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.

Previous Showing 51-60 of 65 results. Next

A065126 Numbers n for which sigma_2(n^2) == 3 (mod 10).

Original entry on oeis.org

11, 19, 22, 29, 31, 33, 38, 41, 44, 55, 57, 58, 59, 61, 62, 66, 71, 76, 77, 79, 82, 87, 88, 89, 93, 95, 99, 101, 109, 110, 114, 116, 118, 122, 123, 124, 131, 132, 133, 139, 142, 143, 145, 149, 151, 152, 154, 155, 158, 164, 165, 171, 174, 176, 177, 178, 179, 181
Offset: 1

Views

Author

Labos Elemer, Nov 21 2001

Keywords

Comments

It appears that sigma_2( m^2 ) = 3 (mod 10) iff m is divisible by a prime p = 1 or 9 (mod 10), else sigma_2( m^2 ) = 1 (mod 10). - M. F. Hasler, May 14 2008
This seems also to be numbers whose square is expressible in only one way as x^2 + 3xy + y^2, with 0 < x < y. - Colin Barker, Dec 24 2014

Examples

			n=29: sigma[2,29^2] = sigma[2,841] = 708123 = 10.70812+3; among the numbers all residues modulo 8 occur.

Links

Crossrefs

Cf. A000290, A001157, A057660, A065803.

Programs

  • Mathematica
    Select[Range[200],Mod[DivisorSigma[2,#^2],10]==3&] (* Harvey P. Dale, Oct 21 2011 *)
  • PARI
    c=0; for( n=1,10^5,sigma(n^2,2)%5==3 & write("b065126.txt",c++" "n)) \\ M. F. Hasler, May 14 2008

Formula

Mod[DivisorSigma[2, n^2], 10]=3.

Extensions

More terms and better description from M. F. Hasler, May 14 2008

A077457 a(n) = sigma_4(n^4)/sigma_2(n^4).

Original entry on oeis.org

1, 205, 5905, 52429, 375601, 1210525, 5649505, 13421773, 38742049, 76998205, 212601841, 309593245, 810932305, 1158148525, 2217923905, 3435973837, 6951703105, 7942120045, 16936647121, 19692384829, 33360327025, 43583377405, 78163228705, 79255569565, 146719125601
Offset: 1

Views

Author

Benoit Cloitre, Nov 30 2002

Keywords

Comments

sigma_y(n^x) divides sigma_x(n^x) for all n if y divides x.

Links

Crossrefs

Cf. A001157, A001159, A013667, A057660, A077454, A077455, A077456.

Programs

  • Mathematica
    f[p_, e_] := (p^(8*e+2) + 1)/(p^2 + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 25] (* Amiram Eldar, Sep 09 2020 *)
  • PARI
    a(n)=sumdiv(n^4,d,d^4)/sumdiv(n^4,d,d^2)
  • PARI
    a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f, 2); \\ Michel Marcus, Sep 09 2020

Formula

a(n) = A001159(n^4)/A001157(n^4).
Multiplicative with a(p^e) = (p^(8*e+2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} (1 - 1/p^3 + 1/p^5 - 1/p^7) = 0.09549806119... . - Amiram Eldar, Oct 28 2022

A086148 Sum of the orders of the elements in the dihedral group D_2n.

Original entry on oeis.org

3, 7, 13, 19, 31, 33, 57, 59, 79, 83, 133, 101, 183, 157, 177, 203, 307, 219, 381, 271, 343, 377, 553, 349, 571, 523, 601, 529, 871, 501, 993, 747, 843, 887, 973, 743, 1407, 1105, 1177, 983, 1723, 987, 1893, 1309, 1371, 1613, 2257, 1293, 2199, 1663, 2013
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 25 2003

Keywords

Links

Crossrefs

Cf. A057660.

Programs

  • Mathematica
    f[p_, e_] := (p^(2*e+1)+1)/(p+1); a[1] = 3; a[n_] := 2*n + Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)
  • PARI
    a(n) = 2*n + sumdivmult(n, d, d*eulerphi(d)); \\ Michel Marcus, Feb 16 2024
  • Python
    from sympy import factorint, prod
a = lambda n: 2*n + prod((p**(2*e+1)+1)//(p+1) for p,e in factorint(n).items()) # Darío Clavijo, Feb 15 2024

Formula

a(n) = 2*n + Sum_{d|n} d*phi(d). - Vladeta Jovovic, Aug 27 2003

Extensions

More terms from Vladeta Jovovic, Aug 27 2003

A127474 Triangle, square of A054522.

