A276378 -id:A276378 - cp's OEIS frontend

A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset: 1

N. J. A. Sloane

1 together with the numbers that are products of distinct primes.
Also smallest sequence with the property that a(m)*a(k) is never a square for k != m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
Numbers k such that there is only one Abelian group with k elements, the cyclic group of order k (the numbers such that A000688(k) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001
Numbers k such that A007913(k) > phi(k). - Benoit Cloitre, Apr 10 2002
a(n) is the smallest m with exactly n squarefree numbers <= m. - Amarnath Murthy, May 21 2002
k is squarefree <=> k divides prime(k)# where prime(k)# = product of first k prime numbers. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004
Numbers k such that omega(k) = Omega(k) = A072047(k). - Lekraj Beedassy, Jul 11 2006
The LCM of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006
This sequence and the Beatty Pi^2/6 sequence (A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). - Ed Pegg Jr, Jul 22 2008
Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1) + ... + alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1) + ... + alpha(r) is sequence (A001222). Function D(n) splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. For D(n)=1/2 we have A048109, for D(n)=3/4 we have A060687. - Ctibor O. Zizka, Sep 21 2008
Numbers k such that gcd(k,k')=1 where k' is the arithmetic derivative (A003415) of k. - Giorgio Balzarotti, Apr 23 2011
Numbers k such that A007913(k) = core(k) = k. - Franz Vrabec, Aug 27 2011
Numbers k such that sqrt(k) cannot be simplified. - Sean Loughran, Sep 04 2011
Indices m where A057918(m)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is a square. - John W. Layman, Sep 08 2011
It appears that these are numbers j such that Product_{k=1..j} (prime(k) mod j) = 0 (see Maple code). - Gary Detlefs, Dec 07 2011. - This is the same claim as Mohammed Bouayoun's Mar 30 2004 comment above. To see why it holds: Primorial numbers, A002110, a subsequence of this sequence, are never divisible by any nonsquarefree number, A013929, and on the other hand, the index of the greatest prime dividing any n is less than n. Cf. A243291. - Antti Karttunen, Jun 03 2014
Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that Sum_{k=1..n} a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). - Zhi-Wei Sun, Mar 26 2013
The probability that a random natural number belongs to the sequence is 6/Pi^2, A059956 (see Cesàro reference). - Giorgio Balzarotti, Nov 21 2013
Booker, Hiary, & Keating give a subexponential algorithm for testing membership in this sequence without factoring. - Charles R Greathouse IV, Jan 29 2014
Because in the factorizations into prime numbers these a(n) (n >= 2) have exponents which are either 0 or 1 one could call the a(n) 'numbers with a fermionic prime number decomposition'. The levels are the prime numbers prime(j), j >= 1, and the occupation numbers (exponents) e(j) are 0 or 1 (like in Pauli's exclusion principle). A 'fermionic state' is then denoted by a sequence with entries 0 or 1, where, except for the zero sequence, trailing zeros are omitted. The zero sequence stands for a(1) = 1. For example a(5) = 6 = 2^1*3^1 is denoted by the 'fermionic state' [1, 1], a(7) = 10 by [1, 0, 1]. Compare with 'fermionic partitions' counted in A000009. - Wolfdieter Lang, May 14 2014
From Vladimir Shevelev, Nov 20 2014: (Start)
The following is an Eratosthenes-type sieve for squarefree numbers. For integers > 1:
1) Remove even numbers, except for 2; the minimal non-removed number is 3.
2) Replace multiples of 3 removed in step 1, and remove multiples of 3 except for 3 itself; the minimal non-removed number is 5.
3) Replace multiples of 5 removed as a result of steps 1 and 2, and remove multiples of 5 except for 5 itself; the minimal non-removed number is 6.
4) Replace multiples of 6 removed as a result of steps 1, 2 and 3 and remove multiples of 6 except for 6 itself; the minimal non-removed number is 7.
5) Repeat using the last minimal non-removed number to sieve from the recovered multiples of previous steps.
Proof. We use induction. Suppose that as a result of the algorithm, we have found all squarefree numbers less than n and no other numbers. If n is squarefree, then the number of its proper divisors d > 1 is even (it is 2^k - 2, where k is the number of its prime divisors), and, by the algorithm, it remains in the sequence. Otherwise, n is removed, since the number of its squarefree divisors > 1 is odd (it is 2^k-1).
(End)
The lexicographically least sequence of integers > 1 such that each entry has an even number of proper divisors occurring in the sequence (that's the sieve restated). - Glen Whitney, Aug 30 2015
0 is nonsquarefree because it is divisible by any square. - Jon Perry, Nov 22 2014, edited by M. F. Hasler, Aug 13 2015
The Heinz numbers of partitions with distinct parts. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} prime(j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] the Heinz number is 2*2*3*7*29 = 2436. The number 30 (= 2*3*5) is in the sequence because it is the Heinz number of the partition [1,2,3]. - Emeric Deutsch, May 21 2015
It is possible for 2 consecutive terms to be even; for example a(258)=422 and a(259)=426. - Thomas Ordowski, Jul 21 2015. [These form a subsequence of A077395 since their product is divisible by 4. - M. F. Hasler, Aug 13 2015]
There are never more than 3 consecutive terms. Runs of 3 terms start at 1, 5, 13, 21, 29, 33, ... (A007675). - Ivan Neretin, Nov 07 2015
a(n) = product of row n in A265668. - Reinhard Zumkeller, Dec 13 2015
Numbers without excess, i.e., numbers k such that A001221(k) = A001222(k). - Juri-Stepan Gerasimov, Sep 05 2016
Numbers k such that b^(phi(k)+1) == b (mod k) for every integer b. - Thomas Ordowski, Oct 09 2016
Boreico shows that the set of square roots of the terms of this sequence is linearly independent over the rationals. - Jason Kimberley, Nov 25 2016 (reference found by Michael Coons).
Numbers k such that A008836(k) = A008683(k). - Enrique Pérez Herrero, Apr 04 2018
The prime zeta function P(s) "has singular points along the real axis for s=1/k where k runs through all positive integers without a square factor". See Wolfram link. - Maleval Francis, Jun 23 2018
Numbers k such that A007947(k) = k. - Kyle Wyonch, Jan 15 2021
The Schnirelmann density of the squarefree numbers is 53/88 (Rogers, 1964). - Amiram Eldar, Mar 12 2021
Comment from Isaac Saffold, Dec 21 2021: (Start)
Numbers k such that all groups of order k have a trivial Frattini subgroup [Dummit and Foote].
Let the group G have order n. If n is squarefree and n > 1, then G is solvable, and thus by Hall's Theorem contains a subgroup H_p of index p for all p | n. Each H_p is maximal in G by order considerations, and the intersection of all the H_p's is trivial. Thus G's Frattini subgroup Phi(G), being the intersection of G's maximal subgroups, must be trivial. If n is not squarefree, the cyclic group of order n has a nontrivial Frattini subgroup. (End)
Numbers for which the squarefree divisors (A206778) and the unitary divisors (A077610) are the same; moreover they are also the set of divisors (A027750). - Bernard Schott, Nov 04 2022
0 = A008683(a(n)) - A008836(a(n)) = A001615(a(n)) - A000203(a(n)). - Torlach Rush, Feb 08 2023
From Robert D. Rosales, May 20 2024: (Start)
Numbers n such that mu(n) != 0, where mu(n) is the Möbius function (A008683).
Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = mu(n)*n, where sigma(n) is the sum of divisors function (A000203). (End)
a(n) is the smallest root of x = 1 + Sum_{k=1..n-1} floor(sqrt(x/a(k))) greater than a(n-1). - Yifan Xie, Jul 10 2024
Number k such that A001414(k) = A008472(k). - Torlach Rush, Apr 14 2025
To elaborate on the formula from Greathouse (2018), the maximum of a(n) - floor(n*Pi^2/6 + sqrt(n)/17) is 10 at indices n = 48715, 48716, 48721, and 48760. The maximum is 11, at the same indices, if floor is taken individually for the two addends and the square root. If the value is rounded instead, the maximum is 9 at 10 indices between 48714 and 48765. - M. F. Hasler, Aug 08 2025

