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 1-5 of 5 results.

A379368 Denominators of the partial sums of the reciprocals of the squarefree kernel function (A007947).

Original entry on oeis.org

1, 2, 6, 3, 15, 10, 70, 35, 105, 210, 2310, 385, 5005, 10010, 30030, 15015, 255255, 510510, 9699690, 4849845, 4849845, 9699690, 223092870, 111546435, 22309287, 44618574, 44618574, 3187041, 92424189, 308080630, 9550499530, 4775249765, 1302340845, 2604681690, 18232771830
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2024

Keywords

References

  • Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 16-17.

Links

Crossrefs

Cf. A007947, A073355, A370896, A379367 (numerators), A379370.

Programs

  • Mathematica
    rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Denominator[Accumulate[Table[1/rad[n], {n, 1, 50}]]]
  • PARI
    rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / rad(k); print1(denominator(s), ", "))};

Formula

a(n) = denominator(Sum_{k=1..n} 1/A007947(k)).

A379367 Numerators of the partial sums of the reciprocals of the squarefree kernel function (A007947).

Original entry on oeis.org

1, 3, 11, 7, 38, 27, 199, 117, 386, 793, 8933, 1553, 20574, 41863, 127591, 71303, 1227166, 2539417, 48759433, 24864701, 25095646, 50632187, 1174239991, 605711068, 125604071, 252924241, 267797099, 19356010, 564511331, 1891973791, 58959268151, 31867258958, 8730535499
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2024

Keywords

Examples

			Fractions begin with 1, 3/2, 11/6, 7/3, 38/15, 27/10, 199/70, 117/35, 386/105, 793/210, 8933/2310, 1553/385, ...

References

  • Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 16-17.

Links

Crossrefs

Cf. A007947, A073355, A370896, A379368 (denominators), A379369.
Cf. A059956, A073004.

Programs

  • Mathematica
    rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Numerator[Accumulate[Table[1/rad[n], {n, 1, 50}]]]
  • PARI
    rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / rad(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} 1/A007947(k)).
a(n)/A379368(n) = exp((1 + o(1)) * sqrt(8*log(n)/log(log(n)))).
a(n)/A379368(n) ~ (1/2) * exp(gamma) * F(log(n)) * log(log(n)), where F(t) = (6/Pi^2) * Sum_{m>=1} min(1,exp(t)/m)/Product_{primes p|m} (p+1).

A379370 Denominators of the partial alternating sums of the reciprocals of the squarefree kernel function (A007947).

Original entry on oeis.org

1, 2, 6, 3, 15, 30, 210, 105, 35, 70, 770, 1155, 15015, 30030, 2002, 1001, 17017, 102102, 1939938, 4849845, 4849845, 9699690, 223092870, 37182145, 37182145, 74364290, 223092870, 111546435, 3234846615, 2156564410, 66853496710, 33426748355, 100280245065, 200560490130
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2024

Keywords

Links

Crossrefs

Cf. A007947, A073355, A370896, A379368, A379369 (numerators).

Programs

  • Mathematica
    rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Denominator[Accumulate[Table[(-1)^(n+1)/rad[n], {n, 1, 50}]]]
  • PARI
    rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / rad(k); print1(denominator(s), ", "))};

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A007947(k)).

A379369 Numerators of the partial alternating sums of the reciprocals of the squarefree kernel function (A007947).

Original entry on oeis.org

1, 1, 5, 1, 8, 11, 107, 1, 12, 17, 257, 193, 3664, 5183, 479, -261, -3436, -37633, -612925, -2017297, -1786352, -4013599, -82613087, -19965872, -12529443, -27919051, -9392863, -12664034, -255710551, -242359181, -5356570201, -19391659278, -55136182529, -116171203003
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2024

Keywords

Examples

			Fractions begin with 1, 1/2, 5/6, 1/3, 8/15, 11/30, 107/210, 1/105, 12/35, 17/70, 257/770, 193/1155, ...

Links

Crossrefs

Cf. A007947, A073355, A370896, A379367, A379368, A379370 (denominators).

Programs

  • Mathematica
    rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Numerator[Accumulate[Table[(-1)^(n+1)/rad[n], {n, 1, 50}]]]
  • PARI
    rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / rad(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A007947(k)).
a(n)/A379370(n) ~ -A379367(n)/A379368(n).

A379714 Partial alternating sums of the number of exponential divisors function (A049419).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, 0, -2, 0, -1, 0, -2, -1, -2, -1, -4, -3, -5, -4, -6, -5, -6, -5, -7, -5, -6, -4, -6, -5, -6, -5, -7, -6, -7, -6, -10, -9, -10, -9, -11, -10, -11, -10, -12, -10, -11, -10, -13, -11, -13, -12, -14, -13, -15, -14, -16, -15, -16, -15, -17, -16
Offset: 1

Views

Author

Amiram Eldar, Dec 30 2024

Keywords

Links

Crossrefs

Cf. A049419, A065442, A145353, A327837.
Similar sequences: A068762, A068773, A307704, A357817, A370895, A370896.

Programs

  • Mathematica
    f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Table[(-1)^(n+1)*ediv[n], {n, 1, 100}]]
  • PARI
    ediv(n) = vecprod(apply(numdiv, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) * ediv(k); print1(s, ", "))};

Formula

a(n) = Sum_{k=1..n} (-1)^(k+1) * A049419(k).
Limit_{n->oo} a(n)/n = A327837 * (2/(A065442 + 1) - 1) = -0.37293122584744001729... .
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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