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 1-5 of 5 results.

A029939 a(n) = Sum_{d|n} phi(d)^2.

Original entry on oeis.org

1, 2, 5, 6, 17, 10, 37, 22, 41, 34, 101, 30, 145, 74, 85, 86, 257, 82, 325, 102, 185, 202, 485, 110, 417, 290, 365, 222, 785, 170, 901, 342, 505, 514, 629, 246, 1297, 650, 725, 374, 1601, 370, 1765, 606, 697, 970, 2117, 430, 1801, 834, 1285, 870, 2705, 730, 1717, 814, 1625
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Equals the inverse Mobius transform (A051731) of A127473. - Gary W. Adamson, Aug 20 2008
Number of (i,j) in {1,2,...,n}^2 such that gcd(n,i) = gcd(n,j). - Benoit Cloitre, Dec 31 2020

Links

Crossrefs

Cf. A051731, A062367, A065463, A127473, A253905.

Programs

  • Maple
    with(numtheory): A029939 := proc(n) local i,j; j := 0; for i in divisors(n) do j := j+phi(i)^2; od; j; end;
# alternative
N:= 1000: # to get a(1)..a(N)
A:= Vector(N,1):
for d from 2 to N do
  pd:= numtheory:-phi(d)^2;
  md:= [seq(i,i=d..N,d)];
  A[md]:= map(`+`,A[md],pd);
od:
seq(A[i],i=1..N); # Robert Israel, May 30 2016
  • Mathematica
    Table[Total[EulerPhi[Divisors[n]]^2],{n,60}] (* Harvey P. Dale, Feb 04 2017 *)
f[p_, e_] := (p^(2*e)*(p-1)+2)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^2); \\ Michel Marcus, Jan 17 2017

Formula

Multiplicative with a(p^e) = (p^(2*e)*(p-1)+2)/(p+1). - Vladeta Jovovic, Nov 19 2001
G.f.: Sum_{k>=1} phi(k)^2*x^k/(1 - x^k), where phi(k) is the Euler totient function (A000010). - Ilya Gutkovskiy, Jan 16 2017
a(n) = Sum_{k=1..n} phi(n/gcd(n, k)). - Ridouane Oudra, Nov 28 2019
Sum_{k>=1} 1/a(k) = 2.3943802654751092440350752246012273573942903149891228695146514601814537713... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/(3*zeta(2))) * Product_{p prime} (1 - 1/(p*(p+1))) = A253905 * A065463 / 3 = 0.171593... . - Amiram Eldar, Oct 25 2022

A065018 a(n) = Sum_{d|n} sigma(d)^2.

Original entry on oeis.org

1, 10, 17, 59, 37, 170, 65, 284, 186, 370, 145, 1003, 197, 650, 629, 1245, 325, 1860, 401, 2183, 1105, 1450, 577, 4828, 998, 1970, 1786, 3835, 901, 6290, 1025, 5214, 2465, 3250, 2405, 10974, 1445, 4010, 3349, 10508, 1765, 11050, 1937, 8555, 6882
Offset: 1

Views

Author

Vladeta Jovovic, Nov 19 2001

Keywords

Links

Crossrefs

Cf. A029939, A062367.

Programs

  • PARI
    { for (n=1, 1000, a=sumdiv(n, d, sigma(d)^2); write("b065018.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 03 2009
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^2*x^k/(1-x^k))) \\ Seiichi Manyama, May 08 2021

Formula

Dirichlet convolution of A072861 and A000012. Dirichlet g.f.: zeta^2(s)*zeta^2(s-1)*zeta(s-2)/zeta(2s-2). - R. J. Mathar, Feb 03 2011
Sum_{k=1..n} a(k) ~ 5 * Zeta(3)^2 * n^3 / 6. - Vaclav Kotesovec, Feb 01 2019
From Seiichi Manyama, May 08 2021: (Start)
G.f.: Sum_{k >= 1} sigma(k)^2 * x^k/(1 - x^k).
If p is prime, a(p) = 1 + (p+1)^2. (End)

A344080 a(n) = Sum_{d|n} tau(d)^n, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 9, 98, 33, 4225, 129, 72354, 20196, 1050625, 2049, 2194099186, 8193, 268468225, 1073807361, 156925970179, 131073, 101629064089930, 524289, 3657261440572306, 4398050705409, 17592194433025, 8388609, 4727105427440383342818, 847322163876, 4503599761588225
Offset: 1

Views

Author

Seiichi Manyama, May 09 2021

Keywords

Links

Crossrefs

Cf. A007425, A062367, A097988, A279789, A344060, A344081.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[0, #]^n &]; Array[a, 26] (* Amiram Eldar, May 09 2021 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d)^n);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (numdiv(k)*x)^k/(1-(numdiv(k)*x)^k)))

Formula

G.f.: Sum_{k >= 1} (tau(k) * x)^k/(1 - (tau(k) * x)^k).
If p is prime, a(p) = 1 + 2^p.

A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 9, 86, 33, 4109, 129, 65622, 19692, 1048613, 2049, 2176786526, 8193, 268435589, 1073741865, 152587956247, 131073, 101559956692208, 524289, 3656158441111670, 4398046511241, 17592186046469, 8388609, 4722366482871822065758
Offset: 1

Views

Author

Seiichi Manyama, May 09 2021

Keywords

Links

Crossrefs

Cf. A007425, A062367, A097988, A279789, A344047, A344080.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[0, #]^# &]; Array[a, 24] (* Amiram Eldar, May 09 2021 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d)^d);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (numdiv(k)*x)^k/(1-x^k)))

Formula

G.f.: Sum_{k >= 1} (tau(k) * x)^k/(1 - x^k).
If p is prime, a(p) = 1 + 2^p.

A224834 a(n) = Sum {d|n, d <= n^(1/2)} tau(d)^2.

Original entry on oeis.org

1, 1, 1, 5, 1, 5, 1, 5, 5, 5, 1, 9, 1, 5, 5, 14, 1, 9, 1, 14, 5, 5, 1, 18, 5, 5, 5, 14, 1, 13, 1, 14, 5, 5, 5, 34, 1, 5, 5, 18, 1, 25, 1, 14, 9, 5, 1, 34, 5, 9, 5, 14, 1, 25, 5, 18, 5, 5, 1, 38, 1, 5, 9, 30, 5, 25, 1, 14, 5, 13, 1, 50, 1, 5, 9
Offset: 1

Views

Author

Michel Marcus, Jul 21 2013

Keywords

Links

Crossrefs

Cf. A000330, A007425, A062367, A224835.

Programs

  • Maple
    f:= proc(n) add(numtheory:-tau(d)^2, d = select(t -> (t^2<=n), numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Nov 30 2016
  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[0, #]^2 &, #^2 <= n &]; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (d<=sqrtn(n, 2))*numdiv(d)^2) \\ Michel Marcus, Jul 21 2013
  • PARI
    a(n)=my(s=sqrtint(n)); sumdiv(n,d,if(d<=s,numdiv(d)^2)) \\ Charles R Greathouse IV, Jul 22 2013

Formula

If p is prime, a(p^k) = A000330(1+floor(k/2)). - Robert Israel, Nov 30 2016
Showing 1-5 of 5 results.

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

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