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris, 2008.
David S. Dummit and Richard M. Foote, Abstract algebra. Vol. 1999. Englewood Cliffs, NJ: David S.Prentice Hall, 1991.
Ivan M. Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.
Michael Pohst and Hans J. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..60794 (first 10000 terms from T. D. Noe)
Zenon B. Batang, Squarefree integers and the abc conjecture, arXiv:2109.10226 [math.GM], 2021.
Andrew R. Booker, Ghaith A. Hiary and Jon P. Keating, Detecting squarefree numbers, Duke Mathematical Journal, Vol. 164, No. 2 (2015), pp. 235-275; arXiv preprint, arXiv:1304.6937 [math.NT], 2013-2015.
Iurie Boreico, Linear independence of radicals, The Harvard College Mathematics Review 2(1), 87-92, Spring 2008.
Ernesto Cesàro, La serie di Lambert in aritmetica assintotica, Rendiconto della Reale Accademia delle Scienze di Napoli, Serie 2, Vol. 7 (1893), pp. 197-204.
Henri Cohen, Francois Dress, and Mohamed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Functiones et Approximatio Commentarii Mathematici 37 (2007), part 1, pp. 51-63.
H. Gent, Letter to N. J. A. Sloane, Nov 27 1975.
Andrew Granville, ABC means we can count squarefrees, International Mathematical Research Notices 19 (1998), 991-1009.
Pentti Haukkanen, Mika Mattila, Jorma K. Merikoski and Timo Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.2.
Aaron Krowne, squarefree number, PlanetMath.org.
Louis Marmet, First occurrences of squarefree gaps and an algorithm for their computation.
Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012.
Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.
J. A. Scott, Square-freedom revisited, The Mathematical Gazette, Vol. 90, No. 517 (2006), pp. 112-113.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
O. Trifonov, On the Squarefree Problem II, Math. Balkanica, Vol. 3 (1989), Fasc. 3-4.
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Squarefree integer.
Index entries for "core" sequences.

