cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A091954 Number of odd proper divisors of n. That is, the number of odd divisors of n that are less than n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 4, 1, 1, 3, 2, 3, 3, 1, 2, 3, 2, 1, 4, 1, 2, 5, 2, 1, 2, 2, 3, 3, 2, 1, 4, 3, 2, 3, 2, 1, 4, 1, 2, 5, 1, 3, 4, 1, 2, 3, 4, 1, 3, 1, 2, 5, 2, 3, 4, 1, 2, 4, 2, 1, 4, 3, 2, 3, 2, 1, 6, 3, 2, 3, 2, 3, 2, 1, 3, 5, 3, 1, 4, 1, 2, 7, 2, 1, 4, 1, 4
Offset: 1

Views

Author

Mohammad K. Azarian, Mar 12 2004

Keywords

Examples

			The odd divisors of 15 that are less than 15 are 1, 3 and 5. Therefore there are three odd divisors of 15 that are less than 15.

Links

Crossrefs

Cf. A000005, A000035, A001620, A032741, A001227, A293214, A293216.
Sum of the k-th powers of the odd proper divisors of n for k=0..10: this sequence (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: this sequence (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Programs

  • Mathematica
    Count[Most[Divisors[#]],?OddQ]&/@Range[100] (* _Harvey P. Dale, Sep 28 2012 *)
a[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]] - Boole[OddQ[n]]; Array[a, 100] (* Amiram Eldar, Jun 11 2022 *)
  • PARI
    A091954(n) = sumdiv(n,d,(dAntti Karttunen, Oct 04 2017
  • PARI
    my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=2, N, x^k/(1-x^(2*k))))) \\ Seiichi Manyama, Jan 23 2021

Formula

From Antti Karttunen, Oct 04 2017: (Start)
a(n) = Sum_{d|n, dA000035(n).
a(n) = A001227(n) - A000035(n).
a(n) = A007814(A293214(n)) = A007814(A293216(n)).
(End)
G.f.: Sum_{k>=2} x^k/(1 - x^(2*k)). - Seiichi Manyama, Jan 23 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 26 2023

Extensions

Corrected and extended by Harvey P. Dale, Sep 28 2012

A091570 Sum of odd proper divisors of n. Sum of the odd divisors of n that are less than n.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 4, 1, 8, 9, 1, 1, 13, 1, 6, 11, 12, 1, 4, 6, 14, 13, 8, 1, 24, 1, 1, 15, 18, 13, 13, 1, 20, 17, 6, 1, 32, 1, 12, 33, 24, 1, 4, 8, 31, 21, 14, 1, 40, 17, 8, 23, 30, 1, 24, 1, 32, 41, 1, 19, 48, 1, 18, 27, 48, 1, 13, 1, 38, 49
Offset: 1

Views

Author

Mohammad K. Azarian, Mar 04 2004

Keywords

Examples

			The sum of odd divisors of 9 that are less than 9 is 1 + 3 = 4.

Links

Crossrefs

Cf. A000035, A000593, A013661, A066191.
Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), this sequence (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n, 0]; Array[a, 75] (* Amiram Eldar, Oct 11 2023 *)
  • PARI
    a(n) = sumdiv(n , d, (d%2) * (dMichel Marcus, Jan 14 2014

Formula

If n is odd, a(n) = A000593(n) - n; if n is even, a(n) = A000593(n). - Michel Marcus, Jan 14 2014
G.f.: Sum_{k>=1} (2*k-1) * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Apr 13 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)-1)/4 = 0.1612335167... . - Amiram Eldar, Oct 11 2023

A351647 Sum of the squares of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 10, 1, 1, 10, 26, 1, 10, 1, 50, 35, 1, 1, 91, 1, 26, 59, 122, 1, 10, 26, 170, 91, 50, 1, 260, 1, 1, 131, 290, 75, 91, 1, 362, 179, 26, 1, 500, 1, 122, 341, 530, 1, 10, 50, 651, 299, 170, 1, 820, 147, 50, 371, 842, 1, 260, 1, 962, 581, 1, 195, 1220, 1, 290
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 26; a(10) = Sum_{d|10, d<10, d odd} d^2 = 1^2 + 5^2 = 26.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), this sequence (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Cf. A000035, A002117, A050999.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(2*e+2) - 1)/(p^2 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^2, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
  • PARI
    a(n) = sumdiv(n, d, if ((d%2) && (dMichel Marcus, Mar 02 2022

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^2 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A050999(n) - n^2*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)-1)/6 = 0.0336761505... . (End)

A352031 Sum of the cubes of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 28, 1, 1, 28, 126, 1, 28, 1, 344, 153, 1, 1, 757, 1, 126, 371, 1332, 1, 28, 126, 2198, 757, 344, 1, 3528, 1, 1, 1359, 4914, 469, 757, 1, 6860, 2225, 126, 1, 9632, 1, 1332, 4257, 12168, 1, 28, 344, 15751, 4941, 2198, 1, 20440, 1457, 344, 6887, 24390, 1, 3528, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 126; a(10) = Sum_{d|10, d<10, d odd} d^3 = 1^3 + 5^3 = 126.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), this sequence (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Cf. A000035, A013662, A051000.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(3*e+3) - 1)/(p^3 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^3, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
Table[Total[Select[Most[Divisors[n]],OddQ]^3],{n,70}] (* Harvey P. Dale, Apr 14 2025 *)
  • PARI
    a(n) = sumdiv(n/2^valuation(n,2), d, if ((dMichel Marcus, Mar 02 2022

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^3 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A051000(n) - n^3*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)-1)/8 = 0.0102904042... . (End)

A352033 Sum of the 5th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 244, 1, 1, 244, 3126, 1, 244, 1, 16808, 3369, 1, 1, 59293, 1, 3126, 17051, 161052, 1, 244, 3126, 371294, 59293, 16808, 1, 762744, 1, 1, 161295, 1419858, 19933, 59293, 1, 2476100, 371537, 3126, 1, 4101152, 1, 161052, 821793, 6436344, 1, 244, 16808, 9768751
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 3126; a(10) = Sum_{d|10, d<10, d odd} d^5 = 1^5 + 5^5 = 3126.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), this sequence (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Cf. A000035, A013664, A051002.

