A000188 -id:A000188 - cp's OEIS frontend

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller, Mar 15 1996

Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]
a(n) = A128651(A129132(n-1) + 2) for n > 1. - Reinhard Zumkeller, Mar 30 2007
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length.  For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017

			G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandache-type multiplicative sequences
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
Neville Holmes, Integer Sequences [Broken link]
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
Wikipedia, Radical of an integer.

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
Range of values is A005117.
Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
Bisections: A099984, A099985.
Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
A003961, A059896 are used to express relationship between terms of this sequence.
Cf. A001615, A008683, A076479, A078615.

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
seq(A007947(i),i=1..69); # Peter Luschny, Mar 22 2011
A:= n -> convert(numtheory:-factorset(n),`*`):
seq(A(n),n=1..100); # Robert Israel, Aug 10 2014
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
From Antti Karttunen, Jun 18 2017: (Start)
a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
a(n) = A019565(A087207(n)). (End)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
From Peter Munn, Jan 01 2020: (Start)
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A003961(n)) = A003961(a(n)).
a(n^2) = a(n).
a(A225546(n)) = A019565(A267116(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

A003557 n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7

Offset: 1

Marc LeBrun

a(n) is the size of the Frattini subgroup of the cyclic group C_n - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001.
Also of the Frattini subgroup of the dihedral group with 2*n elements. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002
Number of solutions to x^m==0 (mod n) provided that n < 2^(m+1), i.e. the sequence of sequences A000188, A000189, A000190, etc. converges to this sequence. - Henry Bottomley, Sep 18 2001
a(n) is the number of nilpotent elements in the ring Z/nZ. - Laszlo Toth, May 22 2009
The sequence of partial products of a(n) is A085056(n). - Peter Luschny, Jun 29 2009
The first occurrence of n in this sequence is at A064549(n). - Franklin T. Adams-Watters, Jul 25 2014
From Hal M. Switkay, Jul 03 2025: (Start)
For n > 1, a(n) is a proper divisor of n. Thus the sequence n, a(n), a(a(n)), ... eventually becomes 1. This yields a minimal factorization of n as a product of squarefree numbers (A005117), each factor dividing all larger factors, in a factorization that is conjugate to the minimal factorization of n as a product of prime powers (A000961), as follows.
Let f(n,0) = n, and let f(n,k) = a(f(n,k-1)) for k > 0. A051903(n) is the minimal value of k such that f(n,k) = 1. A051903(n) <= log(n)/log(2). Since n/a(n) = A007947(n) is always squarefree by definition, n is a product of squarefree factors in the form Product_{i=1..A051903(n)} [f(n,i-1)/f(n,i)].
The two factorizations correspond to conjugate partitions of bigomega(n) = A001222(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A007947, A062378, A062379, A064549, A300717 (Möbius transform), A326306 (inv. Möbius transf.), A328572.
Sequences that are multiples of this sequence (the other factor of a pointwise product is given in parentheses): A000010 (A173557), A000027 (A007947), A001615 (A048250), A003415 (A342001), A007434 (A345052), A057521 (A071773).
Cf. A082695 (Dgf at s=2), A065487 (Dgf at s=3).
Cf. A000961, A001222, A005117, A051903.

a003557 n = product $ zipWith (^)
                      (a027748_row n) (map (subtract 1) $ a124010_row n)
-- Reinhard Zumkeller, Dec 20 2013

using Nemo
function A003557(n)
    n < 4 && return 1
    q = prod([p for (p, e) ∈ Nemo.factor(fmpz(n))])
    return n == q ? 1 : div(n, q)
end
[A003557(n) for n in 1:90] |> println  # Peter Luschny, Feb 07 2021

[(&+[(Floor(k^n/n)-Floor((k^n-1)/n)): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Nov 02 2018

A003557 := n -> n/ilcm(op(numtheory[factorset](n))):
seq(A003557(n), n=1..98); # Peter Luschny, Mar 23 2011
seq(n / NumberTheory:-Radical(n), n = 1..98); # Peter Luschny, Jul 20 2021