Complement of A013929. Subsequence of A072774 and A209061.
Characteristic function: A008966 (mu(n)^2, where mu = A008683).
Subsequences: A000040, A002110, A235488.
Cf. A001414, A008472, A076259 (first differences), A173143 (partial sums), A000688, A003277, A013928, A020753, A020754, A020755, A030059, A030229, A033197, A034444, A039956, A048672, A053797, A057918, A059956, A071403, A072284, A120992, A133466, A136742, A136743, A160764, A243289, A243347, A243348, A243351, A215366, A046660, A265668, A265675.
Cf. A027750, A077610, A206778.
Subsequences: numbers j such that j*a(k) is squarefree where k > 1: A056911 (k = 2), A261034 (k = 3), A274546 (k = 5), A276378 (k = 6).

a005117 n = a005117_list !! (n-1)
a005117_list = filter ((== 1) . a008966) [1..]
-- Reinhard Zumkeller, Aug 15 2011, May 10 2011

[ n : n in [1..1000] | IsSquarefree(n) ];

with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
t:= n-> product(ithprime(k),k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 07 2011
A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc:  # R. J. Mathar, Jan 09 2013

Select[ Range[ 113], SquareFreeQ] (* Robert G. Wilson v, Jan 31 2005 *)
Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
NextSquareFree[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sf = n + sgn; While[c < Abs[k], While[ ! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[ sgn < 0, sf--, sf++]; c++]; sf + If[ sgn < 0, 1, -1]]; NestList[ NextSquareFree, 1, 70] (* Robert G. Wilson v, Apr 18 2014 *)
Select[Range[250], MoebiusMu[#] != 0 &] (* Robert D. Rosales, May 20 2024 *)

bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),L[j]=i; j=j+1)); L

{a(n)= local(m,c); if(n<=1,n==1, c=1; m=1; while( cMichael Somos, Apr 29 2005 */

list(n)=my(v=vectorsmall(n,i,1),u,j); forprime(p=2,sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1,n,v[i])); for(i=1,n,if(v[i],u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012

for(n=1, 113, if(core(n)==n, print1(n, ", "))); \\ Arkadiusz Wesolowski, Aug 02 2016

S(n) = my(s); forsquarefree(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s;
a(n) = my(min=1, max=231, k=0, sc=0); if(n >= 144, min=floor(zeta(2)*n - 5*sqrt(n)); max=ceil(zeta(2)*n + 5*sqrt(n))); while(min <= max, k=(min+max)\2; sc=S(k); if(abs(sc-n) <= sqrtint(n), break); if(sc > n, max=k-1, if(sc < n, min=k+1, break))); while(!issquarefree(k), k-=1); while(sc != n, my(j=1); if(sc > n, j = -1); k += j; sc += j; while(!issquarefree(k), k += j)); k; \\ Daniel Suteu, Jul 07 2022

first(n)=my(v=vector(n),i); forsquarefree(k=1,if(n<268293,(33*n+30)\20,(n*Pi^2/6+0.058377*sqrt(n))\1), if(i++>n, return(v)); v[i]=k[1]); v \\ Charles R Greathouse IV, Jan 10 2023

A5117=[1..3]; A005117(n)={if(n>#A5117, my(N=#A5117); A5117=Vec(A5117, max(n+999, N*5\4)); iferr(forsquarefree(k=A5117[N]+1, #A5117*Pi^2\6+sqrtint(#A5117)\17+11, A5117[N++]=k[1]),E,)); A5117[n]} \\ M. F. Hasler, Aug 08 2025

