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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 1-3 of 3 results.

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).

A348983 a(n) = Sum_{d|n} A038040(d) * A322582(n/d), where A038040(n) = n*d(n), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 7, 1, 14, 1, 31, 11, 20, 1, 80, 1, 26, 23, 111, 1, 109, 1, 122, 29, 38, 1, 328, 19, 44, 76, 164, 1, 250, 1, 351, 41, 56, 35, 565, 1, 62, 47, 514, 1, 334, 1, 248, 208, 74, 1, 1128, 27, 245, 59, 290, 1, 650, 47, 700, 65, 92, 1, 1336, 1, 98, 274, 1023, 53, 502, 1, 374, 77, 490, 1, 2213, 1, 116, 302, 416, 53
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of A322582 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A348980.
Dirichlet convolution of sigma (A000203) with A348981.

Links

Crossrefs

Cf. A000005, A000027, A000203, A003958, A038040, A322582, A348980, A348981, A348982.
Cf. also A349123, A349143.

Programs

  • Mathematica
    f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A038040(n) = (n*numdiv(n));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348983(n) = sumdiv(n,d,A038040(n/d)*A322582(d));

Formula

a(n) = Sum_{d|n} A038040(n/d) * A322582(d).
a(n) = Sum_{d|n} d * A348980(n/d).
a(n) = Sum_{d|n} A000203(d) * A348981(n/d).
For all n >= 1, a(n) <= A349123(n) <= A349143(n).

A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 9, 1, 16, 1, 51, 13, 22, 1, 114, 1, 28, 25, 233, 1, 145, 1, 168, 31, 40, 1, 590, 21, 46, 106, 222, 1, 310, 1, 939, 43, 58, 37, 915, 1, 64, 49, 896, 1, 406, 1, 330, 262, 76, 1, 2570, 29, 297, 61, 384, 1, 1012, 49, 1202, 67, 94, 1, 2040, 1, 100, 340, 3489, 55, 598, 1, 492, 79, 574, 1, 4457, 1, 118, 360, 546, 55
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A349140.
Dirichlet convolution of sigma (A000203) with A349141.
Dirichlet convolution of A060640 with A348971.

Links

Crossrefs

Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.
Cf. also A349123, A348983.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*DivisorSigma[0, #]*(s[n/#] - n/#) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A038040(n) = (n*numdiv(n));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349143(n) = sumdiv(n,d,A038040(d)*A348507(n/d));

Formula

a(n) = Sum_{d|n} A038040(n/d) * A348507(d).
a(n) = Sum_{d|n} d * A349140(n/d).
a(n) = Sum_{d|n} A000203(d) * A349141(n/d).
a(n) = Sum_{d|n} A060640(d) * A348971(n/d).
For all n >= 1, a(n) >= A349123(n) >= A348983(n).
Showing 1-3 of 3 results.

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