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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.

A294649 a(n) = numerator(A206369(n))/n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 5, 7, 2, 10, 1, 12, 3, 8, 11, 16, 7, 18, 3, 4, 5, 22, 5, 21, 6, 20, 9, 28, 4, 30, 21, 20, 8, 24, 7, 36, 9, 8, 1, 40, 2, 42, 15, 28, 11, 46, 11, 43, 21, 32, 9, 52, 10, 8, 15, 12, 14, 58, 2, 60, 15, 2, 43, 48, 10, 66, 12, 44, 12, 70, 35, 72, 18, 14
Offset: 1

Views

Author

Michel Marcus, Nov 06 2017

Keywords

Comments

a(n) = 1 for n in A127724.

Links

Crossrefs

Cf. A127724 (k-imperfect), A206369, A294650 (denominator).
A225679 is a subsequence (on squarefree indices). - Michel Marcus, Dec 22 2017

Programs

  • Mathematica
    (* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
a[n_] := Numerator[b[n]/n];
Array[a, 100] (* Jean-François Alcover, Dec 04 2017 *)
  • PARI
    rhope(p, e) = my(s=1); for(i=1, e, s=s*p + (-1)^i); s;
rho(n) = my(f=factor(n)); prod(i=1, #f~, rhope(f[i, 1], f[i, 2]));
a(n) = numerator(rho(n)/n);

A294650 a(n) = denominator(A206369(n))/n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 2, 13, 7, 15, 16, 17, 18, 19, 5, 7, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 32, 33, 17, 35, 12, 37, 19, 13, 2, 41, 7, 43, 22, 45, 23, 47, 24, 49, 50, 51, 13, 53, 27, 11, 28, 19, 29, 59, 5, 61, 31, 3, 64, 65, 33, 67, 17, 69, 35
Offset: 1

Views

Author

Michel Marcus, Nov 06 2017

Keywords

Links

Crossrefs

Cf. A127724 (k-imperfect), A206369, A294649 (numerator).
A225680 is a subsequence (on squarefree indices). - Michel Marcus, Dec 22 2017

Programs

  • Mathematica
    (* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
a[n_] := Denominator[b[n]/n];
Array[a, 100] (* Jean-François Alcover, Dec 04 2017 *)
  • PARI
    rhope(p, e) = my(s=1); for(i=1, e, s=s*p + (-1)^i); s;
rho(n) = my(f=factor(n)); prod(i=1, #f~, rhope(f[i, 1], f[i, 2]));
a(n) = denominator(rho(n)/n);

A295236 Hemi-imperfect numbers: numbers such that the denominator of k/A206369(k) is equal to 2.

Original entry on oeis.org

3, 10, 42, 60, 63, 840, 1260, 12642, 18480, 18900, 18963, 154350, 228480, 252840, 379260, 3458700, 5562480, 5688900, 68772480, 1041068700, 15032156160, 53621568000, 4524679004160, 9812746944000
Offset: 1

Views

Author

Michel Marcus, Nov 19 2017

Keywords

Comments

This is to rho (A206369) what hemiperfect numbers are to sigma (A000203).
After 3, 10 and 42, whose quotients are resp. 3/2, 5/2 and 7/2, 373316437260251755241798182764378479569038727298776522806597255168000000 is an instance of a term with quotient 9/2. - Michel Marcus, Dec 17 2017
a(25) > 10^13. - Giovanni Resta, Feb 17 2020

Examples

			3 is a term since rho(3) = 2, so 3/rho(3) is 3/2.
10 is a term since rho(10) = 4, so 10/rho(10) is 5/2.
42 is a term since rho(42) = 12, so 42/rho(42) is 7/2.

Links

Crossrefs

Cf. A127724 (k-imperfect), A206369 (rho).
Cf. A159907 (hemiperfect).

Programs

  • Maple
    rho:= proc(n) local f;
  mul((f[1]^(f[2]+1)+(-1)^f[2])/(f[1]+1), f = ifactors(n)[2]);
end proc:
select(t -> denom(t/rho(t)) = 2, [$1..10^6]); # Robert Israel, Nov 20 2017
  • Mathematica
    (* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
Select[Range[10^6], If[Denominator[#/b[#]] == 2, Print[#]; True, False]&] (* Jean-François Alcover, Dec 04 2017 *)
  • PARI
    rho(n) = {my(f = factor(n), res = q = 1); for(i=1, #f~, q = 1; for(j = 1, f[i, 2], q = -q + f[i, 1]^j); res *= q); res;}
isok(n) = denominator(n/rho(n))==2;

Extensions

a(20) from Jinyuan Wang, Feb 15 2020
a(21)-a(24) from Giovanni Resta, Feb 17 2020
Showing 1-3 of 3 results.

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