A046098 -id:A046098 - cp's OEIS frontend

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

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

A064141 Sum of non-unitary divisors of central binomial coefficient C(n, floor(n/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 12, 0, 0, 72, 328, 0, 768, 1344, 4032, 3024, 9072, 0, 36288, 0, 120960, 322560, 967680, 0, 1935360, 6013440, 15966720, 43545600, 104094720, 163296000, 362361600, 149299200, 447897600, 1194393600, 4644864000, 2654208000

Offset: 1

Labos Elemer, Sep 11 2001

nonn

			For n = 6, binomial(6,3) = 20 = 4*5, divisors = {1,2,4,5,10,20} of which the non-unitary divisors are 2 and 10 with sum a(6) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from Harry J. Smith)

Cf. A034448, A001405, A046098, A048146, A048243.

nus[n_] := If[n==1, 0, DivisorSigma[1, n] - Times @@ (1 + Power @@@ FactorInteger[n])]; Table[nus@ Binomial[n, Floor[n/2]], {n, 35}] (* Giovanni Resta, Jun 22 2018 *)

usigma(n)= { my(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
a(n)={my(b=binomial(n, n\2)); sigma(b) - usigma(b);} \\ Harry J. Smith, Sep 08 2009

a(n) = A048146(A001405(n)). [corrected by Amiram Eldar, Mar 07 2025]

If n is in A046098 then a(n)=0.

A064146 Sum of non-unitary prime divisors of binomial(n,floor(n/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 3, 5, 0, 2, 2, 2, 3, 3, 0, 2, 0, 2, 2, 2, 0, 2, 7, 7, 10, 10, 5, 5, 3, 3, 3, 5, 5, 7, 7, 7, 3, 5, 2, 2, 2, 2, 10, 10, 8, 10, 12, 12, 12, 12, 9, 9, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 7, 9, 7, 9, 5, 5, 0, 2, 2, 2, 7, 7, 14, 14, 7, 9, 12, 12, 5, 5, 10, 10, 10, 10, 5, 5, 12, 12, 12

Offset: 1

Labos Elemer, Sep 11 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..7500 (first 1000 terms from Harry J. Smith)

Cf. A008472, A001405, A063958, A034444, A056169, A046098, A048243.

a:= n-> add(`if`(i[2]>1, i[1], 0), i=ifactors(binomial(n, iquo(n,2)))[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 24 2018

a[n_] := Sum[If[i[[2]] > 1, i[[1]], 0], {i, FactorInteger[ Binomial[n, Quotient[n, 2]]]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

a(n) = { my(f=factor(binomial(n, n\2))); sum(i=1, #f~, if (f[i, 2]>1, f[i,1])) } \\ Harry J. Smith, Sep 09 2009

a(n) = A063958(A001405(n)).

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

A048197 Numbers k for which binomial(k, floor(k/2)) has more unitary than non-unitary divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 31, 32, 35, 39, 41, 43, 55, 65, 67, 71, 72, 73, 79, 131, 271, 1567

Offset: 1

Labos Elemer

A048107 is applied to central binomial coefficients. This sequence includes the 12 known squarefree central binomial coefficients, i.e., 1, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71 collected in A046098.
Numbers k such that A034444(A001405(k)) > A048105(A001405(k)).
No more terms below 10^5. - Ivan Neretin, Sep 06 2015

			For k = 59 the corresponding binomial(59,29) has 8192 divisors, of which 4096 are unitary and equally 4096 are non-unitary. So 59 is not in the sequence.

Cf. A001405, A034444, A048107, A046098.

Select[Range[60], Function[n, r = Binomial[n, Floor[n/2]]; 2^(PrimeNu[r] + 1) > DivisorSigma[0, r]]] (* Ivan Neretin, Sep 06 2015 *)

is(n) = apply(x -> 2^(omega(x)+1) - numdiv(x), binomial(n, n\2)) > 0; \\ Amiram Eldar, Jul 22 2024

More terms from Ivan Neretin, Sep 06 2015

A081392 Numbers k such that the central binomial coefficient C(k, floor(k/2)) has only one prime divisor whose exponent is greater than one.

Original entry on oeis.org

6, 9, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 31, 32, 33, 35, 39, 41, 42, 43, 44, 55, 56, 57, 58, 59, 60, 61, 62, 65, 67, 72, 73, 74, 79, 107, 108, 109, 110, 113, 114, 115, 116, 131, 159, 219, 220, 271, 319, 341, 342, 1567, 1568, 1571, 1572

Offset: 1

Labos Elemer, Mar 27 2003

As expected, the (single) non-unitary prime divisors for C(2k, k) and C(k, floor(k/2)) or for Catalan numbers equally come from the smallest prime(s).
Numbers k such that A001405(k) is in A190641. - Michel Marcus, Jul 30 2017
a(56) > 5*10^6 if it exists. - David A. Corneth, Apr 03 2021

			For k=341, binomial(341,170) = 2*2*2*2*M, where M is a squarefree product of 48 further prime factors.

Cf. A000108, A000984, A001405, A046098, A080664, A081386-A081391, A190641.

pde1Q[n_]:=Length[Select[FactorInteger[Binomial[n,Floor[n/2]]],#[[2]]> 1&]] == 1; Select[Range[1600],pde1Q] (* Harvey P. Dale, Jan 21 2019 *)

isok(n) = my(f=factor(binomial(n, n\2))); #select(x->(x>1), f[,2]) == 1; \\ Michel Marcus, Jul 30 2017

is(n) = { my(nf2 = n\2, nmnf2 = n-nf2, t); forprime(p = 2, n, if(val(n, p) - val(nf2, p) - val(nmnf2, p) > 1, t++; if(t > 1, return(0) ) ) ); t==1 }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Apr 03 2021

a(52)-a(55) from Michel Marcus, Jul 30 2017

A106578 First differences of indices of squarefree central binomial numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 6, 2, 4, 48

Offset: 0

Paul Barry, May 09 2005

nonn

First differences of A046098. Next term >= 27929.

Cf. A001405.

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

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

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A064141 Sum of non-unitary divisors of central binomial coefficient C(n, floor(n/2)).

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A064146 Sum of non-unitary prime divisors of binomial(n,floor(n/2)).

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A113543 Numbers both squarefree and triangle-free.

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A113545 Numbers both pentagon-free and squarefree.

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A048197 Numbers k for which binomial(k, floor(k/2)) has more unitary than non-unitary divisors.

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A081392 Numbers k such that the central binomial coefficient C(k, floor(k/2)) has only one prime divisor whose exponent is greater than one.

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