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 1-7 of 7 results.

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

A113508 Pentagon-free numbers: numbers k such that no divisor of k is a pentagonal number > 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 21, 23, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 49, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 81, 82, 83, 86, 87, 89, 91, 93, 94, 97, 98, 99, 101
Offset: 1

Views

Author

Jonathan Vos Post, Jan 11 2006

Keywords

Comments

Pentagonal number analogy of A112886 (the triangle-free positive integers).
Density is empirically close to 12/(Pi^(5/2)). - Richard Peterson, Apr 06 2025

Examples

			10 is not a term, since 10 = 2 * 5 and 5 is the first nontrivial pentagonal number.

Links

Crossrefs

Cf. A000326, A112886, A113502.

Programs

  • Mathematica
    Select[Range[1, 101], {} == Intersection[{5, 12, 22, 35, 51, 70, 92}, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)
  • PARI
    is(n)=fordiv(n, d, if(ispolygonal(d, 5) && d>1, return(0))); 1 \\ Charles R Greathouse IV, Dec 24 2018

Extensions

Data corrected by Giovanni Resta, Jun 13 2016
a(1)=1 inserted by Andrew Howroyd, Sep 08 2024

A113544 Numbers simultaneously pentagon-free, squarefree and triangle-free.

Original entry on oeis.org

1, 2, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 77, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 119, 122, 127, 131, 133, 134, 137, 139, 142, 143, 146, 149, 151, 157, 158, 161, 163
Offset: 1

Views

Author

Jonathan Vos Post, Jan 13 2006

Keywords

References

  • Bellman, R. and Shapiro, H. N. "The Distribution of Squarefree Integers in Small Intervals." Duke Math. J. 21, 629-637, 1954.
  • Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, 2003.
  • Hardy, G. H. and Wright, E. M. "The Number of Squarefree Numbers." Section 18.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 269-270, 1979.

Links

Crossrefs

Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.

Programs

  • Mathematica
    bad = Rest@ Union[# (# + 1)/2 &@ Range[19], Range[14]^2, # (3 # - 1)/2 &@ Range[11]]; Select[Range[200], {} == Intersection[bad, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)
  • PARI
    list(lim)=my(v=List()); forsquarefree(n=1,lim\1, fordiv(n,d, if((ispolygonal(d,3) || ispolygonal(d,5)) && d>1, next(2))); listput(v,n[1])); Vec(v); \\ Charles R Greathouse IV, Dec 24 2018

Formula

a(n) has no factor >1 of form a*(a+1)/2 nor b^2 nor c*(3*c-1)/2. A005117 INTERSECTION A112886 INTERSECTION A113508.

Extensions

Corrected and extended by Giovanni Resta, Jun 13 2016

A113543 Numbers both squarefree and triangle-free.

Original entry on oeis.org

1, 2, 5, 7, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 94, 95, 97, 101, 103, 106, 107, 109, 113, 115, 118, 119, 122, 127, 131, 133, 134, 137, 139, 142, 143, 145
Offset: 1

Views

Author

Jonathan Vos Post, Jan 13 2006

Keywords

Comments

The cardinality (count, enumeration) of these through n equals n - card{squarefree numbers <= n} - card{trianglefree numbers <= n} + card{numbers <= n which are both square and triangular} = n - card{numbers <= n in A005117} - card{numbers <=n in A112886} + card{numbers <= n in A001110}. "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer. In fact, this problem may be no easier than the general problem of integer factorization (obviously, if an integer can be factored completely, is squarefree iff it contains no duplicated factors). This problem is an important unsolved problem in number theory" [Weisstein]. Conjecture: there is no polynomial time algorithm for recognizing numbers which are both squarefree and triangle-free.

References

  • Bellman, R. and Shapiro, H. N. "The Distribution of Squarefree Integers in Small Intervals." Duke Math. J. 21, 629-637, 1954.
  • Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, 2003.
  • Hardy, G. H. and Wright, E. M. "The Number of Squarefree Numbers." Section 18.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 269-270, 1979.

Links

Crossrefs

Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886.

Programs

  • Mathematica
    bad = Rest@Union[# (# + 1)/2 &@ Range[19], Range[14]^2]; Select[ Range[200], {} == Intersection[bad, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)

Formula

a(n) has no factor >1 of form a*(a+1)/2 nor b^2. A005117 INTERSECTION A112886.

