A112886 -id:A112886 - cp's OEIS frontend

A113502 A number n is included if at least one of its divisors > 1 is a triangular number (i.e., is of the form m(m+1)/2, m >= 2).

3, 6, 9, 10, 12, 15, 18, 20, 21, 24, 27, 28, 30, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 56, 57, 60, 63, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 91, 93, 96, 99, 100, 102, 105, 108, 110, 111, 112, 114, 117, 120, 123, 126, 129, 130, 132, 135, 136, 138, 140, 141

Offset: 1

Leroy Quet, Jan 10 2006

nonn

A number n is in the sequence iff it is not a "triangle-free" positive integer.

Multiples of A226863. - Charles R Greathouse IV, Jul 29 2016

			12 is included because its divisors are 1, 2, 3, 4, 6 and 12, two of which (3 and 6) are triangular numbers > 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A112886, A000217, A013929, A226863.

v={};Do[If[b=Select[Divisors[n], #>1 && IntegerQ[(1+8#)^(1/2)]&]; b!={}, AppendTo[v, n]], {n, 200}]; v (* Farideh Firoozbakht, Jan 12 2006 *)
Select[Range[200],AnyTrue[Rest[Divisors[#]],OddQ[Sqrt[8#+1]]&]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2017 *)

is(n)=fordiv(n,d, if(ispolygonal(d,3) && d>1, return(1))); 0 \\ Charles R Greathouse IV, Jul 29 2016

a(n) = A088723(n)/2. - Ray Chandler, May 29 2008

More terms from Farideh Firoozbakht, Jan 12 2006

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

Jonathan Vos Post, Jan 11 2006

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

Density is empirically close to 12/(Pi^(5/2)). - Richard Peterson, Apr 06 2025

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

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000326, A112886, A113502.

Select[Range[1, 101], {} == Intersection[{5, 12, 22, 35, 51, 70, 92}, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)

is(n)=fordiv(n, d, if(ispolygonal(d, 5) && d>1, return(0))); 1 \\ Charles R Greathouse IV, Dec 24 2018

Data corrected by Giovanni Resta, Jun 13 2016

a(1)=1 inserted by Andrew Howroyd, Sep 08 2024

A132895 Even numbers for which all divisors, with the exception of 1 and 2, are isolated. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 50, 52, 58, 62, 64, 68, 70, 74, 76, 82, 86, 88, 92, 94, 98, 104, 106, 116, 118, 122, 124, 128, 130, 134, 136, 142, 146, 148, 152, 154, 158, 164, 166, 170, 172, 176, 178, 184, 188, 190, 194, 196, 202, 206, 208, 212

Offset: 1

Emeric Deutsch, Oct 16 2007, Oct 19 2007

nonn

Obviously, all divisors of an odd number are isolated.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 4, 29, 278, 2782, 27813, 278055, 2780548, 27805234, 278052138, 2780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.27805... . - Amiram Eldar, Apr 20 2025

			28 is a term of the sequence because its divisors are 1, 2, 4, 7, 14, 28 and only 1 and 2 are non-isolated.
30 does not belong to the sequence because its divisors are 1, 2, 3, 4, 6, 8, 12, 24 and 1, 2, 3, 4 are non-isolated.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A112886, A133950, A133951, A088722-A088726.

with(numtheory): b:=proc(n) local div,ISO,i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1,div)=false and member(div[i]+1,div)=false then ISO:=`union`(ISO,{div[i]}) end if end do end proc: a:=proc(n) if nops(b(n))= tau(n)-2 then n else end if end proc: seq(a(n), n=4..200);

Select[2*Range[120],Min[Differences[Rest[Divisors[#]]]]>1&] (* Harvey P. Dale, Jul 13 2022 *)

isok(k) = if(k%2, 0, if(!(k%3), 0, fordiv(k, d, if(d > 1 && !(k % (d+1)), return(0))); 1)); \\ Amiram Eldar, Apr 20 2025

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A132895_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:all(d==1 or not is_square((d<<3)+1) for d in divisors(n>>1,generator=True)), count(max(startvalue+(startvalue&1),2),2))
A132895_list = list(islice(A132895_gen(),40)) # Chai Wah Wu, Jun 07 2025

a(n) = 2*A112886(n). - Ray Chandler, May 29 2008

Corrected and extended by Ray Chandler, May 29 2008

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

Jonathan Vos Post, Jan 13 2006

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.