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) == n
print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, Jul 31 2021

from itertools import count, islice
from sympy import factorint
def A005117_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(x == 1 for x in factorint(n).values()),count(max(startvalue,1)))
A005117_list = list(islice(A005117_gen(),20)) # Chai Wah Wu, May 09 2022

from math import isqrt
from sympy import mobius
def A005117(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 22 2024

Limit_{n->oo} a(n)/n = Pi^2/6 (see A013661). - Benoit Cloitre, May 23 2002
Equals A039956 UNION A056911. - R. J. Mathar, May 16 2008
A122840(a(n)) <= 1; A010888(a(n)) < 9. - Reinhard Zumkeller, Mar 30 2010
a(n) = A055229(A062838(n)) and a(n) > A055229(m) for m < A062838(n). - Reinhard Zumkeller, Apr 09 2010
A008477(a(n)) = 1. - Reinhard Zumkeller, Feb 17 2012
A055653(a(n)) = a(n); A055654(a(n)) = 0. - Reinhard Zumkeller, Mar 11 2012
A008966(a(n)) = 1. - Reinhard Zumkeller, May 26 2012
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012
A056170(a(n)) = 0. - Reinhard Zumkeller, Dec 29 2012
A013928(a(n)+1) = n. - Antti Karttunen, Jun 03 2014
A046660(a(n)) = 0. - Reinhard Zumkeller, Nov 29 2015
Equals {1} UNION A000040 UNION A006881 UNION A007304 UNION A046386 UNION A046387 UNION A067885 UNION A123321 UNION A123322 UNION A115343 ... - R. J. Mathar, Nov 05 2016
|a(n) - n*Pi^2/6| < 0.058377*sqrt(n) for n >= 268293; this result can be derived from Cohen, Dress, & El Marraki, see links. - Charles R Greathouse IV, Jan 18 2018
From Amiram Eldar, Jul 07 2021: (Start)
Sum_{n>=1} (-1)^(a(n)+1)/a(n)^2 = 9/Pi^2.
Sum_{k=1..n} 1/a(k) ~ (6/Pi^2) * log(n).
Sum_{k=1..n} (-1)^(a(k)+1)/a(k) ~ (2/Pi^2) * log(n).
(all from Scott, 2006) (End)

A182318 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 2.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Patrick Devlin, Apr 24 2012

nonn

The prime tower factorization of a number can be recursively defined as follows: the prime tower factorization of 1 is itself; to find the prime tower factorization of an integer n > 1, let n = p_1^e_1 * p_2^e_2 * ... * p_k^e_k be the canonical prime factorization of n, then the prime tower factorization is given by p_1^f_1 * p_2^f_2 * ... * p_k^f_k, where f_i is the prime tower factorization of e_i.

An alternative definition: let I(n) be the indicator function for the set of positive integers whose prime tower factorization does not contain a 2. Then I(n) is the multiplicative function satisfying I(p^k) = I(k) for p prime not equal to 2, and I(2^k) = 0.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251 [math.NT], 2012-2014.

A276378 is a subsequence.

# The integer n is in this sequence if and only if
# containsPrimeInTower(2, n) returns false
containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
  if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
  for i to nops(L) do currentExponent := L[i][2];
    if containsPrimeInTower(q, currentExponent) then return true: end if
  end do;
  return false:
end proc:

Select[Range[150], ! MemberQ[Flatten@ FixedPoint[Map[If[PrimeQ@ Last@ # || Last@ # == 1, #, {First@ #, FactorInteger@ Last@ #}] &, #, {Depth@ # - 2}] &, FactorInteger@ #], 2] &] (* Michael De Vlieger, Feb 17 2017 *)
containsPrimeInTower[q_, n_] := containsPrimeInTower[q, n] = Module[{i, L, currentExponent}, If[n <= 1, Return[False]]; If[IntegerQ[n/q], Return[True] ]; L = FactorInteger[n]; For[i = 1, i <= Length[L], i++, currentExponent = L[[i, 2]]; If[containsPrimeInTower[q, currentExponent], Return[True]]]; Return[False]];
Select[Range[150], !containsPrimeInTower[2, #]&] (* Jean-François Alcover, Jan 22 2019, translated from Maple *)

is(n)=if(n<4, return(n!=2)); if(n%2==0, return(0)); my(f=factor(n)[,2]); for(i=1,#f, if(!is(f[i]), return(0))); 1 \\ Charles R Greathouse IV, May 16 2024

Typo in Maple program corrected by Rémy Sigrist, Dec 13 2016

A104141 Decimal expansion of 3/Pi^2.

