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.

A349173 Dirichlet convolution of A003415 with A003959, where A003415 is the arithmetic derivative and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 7, 1, 12, 1, 33, 10, 16, 1, 68, 1, 20, 18, 131, 1, 87, 1, 96, 22, 28, 1, 296, 16, 32, 67, 124, 1, 167, 1, 473, 30, 40, 26, 449, 1, 44, 34, 428, 1, 215, 1, 180, 147, 52, 1, 1128, 22, 171, 42, 208, 1, 510, 34, 560, 46, 64, 1, 881, 1, 68, 187, 1611, 38, 311, 1, 264, 54, 295, 1, 1871, 1, 80, 203, 292, 38, 359
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Links

Crossrefs

Cf. A003415, A003959, A349129, A349133, A349170, A349171, A349172, A349356.

Programs

  • Mathematica
    f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a1[1] = 0; a1[n_] := n*Plus @@ (f1 @@@ FactorInteger[n]); a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, a1[#] * a2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349173(n) = sumdiv(n,d,A003415(d)*A003959(n/d));

Formula

a(n) = Sum_{d|n} A003415(d) * A003959(n/d).
a(n) = Sum_{d|n} A349133(d) * A349356(n/d). - Antti Karttunen, Nov 16 2021
For all n >= 1, a(n) >= A349133(n).

A349131 a(n) = Sum_{d|n} phi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and phi is Euler totient function.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 12, 8, 14, 16, 20, 16, 24, 24, 32, 16, 32, 28, 36, 32, 48, 40, 44, 32, 52, 48, 46, 48, 56, 64, 60, 32, 80, 64, 96, 56, 72, 72, 96, 64, 80, 96, 84, 80, 112, 88, 92, 64, 114, 104, 128, 96, 104, 92, 160, 96, 144, 112, 116, 128, 120, 120, 168, 64, 192, 160, 132, 128, 176, 192, 140, 112, 144, 144, 208
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Dirichlet convolution of A003958 with Euler totient function phi, A000010.
Möbius transform of A349130.

Links

Crossrefs

Cf. A000010, A003958, A018804, A348981, A349130 (inverse Möbius transform), A349132, A349171.

Programs

  • Mathematica
    f[p_, e_] := (p - 1)*p^e - (p - 2)*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349131(n) = sumdiv(n,d,eulerphi(d)*A003958(n/d));

Formula

a(n) = Sum_{d|n} A000010(d) * A003958(n/d).
a(n) = Sum_{d|n} A008683(d) * A349130(n/d).
a(n) = Sum_{k=1..n} A003958(gcd(n, k)).
a(n) = A018804(n) - A348981(n).
For all n >= 1, a(n) <= A349171(n).
Multiplicative with a(p^e) = (p-1)*p^e - (p-2)*(p-1)^e. - Amiram Eldar, Nov 09 2021
Dirichlet g.f.: (zeta(s-1)/zeta(s)) / Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - Amiram Eldar, Dec 24 2023

A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 6, 1, 9, 1, 26, 9, 13, 1, 44, 1, 17, 15, 98, 1, 57, 1, 68, 19, 25, 1, 176, 15, 29, 57, 92, 1, 105, 1, 342, 27, 37, 23, 252, 1, 41, 31, 280, 1, 141, 1, 140, 111, 49, 1, 636, 21, 125, 39, 164, 1, 309, 31, 384, 43, 61, 1, 480, 1, 65, 147, 1138, 35, 213, 1, 212, 51, 209, 1, 960, 1, 77, 155, 236, 35, 249, 1, 1028
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of Euler phi (A000010) with A348507.
Möbius transform of A349140.

