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 21-27 of 27 results.

A359954 a(n) = Sum_{d|n} tau(d)*phi(d)*mu(n/d).

Original entry on oeis.org

1, 1, 3, 4, 7, 3, 11, 10, 14, 7, 19, 12, 23, 11, 21, 24, 31, 14, 35, 28, 33, 19, 43, 30, 52, 23, 54, 44, 55, 21, 59, 56, 57, 31, 77, 56, 71, 35, 69, 70, 79, 33, 83, 76, 98, 43, 91, 72, 114, 52, 93, 92, 103, 54, 133, 110, 105, 55, 115, 84, 119, 59, 154, 128, 161, 57, 131
Offset: 1

Views

Author

Wesley Ivan Hurt, May 12 2023

Keywords

Comments

Möbius transform of tau(n)*phi(n) (A062355).

Links

Crossrefs

Cf. A000005 (tau), A000010 (phi), A008683 (mu), A062355, A363012.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, 2*p - 3, (e*p + p - e)*(p - 1)*p^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 16 2023 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d)*eulerphi(d)*moebius(n/d)); \\ Michel Marcus, May 13 2023

Formula

Multiplicative with a(p) = 2*p - 3 and a(p^e) = (e*p + p - e)*(p - 1)*p^(e - 2) for e > 1. - Amiram Eldar, May 16 2023

A067892 Numbers k such that tau(k)*phi(k) > prime(k) where tau(k) = A000005(k).

Original entry on oeis.org

720, 864, 1008, 1080, 1120, 1260, 1296, 1440, 1512, 1584, 1680, 1728, 1760, 1800, 1872, 1980, 2016, 2080, 2160, 2240, 2304, 2376, 2400, 2448, 2464, 2520, 2592, 2640, 2688, 2700, 2720, 2736, 2772, 2800, 2808, 2835, 2880, 2912, 3024, 3080, 3120, 3168
Offset: 1

Views

Author

Benoit Cloitre, Mar 02 2002

Keywords

Links

Crossrefs

Cf. A000005 (tau), A000010 (phi), A062355.

Programs

  • Mathematica
    Select[Range[3200],EulerPhi[#]DivisorSigma[0,#]>Prime[#]&] (* Harvey P. Dale, Nov 17 2018 *)
  • PARI
    is(k) = numdiv(k) * eulerphi(k) > prime(k); \\ Amiram Eldar, Apr 17 2024
  • PARI
    lista(pmax) = {my(f, k = 0); forprime(p=1, pmax, k++; f = factor(k); if(numdiv(f) * eulerphi(f) > p, print1(k, ", ")));} \\ Amiram Eldar, Apr 17 2024

A309153 a(n) = A000203(n)*A001227(n).

Original entry on oeis.org

1, 3, 8, 7, 12, 24, 16, 15, 39, 36, 24, 56, 28, 48, 96, 31, 36, 117, 40, 84, 128, 72, 48, 120, 93, 84, 160, 112, 60, 288, 64, 63, 192, 108, 192, 273, 76, 120, 224, 180, 84, 384, 88, 168, 468, 144, 96, 248, 171, 279, 288, 196, 108, 480, 288, 240, 320, 180, 120, 672, 124, 192, 624, 127, 336, 576, 136, 252, 384
Offset: 1

Views

Author

Omar E. Pol, Jul 14 2019

Keywords

Comments

A001227(n) is denoted by Delta_0(n) in Glaisher 1907.
a(n) = A000203(n) iff n is a power of 2.

Links

Crossrefs

Cf. A000079, A000203, A001227, A062354, A062355, A064840.
Cf. A001620, A002117, A244115, A306016

Programs

  • Mathematica
    Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* Michael De Vlieger, Nov 22 2019 *)
f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

Formula

a(n) = sigma(n)*delta(n).
Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - Amiram Eldar, Nov 01 2022
From Amiram Eldar, Dec 04 2023: (Start)
Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A341636 a(n) = Sum_{d|n} phi(d) * tau(d) * tau(n/d).

Original entry on oeis.org

1, 4, 6, 13, 10, 24, 14, 38, 29, 40, 22, 78, 26, 56, 60, 103, 34, 116, 38, 130, 84, 88, 46, 228, 79, 104, 124, 182, 58, 240, 62, 264, 132, 136, 140, 377, 74, 152, 156, 380, 82, 336, 86, 286, 290, 184, 94, 618, 153, 316, 204, 338, 106, 496, 220, 532, 228, 232, 118, 780, 122, 248
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

Comments

Inverse Moebius transform of A062949.

