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 1-6 of 6 results.

A226458 G.f.: exp( Sum_{n>=1} A226459(n)*x^n/n ), where A226459(n) = Sum_{d|n} phi(d^d).

Original entry on oeis.org

1, 1, 2, 8, 41, 547, 3193, 104733, 1159483, 29990445, 431859113, 24050995053, 272382000003, 21806033497537, 362394321610042, 15956110448082190, 592910703485329797, 46410258555248498805, 775743319456458483203, 99472768731785230089041
Offset: 0

Views

Author

Paul D. Hanna, Jun 08 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 547*x^5 + 3193*x^6 +...
where
log(A(x)) = x + 3*x^2/2 + 19*x^3/3 + 131*x^4/4 + 2501*x^5/5 +...+ A226459(n)*x^n/n +...

Crossrefs

Cf. A226459, A226560.

Programs

  • PARI
    {A226459(n)=sumdiv(n,d, eulerphi(d^d))}
{a(n)=polcoeff(exp(sum(m=1, n+1, A226459(m)*x^m/m)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

The logarithmic derivative yields A226459.

A226561 a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).

Original entry on oeis.org

1, 5, 55, 529, 12501, 94835, 4941259, 67240193, 2324562301, 40039063525, 2853116706111, 35668789979107, 3634501279107037, 66676110291801575, 3503151245145885315, 147575078498173255681, 13235844190181388226833, 236079349222711695887225, 35611553801885644604231623
Offset: 1

Views

Author

Paul D. Hanna, Jun 10 2013

Keywords

Comments

Compare formula to the identity: Sum_{d|n} phi(d) = n.

Examples

			L.g.f.: L(x) = x + 5*x^2/2 + 55*x^3/3 + 529*x^4/4 + 12501*x^5/5 + 94835*x^6/6 + ...
where
exp(L(x)) = 1 + x + 3*x^2 + 21*x^3 + 155*x^4 + 2691*x^5 + 18924*x^6 + 732230*x^7 + 9223166*x^8 + ... + A226560(n)*x^n + ...

Links

Crossrefs

Cf. A226560, A226459, A000010, A321349, A332517.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (&+[EulerPhi(k)*(k*x)^k/(1-(k*x)^k): k in [1..2*m]]) )); // G. C. Greubel, Nov 07 2018
  • Maple
    f:= n -> add(d^n * numtheory:-phi(d), d = numtheory:-divisors(n)):
map(f, [$1..40]); # Robert Israel, Jun 16 2017
  • Mathematica
    Table[DivisorSum[n, #*EulerPhi[#^n]  &], {n, 1, 30}]  (* or *) With[{nmax = 30}, Rest[CoefficientList[Series[Sum[EulerPhi[k]*(k*x)^k/(1 - (k*x)^k), {k, 1, 2*nmax}], {x, 0, nmax}], x]]]  (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    {a(n)=sumdiv(n, d, d^n*eulerphi(d))}
for(n=1,30,print1(a(n),", "))
  • PARI
    a(n) = sum(k=1, n, (n/gcd(k, n))^n); \\ Seiichi Manyama, Mar 11 2021
  • Python
    from sympy import totient, divisors
def A226561(n):
    return sum(totient(d)*d**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

Logarithmic derivative of A226560.
a(n) = Sum_{d|n} d * phi(d^n).
a(n) = Sum_{d|n} phi(d^(n+1)).
a(n) = Sum_{d|n} phi(d^(n+2))/d.
a(n) = Sum_{d|n} d^(n-k+1) * phi(d^k) for k >= 1.
G.f.: Sum_{k>=1} phi(k)*(k*x)^k/(1 - (k*x)^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Sum_{k=1..n} (n/gcd(k,n))^n. - Seiichi Manyama, Mar 11 2021
a(n) = Sum_{k=1..n} gcd(n,k)^n*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021

A321349 a(n) = Sum_{d|n} phi(d^n), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 3, 19, 137, 2501, 16071, 705895, 8421505, 258293449, 4007813013, 259374246011, 2972767821815, 279577021469773, 4762869973595499, 233543432626753439, 9223512776490647553, 778579070010669895697, 13115569455375954492093, 1874292305362402347591139
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 06 2018

