A072047 -id:A072047 - 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)

A048672 Binary encoding of squarefree numbers (A005117): A048640(n)/2.

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 5, 16, 32, 9, 6, 64, 128, 10, 17, 256, 33, 512, 7, 1024, 18, 65, 12, 2048, 129, 34, 4096, 11, 8192, 257, 16384, 66, 32768, 20, 130, 513, 65536, 131072, 1025, 36, 19, 262144, 258, 13, 524288, 1048576, 2049, 24, 35, 2097152, 4097, 4194304, 68

Offset: 1

Antti Karttunen, Jul 14 1999

Permutation of nonnegative integers. Note the indexing, the domain starts from 1, although the range includes also 0.
A246353 gives the inverse of this sequence, in a sense that a(A246353(n)) = n for all n >= 0, and A246353(a(n)) = n for all n >= 1. When one is subtracted from the latter, another permutation of nonnegative integers is obtained: A064273. - Antti Karttunen, Aug 23 2014 based on comment from Howard A. Landman, Sep 25 2001
Also index of n-th term of A019565 when its terms are sorted in increasing order. For example: a(6) = 8. The smallest values of A019565 are 1,2,3,5,6,7 . The 6th is 7 which is A019565(8). - Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 28 2008
a(n) is the number whose binary indices are the prime indices of the n-th squarefree number (row n of A329631), where a binary index of n is any position of a 1 in its reversed binary expansion, and a prime index of n is a number m such that prime(m) divides n. The binary indices of n are row n of A048793, while the prime indices of n are row n of A112798. - Gus Wiseman, Nov 30 2019

			From _Gus Wiseman_, Nov 30 2019: (Start)
The sequence of squarefree numbers together with their prime indices (A329631) and the number a(n) with those binary indices begins:
   1 ->  {}      ->   0
   2 ->  {1}     ->   1
   3 ->  {2}     ->   2
   5 ->  {3}     ->   4
   6 ->  {1,2}   ->   3
   7 ->  {4}     ->   8
  10 ->  {1,3}   ->   5
  11 ->  {5}     ->  16
  13 ->  {6}     ->  32
  14 ->  {1,4}   ->   9
  15 ->  {2,3}   ->   6
  17 ->  {7}     ->  64
  19 ->  {8}     -> 128
  21 ->  {2,4}   ->  10
  22 ->  {1,5}   ->  17
  23 ->  {9}     -> 256
  26 ->  {1,6}   ->  33
  29 ->  {10}    -> 512
  30 ->  {1,2,3} ->   7
(End)

Index entries for sequences that are permutations of the natural numbers

Inverse: A246353 (see also A064273).
Cf. A005117, A048639, A048640, A048623.
Cf. A019565.
A similar encoding of set-systems is A329661.
Cf. A000120, A048793, A056239, A070939, A072047, A112798, A329631.
Cf. A087207.

encode_sqrfrees := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(0 <> mobius(i)) then b := [ op(b), bef(i) ]; fi; od: RETURN(b); end; # see A048623 for bef

Join[{0}, Total[2^(PrimePi[FactorInteger[#][[All, 1]]] - 1)]& /@ Select[ Range[2, 100], SquareFreeQ]] (* Jean-François Alcover, Mar 15 2016 *)

lista(nn) = {for (n=1, nn, if (issquarefree(n), if (n==1, x = 0, f = factor(n); x = sum(k=1, #f~, 2^(primepi(f[k, 1])-1))); print1(x, ", "); ); ); } \\ Michel Marcus, Oct 02 2015

from math import isqrt
from sympy import mobius, primepi, primefactors
def A048672(n):
    if n == 1: return 0
    def f(x): return int(n-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return sum(1<Chai Wah Wu, Feb 22 2025

a(n) = 2^(i1-1)+2^(i2-1)+...+2^(iz-1), where A005117(n) = p_i1*p_i2*p_i3*...*p_iz.
A019565(a(n)) = A005117(n). - Peter Munn, Nov 19 2019
A000120(a(n)) = A072047(n). - Gus Wiseman, Nov 30 2019
a(n) = A087207(A005117(n)). - Flávio V. Fernandes, Feb 26 2025

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030

Offset: 0

Gus Wiseman, Dec 02 2020

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021
From David James Sycamore, Jan 09 2025: (Start)
Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.
Permutation of A005117, Squarefree version A379746. (End)

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Alois P. Heinz, Rows n = 0..14, flattened
Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

A011782 gives row lengths.
A339360 gives row sums.
A008578 (shifted) is column k = 1.
A100484 is column k = 2.
A001748 is column k = 3.
A002110 is column k = 2^(n-1).
A070826 is column k = 2^(n-1) - 1.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A261144 divides the n-th row by prime(n), with row sums A054640.
A339116 is the restriction to semiprimes, with row sums A339194.
A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
A006881 lists squarefree semiprimes.
A072047 counts prime factors of squarefree numbers.
A319246 is the sum of prime indices of the n-th squarefree number.
A329631 lists prime indices of squarefree numbers, reversed: A319247.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
Cf. A379746.