Extensions

Corrected and extended by Giovanni Resta, Jun 13 2016

A113545 Numbers both pentagon-free and squarefree.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 53, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 86, 87, 89, 91, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 114, 118, 119, 122, 123, 127, 129
Offset: 1

Views

Author

Jonathan Vos Post, Jan 13 2006

Keywords

References

  • Bellman, R. and Shapiro, H. N. "The Distribution of Squarefree Integers in Small Intervals." Duke Math. J. 21, 629-637, 1954.
  • Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, 2003.
  • Hardy, G. H. and Wright, E. M., Section 18.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 269-270, 1979.

Links

Crossrefs

Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.

Programs

  • Mathematica
    bad = Rest@Union[Range[14]^2, # (3 # - 1)/2 &@ Range[11]]; Select[ Range[200], {} == Intersection[ bad, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)

Formula

a(n) has no factor >1 of form b^2 nor c*(3*c-1)/2. A005117 INTERSECTION A113508.

Extensions

Data corrected by Giovanni Resta, Jun 13 2016

A113619 Heptagon-free numbers: numbers k such that no divisor of k is a heptagonal number > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 71, 73, 74, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 99, 100
Offset: 1

Views

Author

Jonathan Vos Post, Jan 14 2006

Keywords

Comments

Heptagonal number analogy of A112886 (the triangle-free positive integers).

Examples

			7 is the first nontrivial heptagonal number, so no multiple of 7 is a term.

Links

Crossrefs

Cf. A000566, A112886, A113502.

Programs

  • Mathematica
    upto=100;Module[{maxhep=Floor[(3+Sqrt[9+40upto])/10],heps}, heps= Rest[ Table[(n(5n-3))/2,{n,maxhep}]];Complement[Range[upto],Union[ Flatten[ Table[n*heps,{n,Ceiling[upto/7]}]]]]] (* Harvey P. Dale, May 19 2012 *)

Extensions

Corrected by Harvey P. Dale, May 19 2012

A113626 Numbers simultaneously heptagon-free, pentagon-free, squarefree and triangle-free.

Original entry on oeis.org

1, 2, 11, 13, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 143, 146, 149, 151, 157, 158, 163, 166, 167, 173, 178, 179, 181, 187
Offset: 1

Views

Author

Jonathan Vos Post, Jan 14 2006

Keywords

Comments

This sequence is the 5th step in a polygonal-factor sieve, where all integers with k-gonal factors have been eliminated from an initial set of the natural numbers, for k = 3, 4, 5, .... There is no need to specifically sieve out hexagonal numbers, as every hexagonal number is a triangular number and thus is already sieved. Every integer n is sieved out no later than step n-3, as n-gonal number(2) = n (e.g. 7 is eliminated when we sieve out all numbers with heptagonal factors, as 7 = Hep(2); 11 is eliminated when we sieve out all 11-gonal number multiples). After an infinite number of steps, the sequence collapses to {1,2}. If, instead, at each step we eliminate all multiples of n-gonal numbers except {1, n} then the sequence converges on {1,4} UNION {primes}.

Links

Crossrefs

Cf. A000217, A000566, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.

Programs

  • Maple
    isA000217 := proc(n) local discr ; discr := 1+8*n ; if issqr(discr) then if ( sqrt(discr)-1 ) mod 2 = 0 then true; else false ; fi ; else false ; fi ; end: isA000326 := proc(n) local discr ; discr := 1+24*n ; if issqr(discr) then if ( sqrt(discr)+1 ) mod 6 = 0 then true; else false ; fi ; else false ; fi ; end: isA000566 := proc(n) local discr ; discr := 9+40*n ; if issqr(discr) then if ( sqrt(discr)+3 ) mod 10 = 0 then true; else false ; fi ; else false ; fi ; end: isA000290 := proc(n) issqr(n) ; end: isA113626 := proc(n) local d ; for d in numtheory[divisors](n) do if d > 1 then if isA000217(d) or isA000290(d) or isA000326(d) or isA000566(d) then RETURN(false) ; fi ; fi ; od: RETURN(true) ; end: for n from 1 to 500 do if isA113626(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Apr 19 2008
  • Mathematica
    The Mathematica function SquareFreeQ[n] in the Mathematica add-on package NumberTheory`NumberTheoryFunctions` (which can be loaded with the command <

Formula

a(n) has no factor >1 of form b*(b+1)/2, c^2, d*(3*d-1)/2, nor e*(5*e-3)/2.
A113544 INTERSECT A113619. - R. J. Mathar, Jul 24 2009

Extensions

More terms from R. J. Mathar, Apr 19 2008
Extended by R. J. Mathar, Jul 24 2009
Showing 1-7 of 7 results.

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