Programs

  • Mathematica
    Table[Total[Select[Most[Divisors[n]],OddQ]^5],{n,50}] (* Harvey P. Dale, May 01 2023 *)
f[2, e_] := 1; f[p_, e_] := (p^(5*e+5) - 1)/(p^5 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^5, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^5 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A051002(n) - n^5*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)-1)/12 = 0.0014452551... . (End)

A352034 Sum of the 6th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 730, 1, 1, 730, 15626, 1, 730, 1, 117650, 16355, 1, 1, 532171, 1, 15626, 118379, 1771562, 1, 730, 15626, 4826810, 532171, 117650, 1, 11406980, 1, 1, 1772291, 24137570, 133275, 532171, 1, 47045882, 4827539, 15626, 1, 85884500, 1, 1771562, 11938421, 148035890
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 15626; a(10) = Sum_{d|10, d<10, d odd} d^6 = 1^6 + 5^6 = 15626.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), this sequence (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Cf. A013665, A321810.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(6*e+6) - 1)/(p^6 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^6, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
Table[Total[Select[Most[Divisors[n]],OddQ]^6],{n,50}] (* Harvey P. Dale, Sep 15 2024 *)

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^6 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
For odd n >1, a(n) = A321810(n)-n^6; for even n, a(n) = A321810(n). - R. J. Mathar, Aug 15 2023
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)-1)/14 = 0.0005963769... . - Amiram Eldar, Oct 11 2023

A352035 Sum of the 7th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2188, 1, 1, 2188, 78126, 1, 2188, 1, 823544, 80313, 1, 1, 4785157, 1, 78126, 825731, 19487172, 1, 2188, 78126, 62748518, 4785157, 823544, 1, 170939688, 1, 1, 19489359, 410338674, 901669, 4785157, 1, 893871740, 62750705, 78126, 1, 1801914272, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 78126; a(10) = Sum_{d|10, d<10, d odd} d^7 = 1^7 + 5^7 = 78126.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), this sequence (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Cf. A000035, A013666, A321811.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(7*e+7) - 1)/(p^7 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^7, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^7 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A321811(n) - n^7*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)-1)/16 = 0.0002548347... . (End)

A352036 Sum of the 8th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 6562, 1, 1, 6562, 390626, 1, 6562, 1, 5764802, 397187, 1, 1, 43053283, 1, 390626, 5771363, 214358882, 1, 6562, 390626, 815730722, 43053283, 5764802, 1, 2563287812, 1, 1, 214365443, 6975757442, 6155427, 43053283, 1, 16983563042, 815737283, 390626, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 390626; a(10) = Sum_{d|10, d<10, d odd} d^8 = 1^8 + 5^8 = 390626.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), this sequence (k=8), A352037 (k=9), A352038 (k=10).
Cf. A000035, A013667, A321812.

Programs

  • Mathematica
    Table[Total[Select[Most[Divisors[n]],OddQ]^8],{n,45}] (* Harvey P. Dale, Aug 07 2022 *)
f[2, e_] := 1; f[p_, e_] := (p^(8*e+8) - 1)/(p^8 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^8, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^8 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A321812(n) - n^8*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)-1)/18 = 0.0001115773... . (End)

A352037 Sum of the 9th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 19684, 1, 1, 19684, 1953126, 1, 19684, 1, 40353608, 1972809, 1, 1, 387440173, 1, 1953126, 40373291, 2357947692, 1, 19684, 1953126, 10604499374, 387440173, 40353608, 1, 38445332184, 1, 1, 2357967375, 118587876498, 42306733, 387440173, 1, 322687697780
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 1953126; a(10) = Sum_{d|10, d<10, d odd} d^9 = 1^9 + 5^9 = 1953126.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), this sequence (k=9), A352038 (k=10).
Cf. A000035, A013668, A321813.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(9*e+9) - 1)/(p^9 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^9, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^9 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A321813(n) - n^9*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)-1)/20 = 0.0000497287... . (End)

A352038 Sum of the 10th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 59050, 1, 1, 59050, 9765626, 1, 59050, 1, 282475250, 9824675, 1, 1, 3486843451, 1, 9765626, 282534299, 25937424602, 1, 59050, 9765626, 137858491850, 3486843451, 282475250, 1, 576660215300, 1, 1, 25937483651, 2015993900450, 292240875, 3486843451
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

Examples

			a(10) = 9765626; a(10) = Sum_{d|10, d<10, d odd} d^10 = 1^10 + 5^10 = 9765626.

Links

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), this sequence (k=10).
Cf. A000035, A013669, A321814.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(10*e+10) - 1)/(p^10 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^10, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
  • Python
    from math import prod
from sympy import factorint
def A352038(n): return prod((p**(10*(e+1))-1)//(p**10-1) for p, e in factorint(n).items() if p > 2) - (n**10 if n % 2 else 0) # Chai Wah Wu, Mar 01 2022

Formula

a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} (2*k-1)^10 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A321814(n) - n^10*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^11, where c = (zeta(11)-1)/22 = 0.0000224631... . (End)
Showing 1-10 of 10 results.

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