Original entry on oeis.org

1, 2, 1, 3, 0, 4, 4, 3, 0, 4, 5, 0, 0, 0, 16, 6, 3, 8, 0, 0, 4, 7, 0, 0, 0, 0, 0, 36, 8, 7, 0, 12, 0, 0, 0, 16, 9, 0, 16, 0, 0, 0, 0, 0, 36, 10, 5, 0, 0, 32, 0, 0, 0, 0, 16
Offset: 1

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Comments

Right border = A127473, squares of phi(n) terms. Row sums = A057660: (1, 3, 7, 11, 21, ...).

Examples

			First few rows of the triangle:
  1;
  2, 1;
  3, 0, 4;
  4, 3, 0,  4;
  5, 0, 0,  0, 16;
  6, 3, 8,  0,  0, 4;
  7, 0, 0,  0,  0, 0, 36;
  8, 7, 0, 12,  0, 0,  0, 16;
  ...

Crossrefs

Cf. A127473, A057660, A054522.

Formula

(A054522)^2 as an infinite lower triangular matrix.

A328502 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).

Original entry on oeis.org

1, -3, -7, -2, -21, 21, -43, -4, -12, 63, -111, 14, -157, 129, 147, -8, -273, 36, -343, 42, 301, 333, -507, 28, -80, 471, -36, 86, -813, -441, -931, -16, 777, 819, 903, 24, -1333, 1029, 1099, 84, -1641, -903, -1807, 222, 252, 1521, -2163, 56, -252, 240, 1911, 314, -2757, 108, 2331
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 22 2019

Keywords

Comments

Dirichlet inverse of A057660.

Links

Crossrefs

Cf. A000010, A008683, A030230 (positions of negative terms), A057660, A101035.

Programs

  • Mathematica
    a[1] = 1; a[n_] := -Sum[DivisorSigma[2, (n/d)^2]/DivisorSigma[1, (n/d)^2] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
Table[DivisorSum[n, EulerPhi[n/#] MoebiusMu[#] #^2 &], {n, 1, 55}]
f[p_, e_] := If[e == 1, p - 1 - p^2, -p^(e - 1)*(p - 1)^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)
  • PARI
    a(n)={sumdiv(n, d, eulerphi(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA057660(n/d) * a(d).
a(n) = Sum_{d|n} phi(n/d) * mu(d) * d^2.
Multiplicative with a(p) = p - 1 - p^2, and a(p^e) = -p^(e-1) * (p-1)^2, for e > 1. - Amiram Eldar, Dec 03 2022
a(n) = Sum_{k = 1..n} gcd(k, n)^2 * mu(gcd(k, n)) (follows from an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

A333558 a(n) = Sum_{d|n} phi(d) * prime(d).

Original entry on oeis.org

2, 5, 12, 19, 46, 41, 104, 95, 150, 165, 312, 203, 494, 365, 432, 519, 946, 545, 1208, 747, 990, 1105, 1828, 991, 1986, 1709, 2004, 1663, 3054, 1481, 3812, 2615, 3062, 3173, 3724, 2519, 5654, 4145, 4512, 3591, 7162, 3449, 8024, 4979, 5298, 6209, 9708, 4983, 9638, 6685
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 26 2020

Keywords

Crossrefs

Cf. A000010, A000040, A007445, A057660, A061150, A130029.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d] Prime[d], {d, Divisors[n]}], {n, 1, 50}]
Table[Sum[Prime[n/GCD[n, k]], {k, 1, n}], {n, 1, 50}]
  • PARI
    a(n) = sumdiv(n, d, prime(d)*eulerphi(d)); \\ Michel Marcus, Mar 27 2020

Formula

G.f.: Sum_{k>=1} phi(k) * prime(k) * x^k / (1 - x^k).
a(n) = Sum_{k=1..n} prime(n/gcd(n,k)).
a(n) = Sum_{k=1..n} prime(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021

A333695 Numerators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 3, 7, 11, 21, 7, 43, 43, 61, 63, 111, 77, 157, 129, 49, 171, 273, 61, 343, 231, 43, 333, 507, 301, 521, 471, 547, 473, 813, 147, 931, 683, 259, 819, 129, 671, 1333, 1029, 1099, 903, 1641, 43, 1807, 111, 427, 1521, 2163, 399, 2101, 1563, 637, 1727, 2757, 547, 2331
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2020

