A071172 -id:A071172 - cp's OEIS frontend

A113543 Numbers both squarefree and triangle-free.

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

A274043 Number of squarefree integers congruent to {1, 2, 3} mod 8 <= 10^n.

Original entry on oeis.org

4, 28, 300, 3033, 30389, 303947, 3039643, 30396338, 303963527, 3039635535, 30396355364, 303963551074, 3039635509269, 30396355092700, 303963550926732, 3039635509266675, 30396355092702331, 303963550927021020

Offset: 1

Frank M Jackson, Jun 18 2016

Empirically, the limit of a(n)/10^n tends to 3/Pi^2 (A104141) and implies that the asymptotic density of squarefree numbers congruent to {1, 2, 3} mod 8 is half that of the asymptotic density of all squarefree integers (A071172). When this sequence is compared with squarefree numbers congruent to {5, 6, 7} mod 8 (A274264) it contains slightly fewer squarefree integers at each of the sampling points, 10^n for n > 1. It can be argued heuristically that, as {1, 2, 3} mod 8 contains a square residue, its equivalence class should contain fewer squarefree numbers.

Also it has been shown, conditional on the Birch Swinnerton-Dyer conjecture, that all squarefree integers congruent to {5, 6, 7} mod 8 (A273929) are primitive congruent numbers (A006991). However, this property applies only sparsely to squarefree integers congruent to {1, 2, 3} mod 8 (A062695).

Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, (2008).
Eric Weisstein's World of Mathematics, Squarefree
Shou-Wu Zhang, The Congruent Numbers and Heegner Points, Asian Pacific Mathematics Newsletter, Vol 3(2) (2013).

Cf. A006991, A062695, A071172, A104141, A273929, A274264.

Table[Length@Select[Range[10^n], MemberQ[{1, 2, 3}, Mod[#, 8]]&&SquareFreeQ[#] &], {n, 1, 8}]

a(10)-a(11) from Giovanni Resta, Jun 19 2016

a(12)-a(18) from Hiroaki Yamanouchi, Dec 25 2016

A274403 Number of primitive (squarefree) congruent numbers (A006991) <= 10^n.

Original entry on oeis.org

3, 36, 361, 3503, 34065, 332712, 3252966, 31925924

Offset: 1

Frank M Jackson, Jun 20 2016

Conjecture: the limit of a(n)/10^n tends to 3/Pi^2 (A104141). This is based on the assumption, conditional on the Birch Swinnerton-Dyer conjecture, that all squarefree integers congruent to {5, 6, 7} mod 8 (A273929) are a subset of primitive congruent numbers (A006991) and have a natural density of 3/Pi^2. However, squarefree integers congruent to {1, 2, 3} mod 8 are conjecturally sparsely congruent numbers with a natural density of 0. It has been proved without the BSD conjecture that the natural density of congruent numbers is at least 55.9% the natural density of squarefree numbers congruent to {5, 6, 7} mod 8 (see A. Smith link).
The Mathematica program below is a slow implementation of the Tunnell criteria for determining congruent numbers. It will give counts for up to 10^5 in realistic time. Counts for 10^6 and 10^7 have been derived from tables generated by Giovanni Resta (see link).
From Jose Aranda, Jul 04 2024: (Start)
The C++ program I have written calculates a(8) = 31925924 in 75 minutes. The time grows almost exponentially.
Looking at the 8 known terms I think the above conjecture should perhaps refer to A274264 rather than to the present sequence.
From the link "A trillion triangles": "The calculation found of these most mysterious congruent numbers up to a trillion = 3148379694."
That number corresponds to a(10) = 108744287 + A274264(10).
With A274264(10) = 3039635407. Now
3/Pi^2      = 0.303963550927013314...
A274264(08) = 0030396356.
A274264(10) = 003039635407.
A274264(18) = 00303963550927001730.
The sequence A274264 tends to this limit. This sequence may not. (End)

Jose Aranda, C++ program
Estelle Basor and Bill Hart, A trillion triangles, American Institute of Mathematics,
Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, (2008).
Giovanni Resta, Table of primitive congruent numbers {1, 2, 3} mod 8
Alexander Smith, The congruent numbers have positive natural density, arXiv:1603.08479 [math.NT], 2016.
Wikipedia, Congruent number
Shou-Wu Zhang, The Congruent Numbers and Heegner Points, Asian Pacific Mathematics Newsletter, Vol 3(2) (2013).

Cf. A006991, A062695, A071172, A104141, A273929, A274043, A274264.

CongruentQ[n_] := Module[{x, y, z, ok=False}, (Which[!SquareFreeQ[n], Null[], MemberQ[{5,6,7}, Mod[n, 8]], ok=True, OddQ@n&&Length@Solve[x^2 + 2 y^2 + 8 z^2 == n, {x, y, z}, Integers]==2Length@Solve[x^2+2y^2+32z^2==n, {x, y, z}, Integers], ok=True, EvenQ@n&&Length@Solve[x^2+4y^2+8z^2==n/2, {x, y, z}, Integers]==2Length@Solve[x^2+4y^2+32z^2==n/2, {x, y, z}, Integers], ok=True]; ok)]; Table[Length@Select[Range[10^n], CongruentQ], {n, 1, 5}]

a(7) corrected by Frank M Jackson, Jul 25 2016

a(8) from Jose Aranda, Jul 04 2024

A348708 Number of squarefree integers with an odd number of prime factors <= 10^n.

