291, A230292, A230293, A230294, A230295, A230476.

Maple
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See A230290.
```

Accumulate[Table[DivisorSigma[0, 72*n + 1] - DivisorSigma[0, 18*n + 1], {n, 1, 100}]] (* Amiram Eldar, Apr 12 2024 *)

vector(100, n, sum(i=1, n, numdiv(72*i+1)) - sum(i=1, n, numdiv(18*i+1))) \\ Michel Marcus, Oct 09 2014

a(n) = (2*log(2)/3) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

A230476 a(n) = Sum_{i=1..n} d(8i+1) - Sum_{i=1..n} d(2i+1), where d(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 6, 6, 7, 7, 6, 10, 10, 11, 11, 9, 11, 13, 15, 16, 14, 16, 15, 15, 17, 17, 22, 22, 22, 20, 18, 20, 24, 24, 25, 27, 27, 27, 26, 28, 26, 30, 30, 29, 31, 31, 37, 35, 35, 35, 31, 35, 35, 40, 40, 38, 40, 40, 41, 41, 41, 43, 47, 47, 46, 42, 44, 46, 50, 48, 46, 52, 52, 52, 54, 52, 55, 55, 53, 55, 53, 59, 58, 56, 58

Offset: 1

Jonathan Sondow, Oct 20 2013

nonn

Cimadevilla proved that a(n) >= 0. That is surprising because d(8*i+1) - d(2*i+1) < 0 for i = 12, 17, 22, 24, 31, 32, 40, 42, 45, 49, 52, 57, 66, 67, 71, 72, 77, 80, 82, 84, 85, ...

			The divisors of 8*1 + 1 = 9 are 1, 3, 9 and those of 2*1 + 1 = 3 are 1, 3, so a(1) = d(9) - d(3) = 3 - 2 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. See inequalities (1.5).

Cf. A000005, A002162, A230290, A230291, A230292, A230293, A230295, A230296.

Table[Sum[ DivisorSigma[0, 8 i + 1] - DivisorSigma[0, 2 i + 1], {i, n}], {n, 100}]

a(n) = sum(i=1, n, numdiv(8*i+1) - numdiv(2*i+1)); \\ Michel Marcus, Jun 19 2015

a(n) = Sum_{i=1..n} (d(8*i+1) - d(2*i+1)) = A230293(n) + A230294(n).

a(n) = log(2) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

cp's OEIS Frontend

A230290 a(n) = Sum_{i=1..n} d(24*i+1) - Sum_{i=1..n} d(6*i+1), where d(n) = A000005(n).

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A230291 a(n) = Sum_{i=1..n} d(24*i+17) - Sum_{i=1..n} d(6*i+5), where d(n) = A000005(n).

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Magma

Maple

Mathematica

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A230292 a(n) = Sum_{i=1..n} d(24*i+9) - Sum_{i=1..n} d(6*i+3), where d(n) = A000005(n).

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Mathematica

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A230294 a(n) = Sum_{i=1..n} d(4*i+1) - Sum_{i=1..n} d(2*i+1), where d(n) = A000005(n).

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Maple

Mathematica

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A230295 a(n) = Sum_{i=1..n} d(8*i+5) - Sum_{i=1..n} d(4*i+3), where d(n) = A000005(n).

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Programs

Maple

Mathematica

PARI

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A230296 a(n) = Sum_{i=1..n} d(72*i+1) - Sum_{i=1..n} d(18*i+1), where d(n) = A000005(n).

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Maple

Mathematica

PARI

Formula

A230476 a(n) = Sum_{i=1..n} d(8*i+1) - Sum_{i=1..n} d(2*i+1), where d(n) = A000005(n) is the number of divisors of n.

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Mathematica

A230290 a(n) = Sum_{i=1..n} d(24i+1) - Sum_{i=1..n} d(6i+1), where d(n) = A000005(n).

A230291 a(n) = Sum_{i=1..n} d(24i+17) - Sum_{i=1..n} d(6i+5), where d(n) = A000005(n).

A230292 a(n) = Sum_{i=1..n} d(24i+9) - Sum_{i=1..n} d(6i+3), where d(n) = A000005(n).

A230294 a(n) = Sum_{i=1..n} d(4i+1) - Sum_{i=1..n} d(2i+1), where d(n) = A000005(n).

A230295 a(n) = Sum_{i=1..n} d(8i+5) - Sum_{i=1..n} d(4i+3), where d(n) = A000005(n).

A230296 a(n) = Sum_{i=1..n} d(72i+1) - Sum_{i=1..n} d(18i+1), where d(n) = A000005(n).

A230476 a(n) = Sum_{i=1..n} d(8i+1) - Sum_{i=1..n} d(2i+1), where d(n) = A000005(n) is the number of divisors of n.