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 1-4 of 4 results.

A138394 Smallest number with a(n) divisors is in A134865.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 16, 20, 24, 30, 48, 60, 64, 72, 80, 84, 100, 108, 126, 162, 189, 192, 224, 384, 448, 512, 576, 672, 11520
Offset: 1

Views

Author

J. Lowell, May 08 2008

Keywords

Comments

The indices of elements of A005179 that belong to A134865. Sorted version of the sequence A140752. - Max Alekseyev, May 26 2008

Examples

			8 is in this sequence because the smallest number with 8 divisors (24) is a member of A134865.

Crossrefs

Cf. A000005, A005179, A134865, A140752, A140753.

Formula

a(n) = A000005(A140753(n)) - Max Alekseyev, May 26 2008

Extensions

a(31)=11520 from Ray Chandler, Jun 30 2008

A140752 Indices of elements of A005179 that belong to A134865 (in order of their appearance in A134865). The sorted version of this sequence is given by A138394.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 16, 20, 24, 30, 48, 60, 64, 72, 80, 84, 100, 108, 126, 192, 162, 224, 189, 384, 448, 512, 576, 672, 11520
Offset: 1

Views

Author

Max Alekseyev, May 26 2008

Keywords

Crossrefs

Cf. A000005, A005179, A134865, A138394, A140753.

Formula

a(n) = A000005(A134865(n))

Extensions

a(31)=11520 from Ray Chandler, Jun 30 2008

A140753 Subsequence of elements of A005179 that appear in A134865.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 2520, 5040, 7560, 10080, 15120, 20160, 45360, 50400, 100800, 352800, 705600, 332640, 665280, 4324320, 8648640, 17297280, 21621600, 43243200, 13492656777600
Offset: 1

Views

Author

Max Alekseyev, May 26 2008

Keywords

Comments

Rearrangement of elements of A134865 in order of their appearance in A005179. Hence A134865 represents a sorted variant of this sequence.

Crossrefs

Cf. A000005, A005179, A134865, A138394, A140752.

Formula

a(n) = A005179(A138394(n))

Extensions

a(31)=13492656777600 from Ray Chandler, Jun 30 2008

A141586 Strongly refactorable numbers: numbers n such that if n is divisible by d, it is divisible by the number of divisors of d.

Original entry on oeis.org

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 3360, 4320, 5280, 6240, 6720, 8160, 9120, 10080, 11040, 13440, 13920, 14880, 15840, 17760, 18720, 19680, 20160, 20640, 21600, 22560, 24480, 25440, 27360, 28320, 29280, 32160, 33120, 34080
Offset: 1

Views

Author

J. Lowell, Aug 19 2008

Keywords

Comments

Let n = Product_{p} p ^ e_p be the prime factorization of n and let M = max{e_p + 1 }. Then n is in the sequence iff for all primes q in the range 2 <= q <= M we have e_q >= Sum_{r} floor( log_q (e_r + 1) ). - N. J. A. Sloane, Sep 01 2008
All terms > 1 are even. A subsequence of A033950. - N. J. A. Sloane, Aug 27 2008
Contains 480*p for all primes p > 5 (see A109802). - N. J. A. Sloane, Aug 27 2008

Examples

			72 qualifies because its divisors are 1,2,3,4,6,8,9,12,18,24,36,72, which have 1,2,2,3,4,4,3,6,6,8,9,12 divisors respectively and all of those numbers are divisors of 72.

References

  • Dmitriy Kunisky, German Manoim and N. J. A. Sloane, On strongly refactorable numbers, in preparation.

Links

Crossrefs

Cf. A033950, A134865, A109802, A141551, A141756, A141758, A141900, A142593, A142594, A100549, A100762, A082725, A135130, A143718, A143719, A143720.

Programs

  • Maple
    isA141586 := proc(n) local dvs,d ; dvs := numtheory[divisors](n) ; for d in dvs do if not numtheory[tau](d) in dvs then RETURN(false) : fi; od: RETURN(true) ; end: for n from 1 to 100000 do if isA141586(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Aug 26 2008
## A100549: if n = prod_p p^e_p, then pp = largest prime <= 1 + max e_p
with(numtheory):
pp := proc(n) local f,m; option remember; if (n = 1) then return 1; end if; m := 1: for f in op(2..-1,ifactors(n)) do if (f[2] > m) then m := f[2]: end if; end do; prevprime(m+2); end proc;
isA141586 := proc(n) local ff,f,g,p,i; global pp;
ff := op(2..-1,ifactors(n));
for f in ff do
p := f[1];
if (add(floor(log(1+g[2])/log(p)),g in ff) > f[2]) then
return false;
end if;
end do;
for i from 1 to pi(pp(n)) do
p := ithprime(i);
if (n mod p <> 0) then
if (add(floor(log(1+g[2])/log(p)),g in ff) > 0) then
return false;
end if;
end if;
end do;
return true;
end proc; # David Applegate and N. J. A. Sloane, Sep 15 2008
  • Mathematica
    l = {}; For[n = 1, n < 100000, n++, b = DivisorSigma[0, Divisors[n]]; If[Length[Select[b, Mod[n, # ] > 0 &]] == 0, AppendTo[l, n]]]; l (* Stefan Steinerberger, Aug 25 2008 *)
sfnQ[n_]:=AllTrue[DivisorSigma[0,Divisors[n]],Mod[n,#]==0&]; Select[ Range[ 35000],sfnQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 27 2019 *)
  • PARI
    is_A141586(n)={ bittest(n,0) & return(n==1); fordiv(n,d,n % numdiv(d) & return);1 } \\ M. F. Hasler, Dec 05 2010
  • Sage
    is_A141586 = lambda n: all(number_of_divisors(d).divides(n) for d in divisors(n)) # D. S. McNeil, Dec 05 2010

Extensions

More terms from German Manoim (gerrymanoim(AT)gmail.com), Aug 27 2008
Showing 1-4 of 4 results.

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