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42

Offset: 1

Jean-François Alcover, Nov 26 2015

If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021

			Triangle begins:
1, 2;                        squarefree and 2-smooth
1, 2, 3, 6;                  squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6,  7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...

Jean-François Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
Eric Weisstein's MathWorld, Smooth number.
Wikipedia, Smooth number

Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n-1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.

b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Nov 28 2015

primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021

A265668 Table read by rows: prime factors of squarefree numbers; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 5, 2, 3, 7, 2, 5, 11, 13, 2, 7, 3, 5, 17, 19, 3, 7, 2, 11, 23, 2, 13, 29, 2, 3, 5, 31, 3, 11, 2, 17, 5, 7, 37, 2, 19, 3, 13, 41, 2, 3, 7, 43, 2, 23, 47, 3, 17, 53, 5, 11, 3, 19, 2, 29, 59, 61, 2, 31, 5, 13, 2, 3, 11, 67, 3, 23, 2, 5, 7, 71, 73, 2

Offset: 1

Reinhard Zumkeller, Dec 13 2015

For n > 1: A072047(n) = length of row n;
T(n,1) = A073481(n); T(n,A001221(n)) = A073482(n);
for n > 1: A111060(n) = sum of row n;
A005117(n) = product of row n.

			.   n | T(n,*)  A5117(n)    n | T(n,*)  A5117(n)    n | T(n,*)   A5117(n)
. ----+---------+------   ----+---------+------   ----+----------+------
.   1 | [1]     |  1       21 | [3,11]  | 33       41 | [2,3,11] | 66
.   2 | [2]     |  2       22 | [2,17]  | 34       42 | [67]     | 67
.   3 | [3]     |  3       23 | [5,7]   | 35       43 | [3,23]   | 69
.   4 | [5]     |  5       24 | [37]    | 37       44 | [2,5,7]  | 70
.   5 | [2,3]   |  6       25 | [2,19]  | 38       45 | [71]     | 71
.   6 | [7]     |  7       26 | [3,13]  | 39       46 | [73]     | 73
.   7 | [2,5]   | 10       27 | [41]    | 41       47 | [2,37]   | 74
.   8 | [11]    | 11       28 | [2,3,7] | 42       48 | [7,11]   | 77
.   9 | [13]    | 13       29 | [43]    | 43       49 | [2,3,13] | 78
.  10 | [2,7]   | 14       30 | [2,23]  | 46       50 | [79]     | 79
.  11 | [3,5]   | 15       31 | [47]    | 47       51 | [2,41]   | 82
.  12 | [17]    | 17       32 | [3,17]  | 51       52 | [83]     | 83
.  13 | [19]    | 19       33 | [53]    | 53       53 | [5,17]   | 85
.  14 | [3,7]   | 21       34 | [5,11]  | 55       54 | [2,43]   | 86
.  15 | [2,11]  | 22       35 | [3,19]  | 57       55 | [3,29]   | 87
.  16 | [23]    | 23       36 | [2,29]  | 58       56 | [89]     | 89
.  17 | [2,13]  | 26       37 | [59]    | 59       57 | [7,13]   | 91
.  18 | [29]    | 29       38 | [61]    | 61       58 | [3,31]   | 93
.  19 | [2,3,5] | 30       39 | [2,31]  | 62       59 | [2,47]   | 94
.  20 | [31]    | 31       40 | [5,13]  | 65       60 | [5,19]   | 95  .

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Cf. A005117, A027748, A124010, A072047 (row lengths), A073481, A073482, A001221, A111060, A049200, A062822.

import Math.NumberTheory.Primes.Factorisation (factorise)
import Data.Maybe (mapMaybe)
a265668 n k = a265668_tabf !! (n-1) !! (k-1)
a265668_row n = a265668_tabf !! (n-1)
a265668_tabf = [1] : mapMaybe f [2..] where
   f x = if all (== 1) es then Just ps else Nothing
         where (ps, es) = unzip $ factorise x

FactorInteger[#][[All,1]]&/@Select[Range[100],SquareFreeQ]//Flatten (* Harvey P. Dale, Apr 27 2018 *)