G. C. Greubel and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Greubel)
Eric Weisstein's World of Mathematics, Squarefree.

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

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 *)

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

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.

Corrected and extended by Giovanni Resta, Jun 13 2016

A327629 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 - x^(k(k + 1)/2))^2.

Original entry on oeis.org

1, 2, 4, 4, 5, 9, 7, 8, 12, 11, 11, 18, 13, 14, 21, 16, 17, 27, 19, 22, 29, 22, 23, 36, 25, 26, 36, 29, 29, 50, 31, 32, 44, 34, 35, 55, 37, 38, 52, 44, 41, 65, 43, 44, 64, 46, 47, 72, 49, 55, 68, 52, 53, 81, 56, 58, 76, 58, 59, 100, 61, 62, 87, 64, 65, 100, 67, 68, 92, 77

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

nonn

Sum of divisors d of n such that n/d is triangular number.

Cf. A000217, A006463, A007862, A010054, A076752, A112886 (fixed points), A185027.

nmax = 70; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Sqrt[8 n/# + 1]] &]; Table[a[n], {n, 1, 70}]

a(n)={sumdiv(n, d, if(ispolygonal(d,3), n/d))} \\ Andrew Howroyd, Sep 19 2019

from sympy import divisors
from sympy.ntheory.primetest import is_square
def A327629(n): return sum(n//d for d in divisors(n,generator=True) if is_square((d<<3)+1)) # Chai Wah Wu, Jun 07 2025

a(n) = Sum_{d|n} A010054(n/d) * d.

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

Jonathan Vos Post, Jan 13 2006

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.

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.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Squarefree.

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

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

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

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

Jonathan Vos Post, Jan 13 2006

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.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Squarefree.

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

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

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

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

Jonathan Vos Post, Jan 14 2006

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

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

Eric Weisstein's World of Mathematics, Heptagonal Number.

Cf. A000566, A112886, A113502.

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 *)

Corrected by Harvey P. Dale, May 19 2012

A294334 Number of partitions of n into triangular numbers dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 1, 1, 4, 2, 1, 9, 1, 1, 7, 1, 1, 16, 1, 3, 9, 1, 1, 25, 1, 1, 10, 2, 1, 74, 1, 1, 12, 1, 1, 50, 1, 1, 14, 5, 1, 85, 1, 1, 35, 1, 1, 81, 1, 6, 18, 1, 1, 100, 2, 3, 20, 1, 1, 544, 1, 1, 46, 1, 1, 145, 1, 1, 24, 8, 1, 219, 1, 1, 81, 1, 1, 197, 1, 9, 28, 1, 1, 628, 1, 1, 30, 1, 1, 2264, 2, 1, 32, 1, 1

Offset: 0

Ilya Gutkovskiy, Oct 28 2017

			a(6) = 4 because 6 has 4 divisors {1, 2, 3, 6} among which 3 are triangular numbers {1, 3, 6} therefore we have [6], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Cf. A000217, A007294, A007862, A018818, A112886, A284345.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] == 0 && IntegerQ[Sqrt[8 k + 1]]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 95}]

a(n) = 1 if n in A112886.

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

Jonathan Vos Post, Jan 14 2006

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}.

J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

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

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

The Mathematica function SquareFreeQ[n] in the Mathematica add-on package NumberTheory`NumberTheoryFunctions` (which can be loaded with the command <

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

More terms from R. J. Mathar, Apr 19 2008

Extended by R. J. Mathar, Jul 24 2009

cp's OEIS Frontend

A113502 A number n is included if at least one of its divisors > 1 is a triangular number (i.e., is of the form m(m+1)/2, m >= 2).

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

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A132895 Even numbers for which all divisors, with the exception of 1 and 2, are isolated. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

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

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A327629 Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 - x^(k*(k + 1)/2))^2.

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Mathematica

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A113543 Numbers both squarefree and triangle-free.

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Mathematica

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A113545 Numbers both pentagon-free and squarefree.

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A327629 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 - x^(k(k + 1)/2))^2.