Links

Crossrefs

Cf. A000010, A003959, A008683, A018804, A348507, A349140 (inverse Möbius transform), A349142, A349143, A349171.
Cf. also A347131, A348981.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #) * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349141(n) = sumdiv(n,d,eulerphi(d)*A348507(n/d));

Formula

a(n) = Sum_{d|n} A000010(n/d) * A348507(d).
a(n) = Sum_{d|n} A008683(n/d) * A349140(d).
a(n) = Sum_{k=1..n} A348507(gcd(n,k)).
For all n >= 1, a(n) >= A347131(n) >= A348981(n).
a(n) = A349171(n) - A018804(n). - Antti Karttunen, Nov 14 2021

A349170 a(n) = Sum_{d|n} d * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

1, 5, 7, 19, 11, 35, 15, 65, 37, 55, 23, 133, 27, 75, 77, 211, 35, 185, 39, 209, 105, 115, 47, 455, 91, 135, 175, 285, 59, 385, 63, 665, 161, 175, 165, 703, 75, 195, 189, 715, 83, 525, 87, 437, 407, 235, 95, 1477, 169, 455, 245, 513, 107, 875, 253, 975, 273, 295, 119, 1463, 123, 315, 555, 2059, 297, 805, 135, 665
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Dirichlet convolution of A003959 with the identity function, A000027.
Dirichlet convolution of sigma (A000203) with A003968.

Links

Crossrefs

Cf. A000027, A000203, A003959, A003968, A038040, A349129, A349130, A349140, A349171 (Möbius transform), A349172, A349173.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^(e + 1) - p^(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349170(n) = sumdiv(n,d,d*A003959(n/d));

Formula

a(n) = Sum_{d|n} d * A003959(n/d).
a(n) = Sum_{d|n} A349171(d).
a(n) = Sum_{d|n} A000203(d) * A003968(n/d).
a(n) = A038040(n) + A349140(n).
For all n >= 1, a(n) >= A349129(n) >= A349130(n).
Multiplicative with a(p^e) = (p+1)^(e+1) - p^(e+1). - Amiram Eldar, Nov 09 2021

A349172 a(n) = Sum_{d|n} psi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and psi is Dedekind psi function, A001615.

Original entry on oeis.org

1, 6, 8, 24, 12, 48, 16, 84, 44, 72, 24, 192, 28, 96, 96, 276, 36, 264, 40, 288, 128, 144, 48, 672, 102, 168, 212, 384, 60, 576, 64, 876, 192, 216, 192, 1056, 76, 240, 224, 1008, 84, 768, 88, 576, 528, 288, 96, 2208, 184, 612, 288, 672, 108, 1272, 288, 1344, 320, 360, 120, 2304, 124, 384, 704, 2724, 336, 1152, 136
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Dirichlet convolution of A001615 with A003959.

Links

Crossrefs

Cf. A001615, A003959, A327251, A349142, A349170, A349171, A349172, A349173.

Programs

  • Mathematica
    f[p_, e_] := (p + 2)*(p + 1)^e - (p + 1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349172(n) = sumdiv(n,d,A001615(d)*A003959(n/d));

Formula

a(n) = Sum_{d|n} A001615(d) * A003959(n/d).
a(n) = A327251(n) + A349142(n).
For all n >= 1, a(n) >= A349132(n).
Multiplicative with a(p^e) = (p+2)*(p+1)^e - (p+1)*p^e. - Amiram Eldar, Nov 09 2021

A349129 a(n) = Sum_{d|n} A003958(d) * A003959(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

1, 4, 6, 13, 10, 24, 14, 40, 28, 40, 22, 78, 26, 56, 60, 121, 34, 112, 38, 130, 84, 88, 46, 240, 76, 104, 120, 182, 58, 240, 62, 364, 132, 136, 140, 364, 74, 152, 156, 400, 82, 336, 86, 286, 280, 184, 94, 726, 148, 304, 204, 338, 106, 480, 220, 560, 228, 232, 118, 780, 122, 248, 392, 1093, 260, 528, 134, 442, 276
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Dirichlet convolution of A003958 with A003959.

Links

Crossrefs

Cf. A003958, A003959, A168066, A349139.
Cf. also A349130, A349131, A349132, A349133 and A349170, A349171, A349172, A349173.

Programs

  • Mathematica
    f[p_, e_] := ((p + 1)^(e + 1) - (p - 1)^(e + 1))/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349129(n) = sumdiv(n,d,A003958(d)*A003959(n/d));

Formula

Multiplicative with a(p^e) = ((p+1)^(e+1) - (p-1)^(e+1))/2. - Amiram Eldar, Nov 09 2021
For all n >= 1, A349130(n) <= a(n) <= A349170(n).
Showing 1-6 of 6 results.

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