A071561 Numbers with no middle divisors (cf. A071090).
3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 111, 113, 114
Offset: 1
Keywords
Examples
From _Michael B. Porter_, Oct 19 2018: (Start) The divisors of 21 are 1, 3, 7, and 21. Since none of these are between sqrt(21/2) = 3.24... and sqrt(2*21) = 6.48..., 21 is in the sequence. The divisors of 20 are 1, 2, 4, 5, 10, and 20. Since 4 and 5 are both between sqrt(20/2) = 3.16... and sqrt(2*20) = 6.32..., 20 is not in the sequence. (End)
Links
- Iain Fox, Table of n, a(n) for n = 1..10000
- Hartmut F. W. Hoft, On the symmetric spectrum of odd divisors of a number, (2015), (Note that in this paper, A241561 should be replaced with A071561, and A241562 should be replaced with A071562. Also note that "the symmetric spectrum of odd divisors of a number" seems to be an attempt to call with a new name to a diagram known since 2014 as "the symmetric representation of sigma(n)"). - _Omar E. Pol_, Oct 08 2018
- José Manuel Rodríguez Caballero, Elementary number-theoretical statements proved by Language Theory, arXiv:1709.09617 [math.LO], 2017.
- J. M. Rodríguez Caballero, Symmetric Dyck Paths and Hooley's Δ-Function, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
Programs
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Mathematica
f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[125], f[ # ] == 0 &] (* Related to the symmetric representation of sigma *) (* subsequence of even parts of number k for m <= k <= n *) (* Function a237270[] is defined in A237270 *) (* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *) a071561[m_, n_]:=Select[Range[m, n], EvenQ[Length[a237270[#]]]&] a071561[1, 114] (* data *) (* Hartmut F. W. Hoft, Jul 07 2014 *) Select[Range@ 120, Function[n, Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] == {}]] (* Michael De Vlieger, Jan 03 2017 *)
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PARI
is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1 \\ Iain Fox, Dec 19 2017
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PARI
is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2>2*n))); 0 \\ Charles R Greathouse IV, Jan 22 2018
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PARI
list(lim)=my(v=List(),t); forfactored(n=3,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2>2*n[1], listput(v,n[1])); break))); Vec(v) \\ Charles R Greathouse IV, Jan 22 2018
Comments