A319247 Irregular triangle whose n-th row lists the strict integer partition whose Heinz number is the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 6, 4, 1, 3, 2, 7, 8, 4, 2, 5, 1, 9, 6, 1, 10, 3, 2, 1, 11, 5, 2, 7, 1, 4, 3, 12, 8, 1, 6, 2, 13, 4, 2, 1, 14, 9, 1, 15, 7, 2, 16, 5, 3, 8, 2, 10, 1, 17, 18, 11, 1, 6, 3, 5, 2, 1, 19, 9, 2, 4, 3, 1, 20, 21, 12, 1, 5, 4, 6, 2, 1, 22

Offset: 1

Gus Wiseman, Sep 15 2018

			The sequence of strict partitions begins: (), (1), (2), (3), (2,1), (4), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (6,1), (10), (3,2,1), (11), (5,2), (7,1), (4,3), (12), (8,1), (6,2), (13), (4,2,1).

Row lengths are A072047. Rows sums are A319246.

Cf. A000009, A000720, A001222, A005117, A056239, A265668, A296150, A299755.

Table[If[n==1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,Select[Range[100],SquareFreeQ]}]

T(n,k) = A000720(A265668(n,-k)).

A329631 Irregular triangle read by rows where row n lists the prime indices of the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 3, 5, 6, 1, 4, 2, 3, 7, 8, 2, 4, 1, 5, 9, 1, 6, 10, 1, 2, 3, 11, 2, 5, 1, 7, 3, 4, 12, 1, 8, 2, 6, 13, 1, 2, 4, 14, 1, 9, 15, 2, 7, 16, 3, 5, 2, 8, 1, 10, 17, 18, 1, 11, 3, 6, 1, 2, 5, 19, 2, 9, 1, 3, 4, 20, 21, 1, 12, 4, 5, 1, 2, 6

Offset: 1

Gus Wiseman, Nov 18 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			Triangle begins:
   1: {}         33: {2,5}      66: {1,2,5}     97: {25}
   2: {1}        34: {1,7}      67: {19}       101: {26}
   3: {2}        35: {3,4}      69: {2,9}      102: {1,2,7}
   5: {3}        37: {12}       70: {1,3,4}    103: {27}
   6: {1,2}      38: {1,8}      71: {20}       105: {2,3,4}
   7: {4}        39: {2,6}      73: {21}       106: {1,16}
  10: {1,3}      41: {13}       74: {1,12}     107: {28}
  11: {5}        42: {1,2,4}    77: {4,5}      109: {29}
  13: {6}        43: {14}       78: {1,2,6}    110: {1,3,5}
  14: {1,4}      46: {1,9}      79: {22}       111: {2,12}
  15: {2,3}      47: {15}       82: {1,13}     113: {30}
  17: {7}        51: {2,7}      83: {23}       114: {1,2,8}
  19: {8}        53: {16}       85: {3,7}      115: {3,9}
  21: {2,4}      55: {3,5}      86: {1,14}     118: {1,17}
  22: {1,5}      57: {2,8}      87: {2,10}     119: {4,7}
  23: {9}        58: {1,10}     89: {24}       122: {1,18}
  26: {1,6}      59: {17}       91: {4,6}      123: {2,13}
  29: {10}       61: {18}       93: {2,11}     127: {31}
  30: {1,2,3}    62: {1,11}     94: {1,15}     129: {2,14}
  31: {11}       65: {3,6}      95: {3,8}      130: {1,3,6}

Row sums are A319246.
Row lengths are A072047.
Same as A319247 with rows reversed.
Composition of A000720 and A265668.
Looking at all numbers instead of just squarefree numbers gives A112798.
Cf. A000009, A005117, A048672, A056239, A299755, A302590.

Table[PrimePi/@First/@If[k==1,{},FactorInteger[k]],{k,Select[Range[30],SquareFreeQ]}]

A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4, 8, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 2, 8, 4, 2

Offset: 1

Reinhard Zumkeller, Jun 09 2002

nonn

Also the number of cubefree numbers with the same squarefree kernel as the n-th squarefree number, see A073245.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
B. Gordon and K. Rogers, Sums of the divisor function, Canadian Journal of Mathematics, Vol. 16 (1964), pp. 151-158.

Cf. A000005, A001620, A062822, A065473.

