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.

A036455 Numbers n such that d(d(n)) is an odd prime, where d(k) is the number of divisors of k.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Compare with sequence A007422 and A030513 -- the resemblance is rather strong. Still this sequence is different. For example, 36, 100, 120, and 168 are here.

Examples

			a(15) = 39 and d(39) = 4, d(d(39)) = d(4) = 3 and d(d(d(39))) = 2. After 3 iteration the equilibrium is reached.

Links

Crossrefs

Cf. A000005, A007422, A030513, A036450, A036452, A036454, A036456, A036457, A036458.

Programs

  • Maple
    filter:= proc(n) local r;
  r:= numtheory:-tau(numtheory:-tau(n));
  r::odd and isprime(r)
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2016
  • Mathematica
    fQ[n_] := Module[{d2 = DivisorSigma[0, DivisorSigma[0, n]]}, d2 > 2 && PrimeQ[d2]]; Select[Range[200], fQ] (* T. D. Noe, Jan 22 2013 *)
  • PARI
    is(n)=isprime(n=numdiv(numdiv(n))) && n>2 \\ Charles R Greathouse IV, Jan 22 2013

Formula

d(d(d(a(n)))) = 2 for all n.
A036459(a(n)) = 3. - Ivan Neretin, Jan 25 2016

Extensions

Definition clarified by R. J. Mathar and Charles R Greathouse IV, Jan 22 2013

A036456 Numbers k for which exactly 4 applications of A000005 are needed to reach 2.

Original entry on oeis.org

12, 18, 20, 24, 28, 30, 32, 40, 42, 44, 45, 48, 50, 52, 54, 56, 63, 66, 68, 70, 75, 76, 78, 80, 88, 92, 98, 99, 102, 104, 105, 110, 112, 114, 116, 117, 124, 128, 130, 135, 136, 138, 144, 147, 148, 152, 153, 154, 162, 164, 165, 170, 171, 172, 174, 175, 176, 182
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Similar to but different from A007624. Terms like 60, 72, 84, 90, 96, 108, 126, etc. are not present here.

Examples

			a(3)=20 and a(17)=63; for both x=20 and 63, d(x)=6 and d(d(x))=4, the 3rd iterates are 3 and the equilibrium value, i.e., 2 appears as 4th iterates.

Links

Crossrefs

Cf. A000005, A007624, A036450, A036452, A036453, A036455.

Programs

  • PARI
    isok(n) = ((nd=numdiv(n)) != 2) && ((nd=numdiv(nd)) != 2) && ((nd=numdiv(nd)) != 2) && ((nd=numdiv(nd)) == 2); \\ Michel Marcus, Dec 30 2013 & Jan 26 2015

Formula

With d(n) = number of divisors(n), d(d(d(d(a(n))))) = 2 and d(d(d(a(n)))) > 2.
A036459(a(n)) = 4. - Ivan Neretin, Jan 25 2016

Extensions

New name (using new name for A036457 from Robert Israel) from Jon E. Schoenfield, May 12 2018

A036458 For all n, if d is recursively applied to a(n) exactly 6 times then the fixed point of d-iteration is just reached.

Original entry on oeis.org

5040, 7920, 8400, 9360, 10080, 10800, 11088, 11340, 11760, 12240, 12600, 12960, 13104, 13200, 13680, 13860, 15600, 15840, 16200, 16380, 16560, 16800, 17136, 17640, 17820, 18000, 18144, 18720, 18900, 19152, 19440, 19800, 20160, 20400, 20592, 20880, 21060, 21168
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Observe that the values giving stationary value in 6 steps are rather large.
"d" in the definition refers to the number of divisors of n. - Harvey P. Dale, Mar 06 2015

Examples

			a(1)=5040 and the nested d functions are 60,12,6,4,3 and the 6th is 2. a(5)=10080 and iterating d with 10080 initial value, after 6 iterations the convergence takes place through 72,12,6,4,3 transients, i.e., 2 is reached on the 6th step.

Links

Crossrefs

Cf. A000005, A036457, A036459.

Programs

  • Mathematica
    draQ[n_]:=Length[FixedPointList[DivisorSigma[0,#]&,n,7]]==8; Select[ Range[ 21000],draQ] (* Harvey P. Dale, Mar 06 2015 *)
  • PARI
    is(n)=for(i=1,5,n=numdiv(n); if(n<3, return(0))); numdiv(n)==2 \\ Charles R Greathouse IV, Sep 17 2015

Formula

A036459(a(n)) = 6. - Ivan Neretin, Jan 25 2016
Showing 1-3 of 3 results.

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