Original entry on oeis.org

3, 0, 3, 9, 6, 3, 5, 5, 0, 9, 2, 7, 0, 1, 3, 3, 1, 4, 3, 3, 1, 6, 3, 8, 3, 8, 9, 6, 2, 9, 1, 8, 2, 9, 1, 6, 7, 1, 3, 0, 7, 6, 3, 2, 4, 0, 1, 6, 7, 3, 9, 6, 4, 6, 5, 3, 6, 8, 2, 7, 0, 9, 5, 6, 8, 2, 5, 1, 9, 3, 6, 2, 8, 8, 6, 7, 0, 6, 3, 2, 3, 5, 7, 3, 6, 2, 7, 8, 2, 1, 7, 7, 6, 8, 6, 5, 5, 1, 2, 8

Offset: 0

Lekraj Beedassy, Mar 07 2005

3/Pi^2 is the limit of (Sum_{k=1..n} phi(k))/n^2, where phi(k) is Euler's totient A000010(k), i.e., of A002088(n)/A000290(n) as n tends to infinity.
The previous comment in the context of Farey series means that the length of the n-th Farey series can be approximated by multiplying this constant by n^2, "and that the approximation gets proportionally better as n gets larger", according to Conway and Guy. - Alonso del Arte, May 28 2011
The asymptotic density of the sequences of squarefree numbers with even number of prime factors (A030229), odd number of prime factors (A030059), and coprime to 6 (A276378). - Amiram Eldar, May 22 2020

			3/Pi^2 = 0.303963550927013314331638389629...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 156.
L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 184.

Eric Weisstein's World of Mathematics, Farey Sequence
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
Index entries for transcendental numbers.

Cf. A000010, A002088, A000290.
Cf. A046642, A030229, A030059, A039956, A276378.
Cf. A013661, A195055, A306633, A082020, A088246.

l = RealDigits[N[3/Pi^2, 100]]; Prepend[First[l], Last[l]] (* Ryan Propper, Aug 04 2005 *)

3/Pi^2 \\ Charles R Greathouse IV, Mar 08 2013

Equals Sum_{n>=1} 1/A039956(n)^2. - Amiram Eldar, May 22 2020
From Terry D. Grant, Oct 31 2020: (Start)
Equals (-1)*zeta(0)/zeta(2).
Equals 1/(zeta(2)/2).
Equals 1/A195055.
Equals (1/2)*Sum_{k>=1} mu(k)/k^2. (End)
From Hugo Pfoertner, Apr 23 2024: (Start)
Equals A059956/2.
Equals A082020/5. (End)

More terms from Ryan Propper, Aug 04 2005

A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 6, 6, 4, 5, 8, 5, 8, 5, 6, 10, 12, 12, 10, 6, 7, 5, 15, 9, 15, 5, 7, 8, 14, 10, 20, 20, 10, 14, 8, 9, 12, 21, 24, 7, 24, 21, 12, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 15, 27, 18, 35, 15, 35, 18, 27, 15, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24, 33, 40, 45, 20, 11, 20, 45, 40, 33, 24, 13

Offset: 1

Peter Munn, Jan 21 2020

As a binary operation, this sequence defines a commutative monoid over the positive integers that is isomorphic to multiplication. The self-inverse permutation A225546(.) provides an isomorphism. This monoid therefore has unique factorization. Its primes are the even terms of A050376: 2, 4, 16, 256, ..., which in standard integer multiplication are the powers of 2 with powers of 2 as exponents.
In this monoid, in contrast, the powers of 2 run through the squarefree numbers, the k-th power of 2 being A019565(k). 4 is irreducible and its powers are the squares of the squarefree numbers, the k-th power of 4 being A019565(k)^2 (where "^2" denotes standard integer squaring); and so on with powers of 16, 256, ...
In many cases the product of two numbers is the same here as in standard integer multiplication. See the formula section for details.