Keywords

Examples

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Crossrefs

Cf. A000010, A000203, A001157, A018804, A057660, A071708, A072155, A074947, A074949, A333696 (denominators), A346602.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 55}] // Numerator
Table[Sum[1/GCD[n, k], {k, n}], {n, 55}] // Numerator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 55}] // Numerator
  • PARI
    a(n) = numerator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

Formula

a(n) = numerator of Sum_{d|n} phi(n/d) / d.
a(n) = numerator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = numerator of sigma_2(n^2) / (n * sigma_1(n^2)).
a(p) = p^2 - p + 1 where p is prime.
From Amiram Eldar, Nov 21 2022: (Start)
a(n) = numerator(A057660(n)/n).
Sum_{k=1..n} a(k)/A333696(k) ~ c * n^2, where c = zeta(3)/(2*zeta(2)) = 0.365381... (A346602). (End)

A333696 Denominators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 12, 13, 14, 5, 16, 17, 6, 19, 20, 3, 22, 23, 24, 25, 26, 27, 28, 29, 10, 31, 32, 11, 34, 5, 36, 37, 38, 39, 40, 41, 2, 43, 4, 15, 46, 47, 16, 49, 50, 17, 52, 53, 18, 55, 56, 57, 58, 59, 20, 61, 62, 63, 64, 65, 22, 67, 68, 23, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2020

Keywords

Examples

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Crossrefs

Cf. A000010, A000203, A001157, A018804, A057660, A070999, A071000 (fixed points), A071708, A072155, A074947, A074949, A333695 (numerators).

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 70}] // Denominator
Table[Sum[1/GCD[n, k], {k, n}], {n, 70}] // Denominator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 70}] // Denominator
  • PARI
    a(n) = denominator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

Formula

a(n) = denominator of Sum_{d|n} phi(n/d) / d.
a(n) = denominator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = denominator of sigma_2(n^2) / (n * sigma_1(n^2)).

A346770 Expansion of g.f. Product_{k>=1} (1 - x^k)^phi(k), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, -1, -1, -1, 0, 0, 3, 1, 4, 2, 3, -5, 1, -13, -5, -13, -6, -22, 12, -12, 35, 17, 59, 11, 101, -1, 81, -35, 45, -165, 29, -311, -69, -383, -57, -501, 181, -501, 425, -191, 990, -70, 1844, 64, 2305, 183, 2625, -951, 2897, -2701, 1845, -4851, 664, -8824, 670, -12366, 269, -14137, 2884
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2021

Keywords

Links

Crossrefs

Convolution inverse of A061255.
Cf. A000010, A057660, A293116, A299069.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^eulerphi(k)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sigma(k^2, 2)/sigma(k^2)*x^k/k)))

Formula

G.f.: exp(-Sum_{k>=1} A057660(k) * x^k/k).
a(0) = 1, a(n) = -(1/n) * Sum_{k=1..n} A057660(k) * a(n-k) for n > 0.

A349469 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1)*zeta(s-3)/(zeta(s-2))^2.

Original entry on oeis.org

1, 2, 12, 20, 80, 24, 252, 168, 360, 160, 1100, 240, 1872, 504, 960, 1360, 4352, 720, 6156, 1600, 3024, 2200, 11132, 2016, 10400, 3744, 9828, 5040, 22736, 1920, 27900, 10912, 13200, 8704, 20160, 7200, 47952, 12312, 22464, 13440, 65600, 6048, 75852, 22000, 28800, 22264, 99452, 16320, 88200, 20800
Offset: 1

Views

Author

Werner Schulte, Nov 18 2021

Keywords

Links

Crossrefs

Cf. A057660, A068963, A340850.

Programs

  • Mathematica
    f[p_, e_] := (p - 1)*p^e*(p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 18 2021 *)

Formula

Multiplicative with a(p^e) = p^e * (p^(2*e)-1) * (p-1) / (p+1) for e > 0 and prime p.
Dirichlet convolution with A057660 equals A068963.
Equals n * A340850(n) for n > 0.
Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and
b(p^e) = -(p-1)^2 * e * p^(2*e-1) for prime p and e > 0.
Sum_{k=1..n} a(k) ~ c * n^4, where c = 9*zeta(3)/Pi^4 = 0.111062... . - Amiram Eldar, Oct 16 2022
Previous Showing 51-60 of 65 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.