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.

A088725 Numbers having no divisors d>1 such that also d+1 is a divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

Comments

Complement of A088723.
Union of A132895 and A005408, the odd numbers. - Ray Chandler, May 29 2008
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 9, 79, 778, 7782, 77813, 778055, 7780548, 77805234, 778052138, 7780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.77805... . - Amiram Eldar, Jun 14 2022

Examples

			From _Gus Wiseman_, Oct 16 2019: (Start)
The sequence of terms together with their divisors > 1 begins:
   1: {}
   2: {2}
   3: {3}
   4: {2,4}
   5: {5}
   7: {7}
   8: {2,4,8}
   9: {3,9}
  10: {2,5,10}
  11: {11}
  13: {13}
  14: {2,7,14}
  15: {3,5,15}
  16: {2,4,8,16}
  17: {17}
  19: {19}
  21: {3,7,21}
  22: {2,11,22}
  23: {23}
  25: {5,25}
(End)

Crossrefs

Positions of 0's and 1's in A129308.
Positions of 0's and 1's in A328457 (also).
Numbers whose divisors (including 1) have no non-singleton runs are A005408.
The number of runs of divisors of n is A137921(n).
The longest run of divisors of n has length A055874(n).
Cf. A000005, A027750, A060680, A088722, A088723 (complement), A088724, A088726, A328166, A328450.

Programs

  • Mathematica
    Select[Range[100],FreeQ[Differences[Rest[Divisors[#]]],1]&] (* Harvey P. Dale, Sep 16 2017 *)
  • PARI
    isok(n) = {my(d=setminus(divisors(n), [1])); #setintersect(d, apply(x->x+1, d)) == 0;} \\ Michel Marcus, Oct 28 2019

Formula

A088722(a(n)) = 0.

Extensions

Extended by Ray Chandler, May 29 2008

A112886 Positive integers that have no triangular divisors > 1.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 25, 26, 29, 31, 32, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 58, 59, 61, 62, 64, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 98, 101, 103, 104, 106, 107, 109, 113, 115, 116, 118, 119
Offset: 1

Views

Author

Leroy Quet, Jan 10 2006

Keywords

Comments

Sequence consists of those positive integers not in A113502.
(Pi^(3/2))/density is empirically close to 10. - Richard Peterson, Apr 06 2025

Examples

			14 is included because the divisors of 14 are 1, 2, 7 and 14, none of which are triangular numbers > 1.

Links

Crossrefs

Cf. A113502, A000217, A005117, A132895.

Programs

  • Mathematica
    v={};Do[If[b=Select[Divisors[n], #>1 && IntegerQ[(1+8#)^(1/2)]&]; b=={}, AppendTo[v, n]], {n, 138}]; v (Firoozbakht)
tfpnQ[n_]:=Module[{nn=120,trnos},trnos=Rest[Accumulate[ Range[ (Sqrt[8nn+1]-1)/2+1]]]; Intersection[ Divisors[ n],trnos]=={}]; Select[Range[ 120], tfpnQ] (* Harvey P. Dale, Jul 17 2015 *)
  • PARI
    is(n)=fordiv(n, d, if(ispolygonal(d, 3) && d>1, return(0))); 1 \\ Charles R Greathouse IV, Jul 31 2016
  • Python
    from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A112886_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(d==1 or not is_square((d<<3)+1) for d in divisors(n,generator=True)), count(max(startvalue,1)))
A112886_list = list(islice(A112886_gen(),40)) # Chai Wah Wu, Jun 07 2025

Formula

a(n) = A132895(n)/2. - Ray Chandler, May 29 2008

Extensions

More terms from Farideh Firoozbakht, Jan 12 2006
Name edited (based on a suggestion from Michel Marcus) by Jon E. Schoenfield, Jul 02 2021

A088723 Numbers k with at least one divisor d>1 such that d+1 also divides k.

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

Comments

Complement of A088725.
Complement of A132895 relative to A005843, the even numbers. - Ray Chandler, May 29 2008
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 21, 222, 2218, 22187, 221945, 2219452, 22194766, 221947862, 2219480686, ... . Apparently, the asymptotic density of this sequence exists and equals 0.22194... . - Amiram Eldar, Apr 20 2025

Links

Crossrefs

Cf. A088722, A088724, A088725, A088726.
Cf. A027750, A008588 (subsequence).

Programs

  • Haskell
    a088723 n = a088723_list !! (n-1)
a088723_list = filter f [2..] where
   f x = 1 `elem` (zipWith (-) (tail divs) divs)
         where divs = tail $ a027750_row x
-- Reinhard Zumkeller, Dec 16 2013
  • Mathematica
    Select[Range[300],MemberQ[Differences[Select[Divisors[#], #>1&]], 1]&]  (* Harvey P. Dale, Apr 03 2011 *)
  • PARI
    isok(k) = if(k%2, 0, if(!(k%3), 1, fordiv(k, d, if(d > 1 && !(k % (d+1)), return(1))); 0)); \\ Amiram Eldar, Apr 20 2025

Formula

A088722(a(n)) > 0.

Extensions

Extended by Ray Chandler, May 29 2008
Showing 1-3 of 3 results.

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