			From _Antti Karttunen_, Feb 02 2020: (Start)
The top left 16 X 16 corner of the array:
   1,  2,  3,  4,  5,  6,   7,   8,   9,  10,  11,  12,  13,  14,  15,  16, ...
   2,  3,  6,  8, 10,  5,  14,  12,  18,  15,  22,  24,  26,  21,  30,  32, ...
   3,  6,  5, 12, 15, 10,  21,  24,  27,  30,  33,  20,  39,  42,   7,  48, ...
   4,  8, 12,  9, 20, 24,  28,  18,  36,  40,  44,  27,  52,  56,  60,  64, ...
   5, 10, 15, 20,  7, 30,  35,  40,  45,  14,  55,  60,  65,  70,  21,  80, ...
   6,  5, 10, 24, 30, 15,  42,  20,  54,   7,  66,  40,  78,  35,  14,  96, ...
   7, 14, 21, 28, 35, 42,  11,  56,  63,  70,  77,  84,  91,  22, 105, 112, ...
   8, 12, 24, 18, 40, 20,  56,  27,  72,  60,  88,  54, 104,  84, 120, 128, ...
   9, 18, 27, 36, 45, 54,  63,  72,  25,  90,  99, 108, 117, 126, 135, 144, ...
  10, 15, 30, 40, 14,  7,  70,  60,  90,  21, 110, 120, 130, 105,  42, 160, ...
  11, 22, 33, 44, 55, 66,  77,  88,  99, 110,  13, 132, 143, 154, 165, 176, ...
  12, 24, 20, 27, 60, 40,  84,  54, 108, 120, 132,  45, 156, 168,  28, 192, ...
  13, 26, 39, 52, 65, 78,  91, 104, 117, 130, 143, 156,  17, 182, 195, 208, ...
  14, 21, 42, 56, 70, 35,  22,  84, 126, 105, 154, 168, 182,  33, 210, 224, ...
  15, 30,  7, 60, 21, 14, 105, 120, 135,  42, 165,  28, 195, 210,  35, 240, ...
  16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240,  81, ...
(End)

Antti Karttunen, Antidiagonals n = 1..144, flattened
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..80200; (antidiagonals n = 1..400)
Eric Weisstein's World of Mathematics, Monoid

Isomorphic to A003991 with A225546 as isomorphism.
Cf. A003961(main diagonal), A048675, A059895, A059896, A059897.
Rows/columns, sorted in ascending order: 2: A000037, 3: A028983, 4: A252849.
A019565 lists powers of 2 in order of increasing exponent.
Powers of k, sorted in ascending order: k=2: A005117, k=3: A056911, k=4: A062503, k=5: A276378, k=6: intersection of A325698 and A005117, k=7: intersection of A007775 and A005117, k=8: A062838.
Irreducibles are A001146 (even terms of A050376).

up_to = 1275;
A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331590sq(x,y) = if(1==x,y,if(1==y,x, my(fx=factor(x),fy=factor(y),u=max(#binary(vecmax(fx[, 2])),#binary(vecmax(fy[, 2]))),prodsx=vector(u,x,1),m=1); for(i=1,u,for(k=1,#fx~, if(bitand(fx[k,2],m),prodsx[i] *= fx[k,1])); for(k=1,#fy~, if(bitand(fy[k,2],m),prodsx[i] *= fy[k,1])); m<<=1); prod(i=1,u,A019565(A048675(prodsx[i]))^(1<<(i-1)))));
A331590list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A331590sq(col,(a-(col-1))))); (v); };
v331590 = A331590list(up_to);
A331590(n) = v331590[n]; \\ Antti Karttunen, Feb 02 2020

Alternative definition: A(n,1) = n; A(n,k) = A(A059897(n,k), A003961(A059895(n,k))).
Main derived identities: (Start)
A(n,k) = A(k,n).
A(1,n) = n.
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = A003961(m).
A(n^2, k^2) = A(n,k)^2.
A(A003961(n), A003961(k)) = A003961(A(n,k)).
A(A019565(n), A019565(k)) = A019565(n+k).
(End)
Characterization of conditions for A(n,k) = n * k: (Start)
The following 4 conditions are equivalent:
(1) A(n,k) = n * k;
(2) A(n,k) = A059897(n,k);
(3) A(n,k) = A059896(n,k);
(4) A059895(n,k) = 1.
If gcd(n,k) = 1, A(n,k) = n * k.
If gcd(n,k) = 1, A(A225546(n), A225546(k)) = A225546(n) * A225546(k).
The previous formula implies A(n,k) = n * k in the following cases:
(1) for n = A005117(m), k = j^2;
(2) more generally for n = A005117(m_1)^(2^i_1), k = A005117(m_2)^(2^i_2), with A004198(i_1, i_2) = 0.
(End)

A261034 Numbers m such that 3*m is squarefree.

Original entry on oeis.org

1, 2, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 53, 55, 58, 59, 61, 62, 65, 67, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 91, 94, 95, 97, 101, 103, 106, 107, 109, 110, 113, 115, 118, 119, 122, 127, 130, 131

Offset: 1

Juri-Stepan Gerasimov, Aug 07 2015

These are the numbers from A005117 that are not divisible by 3. See the Maple program by Robert Israel. - Wolfdieter Lang, Aug 21 2015

Squarefree numbers divisible by 3: 3, 6, 15, 21, 30, 33, 39, 42, 51, 57, 66, 69, 78, 87, 93, 102, ...

