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 1-7 of 7 results.

A347126 a(n) = A347129(A276086(n)).

Original entry on oeis.org

0, 1, 1, 10, 3, 21, 1, 14, 16, 124, 39, 246, 3, 27, 33, 222, 72, 423, 6, 44, 56, 344, 114, 636, 10, 65, 85, 490, 165, 885, 1, 18, 20, 164, 51, 330, 24, 236, 284, 1976, 636, 3804, 57, 438, 552, 3468, 1143, 6462, 104, 696, 904, 5296, 1776, 9624, 165, 1010, 1340, 7460, 2535, 13290, 3, 33, 39, 282, 90, 549, 51, 414, 516
Offset: 0

Views

Author

Antti Karttunen, Aug 25 2021

Keywords

Links

Crossrefs

Cf. A003415, A003557, A276086, A347129, A347130.
Cf. also A342002, A346471, A347232.

Programs

Formula

a(n) = A347129(A276086(n)).

A347130 a(n) = Sum_{d|n} d * A003415(n/d), where A003415 is the arithmetic derivative.

Original entry on oeis.org

0, 1, 1, 6, 1, 10, 1, 24, 9, 14, 1, 48, 1, 18, 16, 80, 1, 63, 1, 72, 20, 26, 1, 176, 15, 30, 54, 96, 1, 124, 1, 240, 28, 38, 24, 270, 1, 42, 32, 272, 1, 164, 1, 144, 117, 50, 1, 560, 21, 135, 40, 168, 1, 324, 32, 368, 44, 62, 1, 552, 1, 66, 153, 672, 36, 244, 1, 216, 52, 236, 1, 936, 1, 78, 165, 240, 36, 284, 1, 880
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Comments

Dirichlet convolution of the identity function (A000027) with the arithmetic derivative of n (A003415).
Dirichlet convolution of Euler phi (A000010) with A319684.

Links

Crossrefs

Inverse Möbius transform of A347131.
Cf. A000010, A000027, A003415, A003557, A319684, A347129.
Cf. also A300251, A305809, A319684, A347132, A347133.

Programs

  • Mathematica
    Table[DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &], {n, 80}] (* Michael De Vlieger, Oct 21 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));

Formula

a(n) = Sum_{d|n} d * A003415(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A319684(d).
a(n) = Sum_{d|n} A347131(d).
a(n) = A003557(n) * A347129(n).

A348496 a(n) = gcd(A018804(n), A347130(n)) / A003557(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 3, 1, 2, 1, 21, 1, 36, 5, 1, 1, 1, 1, 15, 3, 4, 1, 1, 1, 1, 7, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 12, 3, 5, 1, 10, 1, 3, 5, 4, 1, 9, 1, 1, 1, 1, 1, 12, 1, 3, 1, 1, 9, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 4, 3, 1, 1, 2, 1, 1, 1, 4, 11, 15, 1, 35, 1, 3, 5, 36, 1, 1, 3, 1, 1, 3, 3, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 21 2021

Keywords

Links

Crossrefs

Cf. A003557, A018804, A347128, A347129, A347130, A348495, A348501 [= a(A276086(n))].
Cf. also A348494, A348498.

Programs

  • Mathematica
    Table[GCD[Total@ GCD[n, Range[n]], DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, FactorInteger[n]]], {n, 101}] (* Michael De Vlieger, Oct 21 2021 *)
  • PARI
    \\ Needs also code from A348495:
A003557(n) = (n/factorback(factorint(n)[, 1]));
A348496(n) = (A348495(n)/A003557(n));

Formula

a(n) = A348495(n) / A003557(n).
a(n) = gcd(A347128(n), A347129(n)).

A347128 a(n) = A018804(n) / A003557(n), where A018804 is Pillai's arithmetical function.

Original entry on oeis.org

1, 3, 5, 4, 9, 15, 13, 5, 7, 27, 21, 20, 25, 39, 45, 6, 33, 21, 37, 36, 65, 63, 45, 25, 13, 75, 9, 52, 57, 135, 61, 7, 105, 99, 117, 28, 73, 111, 125, 45, 81, 195, 85, 84, 63, 135, 93, 30, 19, 39, 165, 100, 105, 27, 189, 65, 185, 171, 117, 180, 121, 183, 91, 8, 225, 315, 133, 132, 225, 351, 141, 35, 145, 219, 65, 148
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Links

Crossrefs

Cf. A003557, A018804, A348494 [= gcd(a(n), A342001(n))], A348496 [= gcd(a(n), A347129(n))].
Cf. also A173557, A347127.

