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.

A226560 exp( Sum_{n>=1} A226561(n)*x^n/n ), where A226561(n) = Sum_{d|n} d^n*phi(d).

Original entry on oeis.org

1, 1, 3, 21, 155, 2691, 18924, 732230, 9223166, 269544904, 4308339664, 264486350330, 3252603264996, 283488024709418, 5058264756924275, 239269507574263597, 9478611818612363119, 788664781674375008343, 13928483471031628860556, 1889997256419148641470346
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 155*x^4 + 2691*x^5 + 18924*x^6 +...
where
log(A(x)) = x + 5*x^2/2 + 55*x^3/3 + 529*x^4/4 + 12501*x^5/5 + 94835*x^6/6 + 4941259*x^7/7 + 67240193*x^8/8 +...+ A226561(n)*x^n/n +...

Crossrefs

Cf. A226561, A226458.

Programs

  • PARI
    {A226561(n)=sumdiv(n, d, d*eulerphi(d^n))}
{a(n)=polcoeff(exp(sum(k=1,n,A226561(k)*x^k/k)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

A332517 a(n) = Sum_{k=1..n} gcd(n,k)^n.

Original entry on oeis.org

1, 5, 29, 274, 3129, 47515, 823549, 16843268, 387459861, 10009769725, 285311670621, 8918311856102, 302875106592265, 11112685048729175, 437893951473411261, 18447025557276459016, 827240261886336764193, 39346558373052524325225, 1978419655660313589123997
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 14 2020

Keywords

Comments

If n is prime, a(n) = n-1 + n^n. - Robert Israel, Feb 16 2020

Links

Crossrefs

Cf. A000010, A008683, A018804, A031971, A069097, A226561, A228640, A321294.

Programs

  • Magma
    [&+[Gcd(n,k)^n:k in [1..n]]: n in [1..20]]; // Marius A. Burtea, Feb 15 2020
  • Maple
    f:= n -> add(igcd(n,k)^n,k=1..n):
map(f, [$1..30]); # Robert Israel, Feb 16 2020
  • Mathematica
    Table[Sum[GCD[n, k]^n, {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[n/d] d^n, {d, Divisors[n]}], {n, 1, 19}]
Table[Sum[MoebiusMu[n/d] d DivisorSigma[n - 1, d], {d, Divisors[n]}], {n, 1, 19}]
  • PARI
    a(n) = sum(k=1, n, gcd(n, k)^n); \\ Michel Marcus, Feb 14 2020
  • Python
    from sympy import totient, divisors
def A332517(n):
    return sum(totient(d)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = Sum_{d|n} phi(n/d) * d^n.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-1)(d).
a(n) ~ n^n.
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^n*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k)*sigma_(n-1)(gcd(n,k))/phi(n/gcd(n,k)). (End)

A226459 a(n) = Sum_{d|n} phi(d^d), where phi(n) is the Euler totient function A000010(n).

Original entry on oeis.org

1, 3, 19, 131, 2501, 15573, 705895, 8388739, 258280345, 4000002503, 259374246011, 2972033498453, 279577021469773, 4762288640230761, 233543408203127519, 9223372036863164547, 778579070010669895697, 13115469358432437487707, 1874292305362402347591139
Offset: 1

Views

Author

Paul D. Hanna, Jun 08 2013

Keywords

Comments

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

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 19*x^3/3 + 131*x^4/4 + 2501*x^5/5 + ...
where
exp(L(x)) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 547*x^5 + 3193*x^6 + ... + A226458(n)*x^n + ...

Links

Crossrefs

Cf. A226458, A226561, A000010.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (&+[EulerPhi(k^k)*x^k/(1-x^k): k in [1..3*m]]) )); // G. C. Greubel, Nov 07 2018
  • Mathematica
    ttf[n_]:=Module[{d=Divisors[n]},Total[EulerPhi[d^d]]]; Array[ttf,20] (* Harvey P. Dale, Aug 21 2013 *)
With[{nmax = 30}, Rest[CoefficientList[Series[Sum[EulerPhi[k^k]*x^k/(1 - x^k), {k, 1, 2*nmax}], {x, 0, nmax}], x]]] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    {a(n)=sumdiv(n,d, eulerphi(d^d))}
for(n=1,30,print1(a(n),", "))
  • PARI
    a(n) = sum(k=1, n, (n/gcd(k, n))^(n/gcd(k, n)-1)); \\ Seiichi Manyama, Mar 11 2021
  • Python
    from sympy import totient, divisors
def A226459(n):
    return sum(totient(d)*d**(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = Sum_{d|n} d^(d-1) * phi(d).
Equals the logarithmic derivative of A226458.
G.f.: Sum_{k>=1} phi(k^k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Sum_{k=1..n} (n/gcd(k,n))^(n/gcd(k,n)-1). - Seiichi Manyama, Mar 11 2021
From Richard L. Ollerton, May 08 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)^gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi((n/gcd(n,k))^(n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} gcd(n,k)^(gcd(n,k)-1)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A332621 a(n) = (1/n) * Sum_{k=1..n} n^(n/gcd(n, k)).

Original entry on oeis.org

1, 3, 19, 133, 2501, 15631, 705895, 8389641, 258280489, 4000040011, 259374246011, 2972033984173, 279577021469773, 4762288684702095, 233543408203327951, 9223372037928525841, 778579070010669895697, 13115469358498302735067, 1874292305362402347591139
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Links

Crossrefs

Cf. A000010, A056665, A130586, A226561, A228640, A308814, A321349, A332620.

