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 1-3 of 3 results.

A318878 Sum of A083254(d) for all such divisors d of n for which A083254(d) > 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 4, 10, 2, 12, 6, 6, 1, 16, 5, 18, 4, 10, 10, 22, 2, 19, 12, 14, 6, 28, 6, 30, 1, 18, 16, 22, 5, 36, 18, 22, 4, 40, 10, 42, 10, 12, 22, 46, 2, 41, 19, 30, 12, 52, 14, 38, 6, 34, 28, 58, 6, 60, 30, 22, 1, 46, 18, 66, 16, 42, 22, 70, 5, 72, 36, 26, 18, 58, 22, 78, 4, 41, 40, 82, 10, 62, 42, 54, 10, 88, 12, 70, 22, 58, 46, 70, 2
Offset: 1

Views

Author

Antti Karttunen, Sep 05 2018

Keywords

Examples

			n = 105 has divisors [1, 3, 5, 7, 15, 21, 35, 105]. When A083254 is applied to them, we obtain [1, 1, 3, 5, 1, 3, 13, -9]. Summing the positive numbers present, we get a(105) = 1+1+3+5+1+3+13 = 27.

Links

Crossrefs

Cf. A000010, A033879, A083254, A318879.
Cf. also A318678, A318874, A318876.

Programs

  • PARI
    A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);

Formula

a(n) = Sum_{d|n} [A083254(d) > 0]*A083254(d), where A083254(n) = 2*phi(n) - n, and [ ] are the Iverson brackets.
a(n) = A318879(n) + A033879(n).

A318874 Number of divisors d of n for which 2*phi(d) > d.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 7
Offset: 1

Views

Author

Antti Karttunen, Sep 05 2018

Keywords

Examples

			n = 105 has eight divisors: [1, 3, 5, 7, 15, 21, 35, 105]. When A083254 is applied to them, we obtain [1, 1, 3, 5, 1, 3, 13, -9], and seven of these numbers are positive, thus a(105) = 7.

Links

Crossrefs

Cf. A000010, A083254, A318875, A318876, A318878.
Differs from A001227 for the first time at n=105, where a(105) = 7, while A001227(105) = 8.

Programs

  • Maple
    A318874 := n -> nops(select(d -> (2*numtheory:-phi(d)) > d, divisors(n))):
seq(A318874(n), n=1..99); # Peter Luschny, Sep 05 2018
  • Mathematica
    A318874[n_] := DivisorSum[n, 1 &, 2*EulerPhi[#] > # &];
Array[A318874, 100] (* Paolo Xausa, Jul 08 2024 *)
  • PARI
    A318874(n) = sumdiv(n,d,(2*eulerphi(d))>d);

Formula

a(n) = Sum_{d|n} [A083254(d) > 0].
For all n >= 1, a(n) + A318875(n) + A007814(n) = A000005(n).

A318877 Sum of divisors d of n for which 2*phi(d) < d.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 0, 10, 0, 18, 0, 14, 0, 0, 0, 24, 0, 30, 0, 22, 0, 42, 0, 26, 0, 42, 0, 46, 0, 0, 0, 34, 0, 72, 0, 38, 0, 70, 0, 62, 0, 66, 0, 46, 0, 90, 0, 60, 0, 78, 0, 78, 0, 98, 0, 58, 0, 138, 0, 62, 0, 0, 0, 94, 0, 102, 0, 94, 0, 168, 0, 74, 0, 114, 0, 110, 0, 150, 0, 82, 0, 186, 0, 86, 0, 154, 0, 154, 0, 138, 0, 94, 0
Offset: 1

Views

Author

Antti Karttunen, Sep 05 2018

Keywords

Links

Crossrefs

Cf. A000010, A000203, A006519, A083254, A318875, A318876, A318879.
Cf. also A187795.

Programs

  • Mathematica
    A318877[n_] := DivisorSum[n,  # &, 2*EulerPhi[#] < # &];
Array[A318877, 100] (* Paolo Xausa, Jul 08 2024 *)
  • PARI
    A318877(n) = sumdiv(n,d,((2*eulerphi(d))

Formula

a(n) = Sum_{d|n} [2*phi(d) < d]*d, where [ ] are the Iverson brackets.
For all n >= 1, a(n) + A318876(n) + 2*(A006519(n)-1) = A000203(n).
Showing 1-3 of 3 results.

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