			10 is in this sequence because 3*10 = 30 is squarefree.

Cf. A045344, A088245.

Numbers m such that k*m is squarefree: A005117 (k = 1), A056911 (k = 2), this sequence (k = 3), A274546 (k = 5), A276378 (k = 6).

[n: n in [1..200] | IsSquarefree(3*n)];

select(numtheory:-issqrfree, [seq(seq(3*i+j,j=1..2),i=0..1000)]); # Robert Israel, Aug 07 2015

Select[Range[0, 200], SquareFreeQ[3 #] &] (* Vincenzo Librandi, Aug 08 2015 *)

is(n)=n%3 && issquarefree(n) \\ Charles R Greathouse IV, Aug 07 2015

a(n) ~ 2*Pi^2*n/9. - Charles R Greathouse IV, Aug 07 2015

Sum_{n>=1} 1/a(n)^s = (3^s)*zeta(s)/((1+3^s)*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023

Corrected and extended by Vincenzo Librandi, Aug 08 2015

A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

Original entry on oeis.org

1, 3, 2, 12, 96, 1152, 2304, 41472, 165888, 3981312, 119439360, 3822059520, 7644119040, 321052999680, 1284211998720, 61642175938560, 3328677500682240, 199720650040934400, 399441300081868800, 1597765200327475200, 115039094423578214400, 230078188847156428800, 18406255107772514304000

Offset: 1

Jonathan Sondow, Feb 01 2014

A236436(n)/(a(n)*zeta(2)) is the asymptotic density of the prime(n-1)-rough squarefree numbers (squarefree numbers whose prime factors are all >= prime(n-1)) for n >= 2. E.g., A236436(2)/(a(2)*zeta(2)) = 2/(3*zeta(2)) = 4/Pi^2 (A185199) is the asymptotic density of the odd squarefree numbers (A056911), and A236436(3)/(a(3)*zeta(2)) = 1/(2*zeta(2)) = 3/Pi^2 (A104141) is the asymptotic density of the 5-rough squarefree numbers (A276378). - Amiram Eldar, Aug 26 2025

			(1 + 1/2)*(1 + 1/3)*(1 + 1/5)*(1 + 1/7) = 96/35 has numerator a(5) = 96.
Fractions begin with 1, 3/2, 2, 12/5, 96/35, 1152/385, 2304/715, 41472/12155, 165888/46189, 3981312/1062347, 119439360/30808063, 3822059520/955049953, ...

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979; Theorem 429.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Jonathan Sondow and Eric W. Weisstein, MathWorld: Mertens Theorem.

Cf. A038110, A060753, A072044, A072045, A073004, A236436 (denominators), A335004.

Cf. A013661, A056911, A104141, A185199, A276378.

Numerator@Table[ Product[ 1 + 1/Prime[ k], {k, 1, n-1}], {n, 1, 23}]

a(n+1) / A236436(n+1) = (A072045(n)/A072044(n)) / (A038110(n+1)/A060753(n+1)) because 1+x = (1-x^2) / (1-x).

a(n) / A236436(n) = Product_{k=1..n-1} (1 + 1/prime(k)) ~ (6/Pi^2)*exp(gamma)*log(n) as n -> infinity, by Mertens's theorem.

A063880 Numbers k such that sigma(k) = 2*usigma(k).

Original entry on oeis.org

108, 540, 756, 1188, 1404, 1836, 2052, 2484, 3132, 3348, 3780, 3996, 4428, 4644, 5076, 5724, 5940, 6372, 6588, 7020, 7236, 7668, 7884, 8316, 8532, 8964, 9180, 9612, 9828, 10260, 10476, 10908, 11124, 11556, 11772, 12204, 12420, 12852, 13716, 14148

Offset: 1

Jason Earls, Aug 27 2001

Numbers so far are all == 108 (mod 216). - Ralf Stephan, Jul 07 2003 [Confirmed up to 10^7 by Robert G. Wilson v.]
Also numbers whose unitary and nonunitary divisors have equal sum. - Amiram Eldar, Sep 30 2019
From Amiram Eldar, Aug 31 2024: (Start)
The primitive terms of this sequence (terms whose proper divisors are not in this sequence) are all powerful numbers (A001694).
All the other terms are of the form m*s, where m is primitive (powerful) and s is a squarefree number coprime to m.
The only primitive term below 10^18 is 108.
If there are no other primitive terms, then a(n) = 108 * A276378(n). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A001694, A034448, A048146, A097702, A276378.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Select[ Range[14363], DivisorSigma[1, # ] == 2 usigma[ # ] &] (* Robert G. Wilson v, Aug 28 2004 *)

u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
j=[];  for(n=1,30000, if(sigma(n) == 2*u(n),j=concat(j,n))); j

A274546 Numbers m such that 5*m is squarefree.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 86, 87, 89, 91, 93, 94, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114, 118, 119, 122, 123, 127, 129, 131, 133, 134

Offset: 1

Juri-Stepan Gerasimov, Sep 05 2016

Squarefree numbers coprime to 5.