Cf. A000079, A001221, A005117, A072047.

a072048 = (2 ^) . a072047  -- Reinhard Zumkeller, Dec 13 2015

A072048:=n->`if`(numtheory[issqrfree](n) = true, numtheory[tau](n), NULL); seq(A072048(k), k=1..100); # Wesley Ivan Hurt, Oct 13 2013

DivisorSigma[0, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 29 2022 *)

from math import isqrt
from sympy import mobius, divisor_count
def A072048(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 divisor_count(m) # Chai Wah Wu, Aug 12 2024

a(n) = A000005(A005117(n)).
a(n) = 2^A072047(n) = 2^A001221(A005117(n)).
Sum_{k=1..n} a(k) ~ A * n * log(n) + B * n + O(n^(1/2+eps)), where  A = A065473, B = A * ((2*gamma-1) + 6 * Sum_{p prime} (p-1)*log(p)/(p^2*(p+2)) = 0.236184..., and gamma = A001620 (Gordon and Rogers, 1964). - Amiram Eldar, Oct 29 2022

A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

Original entry on oeis.org

1, 2, 6, 30, 180, 1260, 12600, 138600, 1801800, 25225200, 378378000, 6432426000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 33765373585944000, 979195833992376000, 29375875019771280000

Offset: 1

Leroy Quet, Oct 07 2005

nonn

Do all terms belong to A242031 (weakly decreasing prime signature)? - Gus Wiseman, May 14 2021

			Since the first 6 squarefree positive integers are 1, 2, 3, 5, 6, 7, the 6th term of the sequence is 1*2*3*5*6*7 = 1260.
From _Gus Wiseman_, May 14 2021: (Start)
The sequence of terms together with their prime signatures begins:
             1: ()
             2: (1)
             6: (1,1)
            30: (1,1,1)
           180: (2,2,1)
          1260: (2,2,1,1)
         12600: (3,2,2,1)
        138600: (3,2,2,1,1)
       1801800: (3,2,2,1,1,1)
      25225200: (4,2,2,2,1,1)
     378378000: (4,3,3,2,1,1)
    6432426000: (4,3,3,2,1,1,1)
  122216094000: (4,3,3,2,1,1,1,1)
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..417

A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A072047 applies Omega to each squarefree number.
A246867 groups squarefree numbers by Heinz weight (row sums: A147655).
A261144 groups squarefree numbers by smoothness (row sums: A054640).
A319246 gives the sum of prime indices of each squarefree number.
A329631 lists prime indices of squarefree numbers (reversed: A319247).
Cf. A001221, A002110, A014466, A070826, A338901, A339360.

Rest[FoldList[Times,1,Select[Range[40],SquareFreeQ]]] (* Harvey P. Dale, Jun 14 2011 *)

m=30;k=1;for(n=1,m,if(issquarefree(n),print1(k=k*n,",")))

More terms from Klaus Brockhaus, Oct 08 2005

A339360 Sum of all squarefree numbers with greatest prime factor prime(n).

Original entry on oeis.org

1, 2, 9, 60, 504, 6336, 89856, 1645056, 33094656, 801239040, 24246190080, 777550233600, 29697402470400, 1250501433753600, 55083063155097600, 2649111037319577600, 143390180403000115200, 8619643674791667302400, 534710099148093259776000, 36412881178052121329664000

Offset: 0

Gus Wiseman, Dec 04 2020

nonn

			The initial terms are:
   1 = 1,
   2 = 2,
   9 = 3 + 6,
  60 = 5 + 10 + 15 + 30.

Robert Israel, Table of n, a(n) for n = 0..349

A010036 takes prime indices here to binary indices, row sums of A209862.
A048672 takes prime indices to binary indices in squarefree numbers.
A054640 divides the n-th term by prime(n), row sums of A261144.
A072047 counts prime factors of squarefree numbers.
A339194 is the restriction to semiprimes, row sums of A339116.
A339195 has this as row sums.
A002110 lists primorials.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A056239 is the sum of prime indices of n (Heinz weight).
A246867 groups squarefree numbers by weight, with row sums A147655.
A319246 is the sum of prime indices of the n-th squarefree number.
A319247 lists reversed prime indices of squarefree numbers.
A329631 lists prime indices of squarefree numbers.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014466, A098350, A168472, A282935, A302590, A326878, A338901.

f:= proc(n) local i;
  `if`(n=0, 1, ithprime(n)) *mul(1+ithprime(i),i=1..n-1)
end proc:
map(f, [$0..20]); # Robert Israel, Dec 08 2020

Table[Sum[Times@@Prime/@stn,{stn,Select[Subsets[Range[n]],MemberQ[#,n]&]}],{n,10}]

For n >= 1, a(n) = A054640(n-1) * prime(n).

a(0)=1 prepended by Alois P. Heinz, Jan 08 2025

cp's OEIS Frontend

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

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A048672 Binary encoding of squarefree numbers (A005117): A048640(n)/2.

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A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

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A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

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A265668 Table read by rows: prime factors of squarefree numbers; a(1) = 1 by convention.

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A319247 Irregular triangle whose n-th row lists the strict integer partition whose Heinz number is the n-th squarefree number.

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