Displaying 1-10 of 11 results found.
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).
+10
8
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
COMMENTS
A number n is in the sequence iff it is not a "triangle-free" positive integer.
EXAMPLE
12 is included because its divisors are 1, 2, 3, 4, 6 and 12, two of which (3 and 6) are triangular numbers > 1.
MATHEMATICA
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 *)
Pentagon-free numbers: numbers k such that no divisor of k is a pentagonal number > 1.
+10
4
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
COMMENTS
Pentagonal number analogy of A112886 (the triangle-free positive integers).
EXAMPLE
10 is not a term, since 10 = 2 * 5 and 5 is the first nontrivial pentagonal number.
24 is not a term, since 12|24 and 12 is a pentagonal number.
44 is not a term, since 22|44 and 22 is a pentagonal number.
MATHEMATICA
Select[Range[1, 101], {} == Intersection[{5, 12, 22, 35, 51, 70, 92}, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)
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.
+10
3
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
COMMENTS
Obviously, all divisors of an odd number are isolated.
EXAMPLE
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.
MAPLE
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);
MATHEMATICA
Select[2*Range[120], Min[Differences[Rest[Divisors[#]]]]>1&] (* Harvey P. Dale, Jul 13 2022 *)
Numbers simultaneously pentagon-free, squarefree and triangle-free.
+10
2
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
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.
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.
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 *)
PROG
(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
CROSSREFS
Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.
Numbers both squarefree and triangle-free.
+10
1
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
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.
FORMULA
a(n) has no factor >1 of form a*(a+1)/2 nor b^2. A005117 INTERSECTION A112886.
MATHEMATICA
bad = Rest@Union[# (# + 1)/2 &@ Range[19], Range[14]^2]; Select[ Range[200], {} == Intersection[bad, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)
CROSSREFS
Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886.
Numbers both pentagon-free and squarefree.
+10
1
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
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.
FORMULA
a(n) has no factor >1 of form b^2 nor c*(3*c-1)/2. A005117 INTERSECTION A113508.
MATHEMATICA
bad = Rest@Union[Range[14]^2, # (3 # - 1)/2 &@ Range[11]]; Select[ Range[200], {} == Intersection[ bad, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)
CROSSREFS
Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.
Heptagon-free numbers: numbers k such that no divisor of k is a heptagonal number > 1.
+10
1
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
COMMENTS
Heptagonal number analogy of A112886 (the triangle-free positive integers).
EXAMPLE
7 is the first nontrivial heptagonal number, so no multiple of 7 is a term.
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 *)
Number of partitions of n into triangular numbers dividing n.
+10
1
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
EXAMPLE
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].
MATHEMATICA
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}]
Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 - x^(k*(k + 1)/2))^2.
+10
1
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
COMMENTS
Sum of divisors d of n such that n/d is triangular number.
MATHEMATICA
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}]
PROG
(PARI) a(n)={sumdiv(n, d, if(ispolygonal(d, 3), n/d))} \\ Andrew Howroyd, Sep 19 2019
Numbers simultaneously heptagon-free, pentagon-free, squarefree and triangle-free.
+10
0
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
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 (i.e. 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}.
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.
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 <<NumberTheory`) determines whether a number is squarefree.
CROSSREFS
Cf. A000217, A000566, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.
Search completed in 0.009 seconds
|