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.

A349133 Dirichlet convolution of A003415 with A003958, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 5, 1, 8, 1, 17, 8, 12, 1, 32, 1, 16, 14, 49, 1, 43, 1, 52, 18, 24, 1, 100, 14, 28, 43, 72, 1, 87, 1, 129, 26, 36, 22, 151, 1, 40, 30, 168, 1, 119, 1, 112, 91, 48, 1, 276, 20, 103, 38, 132, 1, 194, 30, 236, 42, 60, 1, 323, 1, 64, 123, 321, 34, 183, 1, 172, 50, 183, 1, 443, 1, 76, 131, 192, 34, 215, 1, 472
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Links

Crossrefs

Cf. A003415, A003958, A349129, A349130, A349131, A349132, A349173.

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]));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349133(n) = sumdiv(n,d,A003415(d)*A003958(n/d));

Formula

a(n) = Sum_{d|n} A003415(d) * A003958(n/d).
For all n >= 1, a(n) <= A349173(n).

A348982 a(n) = Sum_{d|n} psi(n/d) * A322582(d), where psi is Dedekind psi (A001615), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 6, 1, 11, 1, 22, 9, 15, 1, 52, 1, 19, 17, 66, 1, 69, 1, 76, 21, 27, 1, 176, 15, 31, 51, 100, 1, 145, 1, 178, 29, 39, 25, 288, 1, 43, 33, 264, 1, 189, 1, 148, 123, 51, 1, 508, 21, 145, 41, 172, 1, 339, 33, 352, 45, 63, 1, 632, 1, 67, 159, 450, 37, 277, 1, 220, 53, 265, 1, 924, 1, 79, 175, 244, 37, 321
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of A001615 with A322582.

Links

Crossrefs

Cf. A001615, A003958, A322582, A327251, A348980, A348981, A348982, A348983, A349132.
Cf. also A347132, A349142.

Programs

  • Mathematica
    f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*psi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 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))); \\ After code in A001615
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348982(n) = sumdiv(n,d,A001615(n/d)*A322582(d));

Formula

a(n) = Sum_{d|n} A001615(n/d) * A322582(d).
For all n >= 1, a(n) <= A347132(n) <= A349142(n).
a(n) = A327251(n) - A349132(n). - Antti Karttunen, Nov 14 2021

A349130 a(n) = Sum_{d|n} d * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 13, 15, 19, 27, 21, 35, 25, 39, 45, 31, 33, 57, 37, 63, 65, 63, 45, 75, 61, 75, 65, 91, 57, 135, 61, 63, 105, 99, 117, 133, 73, 111, 125, 135, 81, 195, 85, 147, 171, 135, 93, 155, 127, 183, 165, 175, 105, 195, 189, 195, 185, 171, 117, 315, 121, 183, 247, 127, 225, 315, 133, 231, 225, 351, 141
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Dirichlet convolution of A003958 with the identity function, A000027.
Dirichlet convolution of sigma (A000203) with A003966.

Links

Crossrefs

Cf. A000203, A003958, A003966, A038040, A348980, A349129, A349131 (Möbius transform), A349132, A349133, A349170.

Programs

  • Mathematica
    f[p_, e_] := p^(e + 1) - (p - 1)^(e + 1); 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); };
A349130(n) = sumdiv(n,d,d*A003958(n/d));

Formula

a(n) = Sum_{d|n} d * A003958(n/d).
a(n) = Sum_{d|n} A349131(d).
a(n) = Sum_{d|n} A000203(d) * A003966(n/d).
a(n) = A038040(n) - A348980(n).
For all n >= 1, a(n) <= A349129(n) <= A349170(n).
Multiplicative with a(p^e) = p^(e+1) - (p-1)^(e+1). - Amiram Eldar, Nov 09 2021

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

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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