			6 is in this sequence because 5*6 = 30 = 2*3*5 is squarefree.

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

Numbers m such that m*A005117(k) is squarefree: A005117 (k = 1), A056911 (k = 2), A261034 (k = 3), this sequence (k = 5), A276378 (k = 6).

Cf. A008587.

[n: n in [1..135] | IsSquarefree(5*n)];

Select[Range[135], !Divisible[#, 5] && SquareFreeQ[#] &] (* Amiram Eldar, Sep 26 2023 *)

lista(nn) = for(n=1, nn, if(issquarefree(5*n), print1(n, ", "))); \\ Altug Alkan, Sep 05 2016

is(n)=n%5 && issquarefree(n) \\ Charles R Greathouse IV, Sep 06 2016

a(n) ~ Pi^2*n/5. - Charles R Greathouse IV, Sep 06 2016 [corrected by Amiram Eldar, Sep 26 2023]

Sum_{n>=1} 1/a(n)^s = (5^s)*zeta(s)/((1+5^s)*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A348753 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k))) < A064989(A064989(k)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211, 215, 217, 221, 223, 227

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Sequence A003961(A003961(A348751(n))), n>=1, sorted into ascending order.
a(38) = 125 is the first term not in A276378.
Not a subsequence of A348748. The first terms that occur here but not there are: 529, 605, 2825, 6425, 7025, 8425, 10825, 15425, 16025, 16325, 16925, 17689, ...
The first squares in this sequence are: 361, 529, 961, 1369, 1849, 2209, 2809, 3721, etc., see A348935 for their square roots.
Of the natural numbers < 2^20, 347712 are in this sequence and only 1812 in A348754.

Cf. A003961, A007310, A064989, A276378, A348750, A348751, A348754.

Cf. also A348748, A348931, A348935.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[250], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #]]] < s[s[#]] &] (* Amiram Eldar, Nov 04 2021 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
isA348753(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n))) < A064989(A064989(n))));

A097703 Numbers j such that m = 216j + 108 satisfies sigma(m) != 2usigma(m).

Original entry on oeis.org

1, 4, 7, 10, 12, 13, 16, 19, 22, 24, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 60, 61, 62, 64, 67, 70, 73, 76, 79, 82, 84, 85, 87, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 122, 124, 127, 130, 133, 136, 137, 139, 142, 144, 145, 148, 151, 154, 157

Offset: 1

Ralf Stephan, Aug 26 2004

nonn

Conjecture: all numbers of form 3k + 1 are here. Other terms are listed in A097704.
From Amiram Eldar, Aug 31 2024: (Start)
The conjecture is true. If j = 3*k+1, then m = 324*(2*k+1). Let e = A007949(2*k+1) >= 0, so 2*k+1 = 3^e * i and i coprime to 6. Then sigma(m)/(2 * usigma(m)) = (7/20) * (3^(e+5)-1)/(3^(e+4)+1) * sigma(i)/usigma(i) >= 847/820 > 1, because sigma(i)/usigma(i) >= 1 for all i.
If m = 216*j + 108 = 108*(2*j+1) then sigma(m) = 2*usigma(m) if and only if 2*j+1 is a squarefree number coprime to 3 (see A097702), i.e., 2*j+1 is a term of A276378. Therefore this sequence consists of numbers j such that 2*j+1 is either a multiple of 3 or nonsquarefree (or both). (End)

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

Complement of A097702.

Cf. A000203, A007949, A013929, A034448, A063880, A097704, A276378.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Complement[ Range[157], (Select[ Range[37000], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216] (* Robert G. Wilson v, Aug 28 2004 *)

is(k) = {my(f = factor(216*k + 108)); sigma(f) != 2 * prod(i = 1, #f~, 1 + f[i,1]^f[i,2]);} \\ Amiram Eldar, Aug 31 2024

cp's OEIS Frontend

A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

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A182318 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 2.

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A104141 Decimal expansion of 3/Pi^2.

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A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

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A261034 Numbers m such that 3*m is squarefree.

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A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

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A097703 Numbers j such that m = 216j + 108 satisfies sigma(m) != 2usigma(m).