Programs

  • Mathematica
    f[p_, e_] := (e*(p - 1)/p + 1)*p^e; A347128[n_] := (Times @@ (f @@@ FactorInteger[n]))/(n/Times @@ (First[Transpose[FactorInteger[n]]]));Table[A347128[n], {n, 1, 76}] (* Robert P. P. McKone, Aug 23 2021, after Amiram Eldar *)
  • PARI
    A347128(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i, 1]-1)*f[i, 2] + f[i, 1])); };

Formula

Multiplicative with a(p^e) = ((p-1)*e + p).
a(n) = A018804(n) / A003557(n).

A347127 a(n) = A327251(n) / A003557(n).

Original entry on oeis.org

1, 5, 7, 8, 11, 35, 15, 11, 11, 55, 23, 56, 27, 75, 77, 14, 35, 55, 39, 88, 105, 115, 47, 77, 17, 135, 15, 120, 59, 385, 63, 17, 161, 175, 165, 88, 75, 195, 189, 121, 83, 525, 87, 184, 121, 235, 95, 98, 23, 85, 245, 216, 107, 75, 253, 165, 273, 295, 119, 616, 123, 315, 165, 20, 297, 805, 135, 280, 329, 825, 143, 121
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Links

Crossrefs

Cf. A001615, A003557, A327251.
Cf. also A048250, A347128, A347129.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)*e + p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)
  • PARI
    A347127(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i, 1]+1)*f[i, 2] + f[i, 1])); };
  • 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
A003557(n) = (n/factorback(factorint(n)[, 1]));
A327251(n) = sumdiv(n, d, A001615(n/d)*d);
A347127(n) = (A327251(n) / A003557(n));

Formula

Multiplicative with a(p^e) = ((p+1)*e + p) for prime p.
a(n) = A327251(n) / A003557(n).

A348498 a(n) = gcd(A003415(n), A347130(n)) / A003557(n), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 3, 1, 7, 1, 8, 1, 9, 8, 2, 1, 7, 1, 12, 10, 13, 1, 11, 1, 15, 3, 16, 1, 31, 1, 5, 14, 19, 12, 5, 1, 21, 16, 17, 1, 41, 1, 24, 13, 25, 1, 14, 1, 9, 20, 28, 1, 9, 16, 23, 22, 31, 1, 46, 1, 33, 17, 3, 18, 61, 1, 36, 26, 59, 1, 13, 1, 39, 11, 40, 18, 71, 1, 22, 2, 43, 1, 62, 22, 45, 32, 35, 1, 41
Offset: 1

Views

Author

Antti Karttunen, Oct 23 2021

Keywords

Links

Crossrefs

Cf. A003415, A003557, A342001, A347129, A347130, A348497, A348502 [= a(A276086(n))].
Cf. also A348494, A348496.

Programs

  • Mathematica
    f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Table[GCD[f[n], DivisorSum[n, # f[n/#] &]]*Apply[Times, FactorInteger[n][[All, 1]]]/n, {n, 90}] (* Michael De Vlieger, Oct 25 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A348498(n) = (gcd(A003415(n), A347130(n))/A003557(n));

Formula

a(n) = A348497(n) / A003557(n).
a(n) = gcd(A342001(n), A347129(n)).

A349124 a(n) = A349123(n) / A003557(n), where A349123 is the Dirichlet convolution of the arithmetic derivative with n*tau(n).

Original entry on oeis.org

0, 1, 1, 4, 1, 15, 1, 10, 4, 21, 1, 48, 1, 27, 24, 20, 1, 42, 1, 72, 30, 39, 1, 110, 4, 45, 10, 96, 1, 279, 1, 35, 42, 57, 36, 120, 1, 63, 48, 170, 1, 369, 1, 144, 78, 75, 1, 210, 4, 54, 60, 168, 1, 90, 48, 230, 66, 93, 1, 828, 1, 99, 102, 56, 54, 549, 1, 216, 78, 531, 1, 260, 1, 117, 66, 240, 54, 639, 1, 330, 20, 129
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Links

Crossrefs

Cf. A003415, A003557, A038040, A349123.
Cf. also A347129.

Programs

  • Mathematica
    f[p_, e_] := p^(e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &] / s[n]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A003557(n) = (n/factorback(factorint(n)[, 1]));
A349124(n) = (A349123(n) / A003557(n)); \\ Needs also code from A349123.

Formula

a(n) = A349123(n) / A003557(n).
Showing 1-7 of 7 results.

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