Original entry on oeis.org

0, 4, 30, 303, 3053, 30421, 303857, 3039127, 30395383, 303963673, 3039652332, 30396399068, 303963519954, 3039635209356, 30396355530761, 303963552535238, 3039635510867921, 30396355103617024, 303963550950390745, 3039635508820145344, 30396355092470750098, 303963550928711270447

Offset: 0

Giorgos Kalogeropoulos, Oct 30 2021

nonn

a(n) is the number of integers k <= 10^n with mu(k)=-1 where mu(k) is the Möbius function.

			a(1) = 4 because there are 4 squarefree integers with an odd number of prime factors <= 10: 2, 3, 5, 7.

Cf. A071172, A084237, A008683, A030059, A104141, A348709.

Table[Length@Select[Range[10^n],MoebiusMu@#==-1&],{n,0,6}]

a(n) = (A071172(n) - A084237(n)) / 2.

Lim_{n->oo} a(n)/10^n = 3/Pi^2 (A104141).

A348709 Number of squarefree integers with an even number of prime factors <= 10^n.

Original entry on oeis.org

1, 3, 31, 305, 3030, 30373, 304069, 3040164, 30397311, 303963451, 3039618610, 30396311212, 303963582320, 3039635808938, 30396354655186, 303963549318865, 3039635507672484, 30396355081786770, 303963550903632005, 3039635509720135531, 30396355092931863204, 303963550925315375170

Offset: 0

Giorgos Kalogeropoulos, Oct 30 2021

nonn

a(n) is the number of integers k <= 10^n with mu(k)=1 where mu(k) is the Möbius function.

			a(1) = 3 because there are 3 squarefree integers with an even number of prime factors <= 10: 1, 6, 10.

Cf. A071172, A084237, A008683, A030229, A104141, A348708.

Table[Length@Select[Range[10^n],MoebiusMu@#==1&],{n,0,6}]

a(n) = (A071172(n) + A084237(n)) / 2.

Lim_{n->oo} a(n)/10^n = 3/Pi^2 (A104141).

A365435 Position of A002110(n) in A005117.

Original entry on oeis.org

1, 2, 5, 19, 129, 1405, 18262, 310347, 5896728, 135624240, 3933101824, 121926157641, 4511267827532, 184961980943493, 7953365180610401, 373808163488684050, 19811832664899731266, 1168898127229083969893, 71302785760974119699474

Offset: 0

Michael De Vlieger, Nov 19 2023

Primorial A002110(n) is the a(n)-th squarefree number.

			Let b(n) = A002110(n) and c(n) = A005117(n).
a(0) = 1 since b(0) = 1, and c(1) = 1.
a(1) = 2 since b(1) = 2 = c(2).
a(2) = 5 since b(2) = 6 = c(5).
a(3) = 19 since b(3) = 30 = c(19), etc.

Cf. A002110, A005117, A071172.

s = Select[Range[2^20], SquareFreeQ]; Map[FirstPosition[s, #][[1]] &, FoldList[Times, 1, Prime@ Range[7]] ]

A071172(n) = my(s); forsquarefree(d=1, sqrtint(n), s += n\d[1]^2 * moebius(d)); s; \\ Charles R Greathouse IV at A071172
a(n) = my(p = prod(i = 1, n, prime(i))); A071172(p); \\ Amiram Eldar, Nov 19 2023

from math import isqrt
from sympy import primorial, mobius
def A365435(n):
    if n == 0: return 1
    p = primorial(n)
    return sum(mobius(k)*(p//k**2) for k in range(1,isqrt(p)+1)) # Chai Wah Wu, Aug 12 2024

a(8)-a(16) from Amiram Eldar, Nov 19 2023
a(17) from Chai Wah Wu, Aug 12 2024
a(18) from Chai Wah Wu, Aug 13 2024

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

A256218 The sequence Q(10^n) arising in connection with the study of a generalized Benford's law for squarefree integer powers.

Original entry on oeis.org

5, 58, 604, 6074, 60786, 607919, 6079261, 60792699, 607927089, 6079271005, 60792710170, 607927101837, 6079271018522, 60792710185383, 607927101854006

Offset: 1

N. J. A. Sloane, Mar 27 2015

See link for precise definition.

W. Hürlimann, A First Digit Theorem for Square-Free Integer Powers, Pure Mathematical Sciences, Vol. 3, 2014, no. 3, 129 - 139 HIKARI Ltd.
Index entries for sequences related to Benford's law

Cf. A071172.

cp's OEIS Frontend

A113543 Numbers both squarefree and triangle-free.

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A113545 Numbers both pentagon-free and squarefree.

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A274043 Number of squarefree integers congruent to {1, 2, 3} mod 8 <= 10^n.

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A274403 Number of primitive (squarefree) congruent numbers (A006991) <= 10^n.

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A348708 Number of squarefree integers with an odd number of prime factors <= 10^n.

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A348709 Number of squarefree integers with an even number of prime factors <= 10^n.

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A365435 Position of A002110(n) in A005117.

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

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