Programs

  • Magma
    [(1/n)*&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[(1/n) Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[(1/n) Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^(j - 1) x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
  • PARI
    a(n) = sum(k=1, n, n^(n/gcd(n, k)))/n; \\ Michel Marcus, Mar 10 2021

Formula

a(n) = [x^n] Sum_{k>=1} Sum_{j>=1} phi(j) * n^(j-1) * x^(k*j).
a(n) = (1/n) * Sum_{k=1..n} n^(lcm(n, k)/k).
a(n) = (1/n) * Sum_{d|n} phi(d) * n^d.
a(n) = A332620(n) / n.

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)

A332620 a(n) = Sum_{k=1..n} n^(n/gcd(n, k)).

Original entry on oeis.org

1, 6, 57, 532, 12505, 93786, 4941265, 67117128, 2324524401, 40000400110, 2853116706121, 35664407810076, 3634501279107049, 66672041585829330, 3503151123049919265, 147573952606856413456, 13235844190181388226849, 236078448452969449231206, 35611553801885644604231641
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Links

Crossrefs

Cf. A000010, A056665, A066108, A226561, A228640, A321349, A332621.

Programs

  • Magma
    [&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^j x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
  • PARI
    a(n) = sum(k=1, n, n^(n/gcd(n, k))); \\ Michel Marcus, Mar 10 2021

Formula

a(n) = [x^n] Sum_{k>=1} Sum_{j>=1} phi(j) * n^j * x^(k*j).
a(n) = Sum_{k=1..n} n^(lcm(n, k)/k).
a(n) = Sum_{d|n} phi(d) * n^d.
a(n) = n * A332621(n).

A332653 a(n) = (1/n) * Sum_{k=1..n} n^(k/gcd(n, k)).

Original entry on oeis.org

1, 2, 5, 19, 157, 1306, 19609, 266372, 5321721, 101001214, 2593742461, 61920391842, 1941507093541, 56984643437138, 2076518238897649, 72340172854919941, 3041324492229179281, 121440691499123469858, 5784852794328402307381, 262799364106291328009626
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 18 2020

Keywords

Crossrefs

Cf. A023037, A056665, A057661, A226561, A228640, A332620, A332621, A332652, A332655.

Programs

  • Magma
    [(1/n)*&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020
  • Mathematica
    Table[(1/n) Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^(k - 1), 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

Formula

a(n) = (1/n) * Sum_{k=1..n} n^(lcm(n, k)/n).
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} n^(k-1).
a(n) = A332652(n) / n.

A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

Original entry on oeis.org

1, 2, 10, 84, 1301, 15693, 376762, 6168552, 176787631, 3770427352, 142364319626, 3152758480715, 154718778284149, 4340093860950619, 210971170836848270, 7281694486114555088, 435659030617933827137, 14181121059071691716406, 1052864393300587929716722, 41673907052879908244100770
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 18 2020

Keywords

Crossrefs

Cf. A031971, A057661, A226561, A332517, A332621, A332652, A332653, A332654.

Programs

  • Magma
    [&+[(k div Gcd(n,k))^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 18 2020
  • Mathematica
    Table[Sum[(k/GCD[n, k])^n, {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^n, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

Formula

a(n) = Sum_{k=1..n} (lcm(n, k)/n)^n.
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} k^n.

A332652 a(n) = Sum_{k=1..n} n^(k/gcd(n, k)).

Original entry on oeis.org

1, 4, 15, 76, 785, 7836, 137263, 2130976, 47895489, 1010012140, 28531167071, 743044702104, 25239592216033, 797785008119932, 31147773583464735, 1157442765678719056, 51702516367896047777, 2185932446984222457444, 109912203092239643840239, 5255987282125826560192520
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 18 2020

Keywords

Crossrefs

Cf. A031972, A056665, A057661, A226561, A228640, A332620, A332621, A332653, A332655.

Programs

  • Magma
    [&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020
  • Mathematica
    Table[Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^k, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

Formula

a(n) = Sum_{k=1..n} n^(lcm(n, k)/n).
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} n^k.
a(n) = n * A332653(n).

A342412 a(n) = Sum_{k=1..n} (n/gcd(k,n))^(n-2).

Original entry on oeis.org

1, 2, 7, 37, 501, 2771, 100843, 1056833, 28702189, 401562757, 23579476911, 247792605523, 21505924728445, 340246521979079, 15569565432876147, 576478345026355201, 45798768824157052689, 728648310343004595593, 98646963440126439346903
Offset: 1

Views

Author

Seiichi Manyama, Mar 11 2021

Keywords

Crossrefs

Cf. A000010, A226561, A321349, A342411.

Programs

  • Mathematica
    a[n_] := Sum[(n/GCD[k, n])^(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-2));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d^(n-1)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*d^(n-2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k^(k-1))*x^k/(1-(k*x)^k)))

Formula

a(n) = Sum_{d|n} phi(d^(n-1)) = Sum_{d|n} phi(d) * d^(n-2).
G.f.: Sum_{k>=1} phi(k^(k-1))*x^k/(1 - (k*x)^k).
Showing 1-10 of 10 results.

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