Links

Crossrefs

Cf. A000005, A000010, A000203, A007426, A055507, A062355, A062949.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d] DivisorSigma[0, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 62}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[0, n/GCD[n, k]], {k, n}], {n, 62}]
f[p_, e_] := (p + 1 + e*(p - 1) + p^(e + 1)*(e*(p - 1) + p - 3))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

Formula

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * tau(n/gcd(n,k)).
a(n) = Sum_{d|n} A062949(d).
Multiplicative with a(p^e) = (p + 1 + e*(p-1) + p^(e+1)*(e*(p-1)+p-3))/(p-1)^2. - Amiram Eldar, Sep 15 2023

A348060 a(n) = Sum_{k=1..n, gcd(n,k) = 1} (k-1) / gcd(n,k-1).

Original entry on oeis.org

0, 0, 1, 1, 6, 2, 15, 5, 15, 8, 45, 7, 66, 18, 30, 21, 120, 18, 153, 23, 70, 50, 231, 24, 190, 72, 147, 56, 378, 40, 435, 85, 194, 128, 280, 55, 630, 162, 284, 87, 780, 68, 861, 153, 264, 242, 1035, 91, 861, 200, 502, 213, 1326, 156, 788, 205, 642, 392, 1653, 110, 1770, 450, 606, 341, 1136
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 26 2021

Keywords

Crossrefs

Cf. A057661, A062355, A348061.

Programs

  • Mathematica
    Table[Sum[If[GCD[n, k] == 1, (k - 1)/GCD[n, k - 1], 0], {k, n}], {n, 65}]
  • PARI
    a(n) = sum(k=1, n, if (gcd(n, k)==1, (k-1)/gcd(n, k-1))); \\ Michel Marcus, Sep 27 2021

A127504 Triangle T(n,k) = phi(n) if k|n, =0 otherwise.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 2, 2, 0, 2, 4, 0, 0, 0, 4, 2, 2, 2, 0, 0, 2, 6, 0, 0, 0, 0, 0, 6, 4, 4, 0, 4, 0, 0, 0, 4, 6, 0, 6, 0, 0, 0, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 4
Offset: 1

Views

Author

Gary W. Adamson, Jan 17 2007

Keywords

Examples

			First few rows of the triangle are:
1;
1, 1;
2, 0, 2;
2, 2, 0, 2;
4, 0, 0, 0, 4;
2, 2, 2, 0, 0, 2;
6, 0, 0, 0, 0, 0, 6;
4, 4, 0, 4, 0, 0, 0, 4;
...

Crossrefs

Cf. A051731, A062355 (row sums), A000010 (phi).

Formula

T(n,k) = A000010(n)*A051731(n,k), 1<=k<=n.

A348011 a(n) = phi(n^2) * Sum_{d|n} 2^omega(d) / d.

Original entry on oeis.org

1, 4, 10, 20, 28, 40, 54, 88, 102, 112, 130, 200, 180, 216, 280, 368, 304, 408, 378, 560, 540, 520, 550, 880, 740, 720, 954, 1080, 868, 1120, 990, 1504, 1300, 1216, 1512, 2040, 1404, 1512, 1800, 2464, 1720, 2160, 1890, 2600, 2856, 2200, 2254, 3680, 2730, 2960
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 24 2021

Keywords

Crossrefs

Cf. A000010, A001221, A002618, A007434, A034444, A047994, A060648, A062355, A069097, A191414, A341772.

Programs

  • Mathematica
    Table[EulerPhi[n^2] DivisorSum[n, 2^PrimeNu[#]/# &], {n, 50}]
f[p_, e_] := p^(e - 1) ((p + 1) p^e - 2); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50]
  • PARI
    a(n) = eulerphi(n^2)*sumdiv(n, d, 2^omega(d)/d); \\ Michel Marcus, Sep 24 2021

Formula

Multiplicative with a(p^e) = p^(e-1) * ((p + 1) * p^e - 2).
a(n) = Sum_{k=1..n, gcd(n,k) = 1} gcd(n,k-1)^2.
a(n) = Sum_{k=1..n} uphi(gcd(n,k)^2).
a(n) = Sum_{d|n} phi(n/d) * uphi(d^2).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.4083249979... . - Amiram Eldar, Nov 05 2022
Previous Showing 21-27 of 27 results.

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