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.

A265393 a(n) = the smallest number k such that floor(Sum_{d|k} 1/tau(d)) = n.

Original entry on oeis.org

1, 6, 24, 60, 180, 420, 840, 2520, 4620, 9240, 13860, 27720, 60060, 55440, 110880, 166320, 180180, 480480, 360360, 900900, 720720, 1441440, 1801800, 2162160, 3063060, 4084080, 7207200, 12612600, 6126120, 27027000, 12252240, 18378360, 43243200, 24504480
Offset: 1

Views

Author

Jaroslav Krizek, Dec 08 2015

Keywords

Comments

Further known terms: a(29) = 6126120, a(31) = 12252240.
Are there numbers n > 1 such that Sum_{d|n} 1/tau(d) is an integer?
Sequences of numbers n such that floor(Sum_{d|n} 1/tau(d)) = k for k = 1..6:
k=1: 1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, ... (A166684);
k=2: 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, ...;
k=3: 24, 30, 36, 40, 42, 48, 54, 56, 66, 70, 72, 78, 80, 88, 96, 100, ...;
k=4: 60, 84, 90, 120, 126, 132, 140, 144, 150, 156, 168, 198, 204, 216, ...;
k=5: 180, 210, 240, 252, 300, 330, 336, 360, 390, 396, 450, 462, 468, ...;
k=6: 420, 630, 660, 720, 780, 900, 924, 990, 1008, 1020, 1050, 1080, ....
Values of function F = Sum_{d|n} 1/tau(d) for some numbers according to their prime signature: F{} = 1; F{1} = 3/2; F{2} = 11/6; F{1, 1} = 9/4; F{3} = 25/12; F{2, 1} = 11/4; F{4} = 137/60; F{3, 1} = 25/8, ...

Examples

			For n = 2; a(2) = 6 because 6 is the smallest number with floor(Sum_{d|6} 1/tau(d)) = floor(1/1 + 1/2 + 1/2 + 1/4) = floor(9/4) = 2.

Links

Crossrefs

Cf. A000005, A253139, A265390, A265391, A265392.
Cf. A237350 (a(n) = the smallest number k such that Sum_{d|k} 1/tau(d) >= n).

Programs

  • Magma
    a:=1; S:=[a]; for n in [2..14] do k:=0; flag:= true; while flag do k+:=1; if Floor(&+[1/NumberOfDivisors(d): d in Divisors(k)]) eq n then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;
  • Mathematica
    Table[k = 1; While[Floor@ Sum[1/DivisorSigma[0, d], {d, Divisors@ k}] != n, k++]; k, {n, 17}] (* Michael De Vlieger, Dec 09 2015 *)
  • PARI
    a(n) = {k=1; while(k, if(floor(sumdiv(k, d, 1/numdiv(d))) == n, return(k)); k++)} \\ Altug Alkan, Dec 09 2015

Extensions

More terms from Michel Marcus, Dec 23 2015
a(33)-a(34) from Hiroaki Yamanouchi, Dec 31 2015

A266226 a(n) = floor(Sum_{d|n} 1 / tau(d)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3
Offset: 1

Views

Author

Jaroslav Krizek, Dec 24 2015

Keywords

Comments

a(n) = floor(Sum_{d|n} 1 / A000005(d)).
Sequences of numbers n such that floor(Sum_{d|n} 1/tau(d)) = k for k = 1..6:
k=1: 1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, ... (A166684);
k=2: 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, ...;
k=3: 24, 30, 36, 40, 42, 48, 54, 56, 66, 70, 72, 78, 80, 88, 96, 100, ...;
k=4: 60, 84, 90, 120, 126, 132, 140, 144, 150, 156, 168, 198, 204, 216, ...;
k=5: 180, 210, 240, 252, 300, 330, 336, 360, 390, 396, 450, 462, 468, ...;
k=6: 420, 630, 660, 720, 780, 900, 924, 990, 1008, 1020, 1050, 1080, ....
See A265393 - the smallest number n such that a(n) = k for k>= 1.

Examples

			For n = 6; a(6) = floor(Sum_{d|6} 1/tau(d)) = floor(1/1 + 1/2 + 1/2 + 1/4) = floor(9/4) = 2.

Links

Crossrefs

Cf. A000005, A237350, A253139, A265390, A265391, A265392, A265393

Programs

  • Magma
    [Floor(&+[1/NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]];
  • Mathematica
    Table[Floor[Sum[1/DivisorSigma[0, d], {d, Divisors[ n]}]], {n, 1, 100}] (* G. C. Greubel, Dec 24 2015 *)

A265719 Numbers n such that Sum_{d|n} 1/tau(d) > Sum_{d|m} 1/tau(d) for all m < n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 48, 60, 120, 180, 210, 240, 360, 420, 720, 840, 1260, 1680, 2520, 4620, 5040, 7560, 9240, 13860, 18480, 27720, 55440, 83160, 110880, 120120, 166320, 180180, 240240, 360360, 720720, 1081080, 1441440, 1801800, 2042040, 2162160, 3063060, 3603600, 4084080
Offset: 1

Views

Author

Jaroslav Krizek, Dec 14 2015

Keywords

Comments

Where record values of Sum_{d|n} 1/tau(d) occur.
Terms a(n) are the smallest number from sequences numbers with following prime signatures: {}, {1}, {2}, {1, 1}, {2, 1}, {3, 1}, {1, 1, 1}, {4, 1}, {2, 1, 1}, {3, 1, 1}, {2, 2, 1}, {1, 1, 1, 1}, {4, 1, 1}, {3, 2, 1}, ...

Examples

			For n = 4; a(4) = 6 because 6 is the smallest number such that Sum_{d|a(4)} 1/tau(d) = Sum_{d|6} 1/tau(d) = 9/4 > Sum_{d|a(3)} 1/tau(d) = Sum_{d|4} 1/tau(d) = 11/6.

Crossrefs

Cf. A000005, A237350, A253139, A265390, A265391, A265392, A265393.

Programs

  • Magma
    a:=1; S:=[a]; for n in [2..25] do k:=0; flag:= true; while flag do k+:=1; if &+[1/NumberOfDivisors(d): d in Divisors(a)] lt &+[1/NumberOfDivisors(d): d in Divisors(k)] then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;
  • PARI
    lista(nn) = {m = 0; for (n=1, nn, if ((mm = sumdiv(n, d, 1/numdiv(d))) > m, print1(n, ", "); m = mm););} \\ Michel Marcus, Dec 22 2015

Extensions

More terms from Michel Marcus, Dec 22 2015
Showing 1-3 of 3 results.

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