Keywords

Links

Crossrefs

Cf. A000010, A057660, A068963, A068970, A226459, A226561.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d^n], {d, Divisors[n]}], {n, 19}]
nmax = 19; Rest[CoefficientList[Series[Sum[k^(k - 1) EulerPhi[k] x^k/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(n/GCD[n, k])^(n - 1), {k, n}], {n, 19}]
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d^n)); \\ Michel Marcus, Nov 06 2018

Formula

G.f.: Sum_{k>=1} k^(k-1)*phi(k)*x^k/(1 - (k*x)^k).
a(n) = Sum_{d|n} d^(n-1)*phi(d).
a(n) = Sum_{k=1..n} (n/gcd(n,k))^(n-1).
From Richard L. Ollerton, May 08 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)^n)/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} gcd(n,k)^(n-1)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A342411 a(n) = Sum_{k=1..n} (n/gcd(k,n))^(n/gcd(k,n) - 2).

Original entry on oeis.org

1, 2, 7, 34, 501, 2600, 100843, 1048610, 28697821, 400000502, 23579476911, 247669459528, 21505924728445, 340163474352620, 15569560546875507, 576460752304472098, 45798768824157052689, 728637186579594211070, 98646963440126439346903
Offset: 1

Views

Author

Seiichi Manyama, Mar 11 2021

Keywords

Crossrefs

Cf. A000010, A226459, A342412.

Programs

  • Mathematica
    a[n_] := Sum[(n/GCD[k, n])^(n/GCD[k, n] - 2), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 11 2021 *)
  • PARI
    a(n) = sum(k=1, n, (n/gcd(k, n))^(n/gcd(k, n)-2));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d^(d-1)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*d^(d-2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k^(k-1))*x^k/(1-x^k)))

Formula

a(n) = Sum_{d|n} phi(d^(d-1)) = Sum_{d|n} phi(d) * d^(d-2).
G.f.: Sum_{k>=1} phi(k^(k-1))*x^k/(1 - x^k).

A342420 a(n) = Sum_{k=1..n} (n/gcd(k,n))^(n/gcd(k,n)).

Original entry on oeis.org

1, 5, 55, 517, 12501, 93371, 4941259, 67109381, 2324522989, 40000012505, 2853116706111, 35664401886907, 3634501279107037, 66672040958289359, 3503151123046887555, 147573952589743522309, 13235844190181388226833, 236078448451781550068849, 35611553801885644604231623
Offset: 1

Views

Author

Seiichi Manyama, Mar 11 2021

Keywords

Crossrefs

Cf. A000010, A226459, A342411.

Programs

  • Mathematica
    a[n_] := Sum[(n/GCD[k, n])^(n/GCD[k, n]), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 11 2021 *)
  • PARI
    a(n) = sum(k=1, n, (n/gcd(k, n))^(n/gcd(k, n)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d^(d+1)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*d^d);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k^(k+1))*x^k/(1-x^k)))

Formula

a(n) = Sum_{d|n} phi(d^(d+1)) = Sum_{d|n} phi(d) * d^d.
G.f.: Sum_{k>=1} phi(k^(k+1))*x^k/(1 - x^k).

A342437 a(n) = Sum_{k=1..n} gcd(k,n)^(n/gcd(k,n) - 1).

Original entry on oeis.org

1, 2, 3, 5, 5, 14, 7, 25, 25, 74, 11, 161, 13, 398, 383, 657, 17, 2110, 19, 3341, 4485, 10262, 23, 19569, 2521, 49178, 39547, 74441, 29, 221462, 31, 328737, 590753, 1048610, 103379, 1905565, 37, 4718630, 6377655, 5573801, 41, 22462826, 43, 31459985, 40634221, 92274734, 47
Offset: 1

Views

Author

Seiichi Manyama, Mar 12 2021

Keywords

Links

Crossrefs

Cf. A000010, A226459, A342421, A342424.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^(n/GCD[k, n] - 1), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Mar 12 2021 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^(n/gcd(k, n)-1));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n/d-1));

Formula

a(n) = Sum_{d|n} phi(n/d) * d^(n/d-1).
If p is prime, a(p) = p.
Showing 1-6 of 6 results.

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

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