Prepend[ Array[ #/Times@@(First[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 1 ] (* Olivier Gérard, Apr 10 1997 *)

a(n)=n/factorback(factor(n)[,1]) \\ Charles R Greathouse IV, Nov 17 2014

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 20 2020

from sympy.ntheory.factor_ import core
from sympy import divisors
def a(n): return n / max(i for i in divisors(n) if core(i) == i)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 16 2017

from math import prod
from sympy import primefactors
def A003557(n): return n//prod(primefactors(n)) # Chai Wah Wu, Nov 04 2022

def A003557(n) : return n*mul(1/p for p in prime_divisors(n))
[A003557(n) for n in (1..98)] # Peter Luschny, Jun 10 2012

Multiplicative with a(p^e) = p^(e-1). - Vladeta Jovovic, Jul 23 2001
a(n) = n/rad(n) = n/A007947(n) = sqrt(J_2(n)/J_2(rad(n))), where J_2(n) is A007434. - Enrique Pérez Herrero, Aug 31 2010
a(n) = (J_k(n)/J_k(rad(n)))^(1/k), where J_k is the k-th Jordan Totient Function: (J_2 is A007434 and J_3 A059376). - Enrique Pérez Herrero, Sep 03 2010
Dirichlet convolution of A000027 and A097945. - R. J. Mathar, Dec 20 2011
a(n) = A000010(n)/|A023900(n)|. - Eric Desbiaux, Nov 15 2013
a(n) = Product_{k = 1..A001221(n)} (A027748(n,k)^(A124010(n,k)-1)). - Reinhard Zumkeller, Dec 20 2013
a(n) = Sum_{k=1..n}(floor(k^n/n)-floor((k^n-1)/n)). - Anthony Browne, May 11 2016
a(n) = e^[Sum_{k=2..n} (floor(n/k)-floor((n-1)/k))*(1-A010051(k))*Mangoldt(k)] where Mangoldt is the Mangoldt function. - Anthony Browne, Jun 16 2016
a(n) = Sum_{d|n} mu(d) * phi(d) * (n/d), where mu(d) is the Moebius function and phi(d) is the Euler totient function (rephrases formula of Dec 2011). - Daniel Suteu, Jun 19 2018
G.f.: Sum_{k>=1} mu(k)*phi(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Dirichlet g.f.: Product_{primes p} (1 + 1/(p^s - p)). - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k).
a(n) = Sum_{k=1..n} mu(gcd(n,k))*(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = A001615(n)/A048250(n) = A003415/A342001(n) = A057521(n)/A071773(n). - Antti Karttunen, Jun 08 2021

Secondary definition added to the name by Antti Karttunen, Jun 08 2021

A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 2, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77

Offset: 1

R. Muller, Mar 15 1996

Also called core(n). [Not to be confused with the squarefree kernel of n, A007947.]
Sequence read mod 4 gives A065882. - Philippe Deléham, Mar 28 2004
This is an arithmetic function and is undefined if n <= 0.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), lcm(A007947(b),c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n. [Corrected by M. F. Hasler, Mar 01 2018]
If n > 1, the quantity f(n) = log(n/core(n))/log(n) satisfies 0 <= f(n) <= 1; f(n) = 0 when n is squarefree and f(n) = 1 when n is a perfect square. One can define n as being "epsilon-almost squarefree" if f(n) < epsilon. - Kurt Foster (drsardonicus(AT)earthlink.net), Jun 28 2008
a(n) is the smallest natural number m such that product of geometric mean of the divisors of n and geometric mean of the divisors of m are integers. Geometric mean of the divisors of number n is real number b(n) = Sqrt(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A000290: a(A000290(n)) = 1. For n = 8; b(8) = sqrt(8), a(n) = 2 because b(2) = sqrt(2); sqrt(8) * sqrt(2) = 4 (integer). - Jaroslav Krizek, Apr 26 2010
Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011
Booker, Hiary, & Keating outline a method for bounding (on the GRH) a(n) for large n using L-functions. - Charles R Greathouse IV, Feb 01 2013
According to the formula a(n) = n/A000188(n)^2, the scatterplot exhibits the straight lines y=x, y=x/4, y=x/9, ..., i.e., y=x/k^2 for all k=1,2,3,... - M. F. Hasler, May 08 2014
The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - Álvar Ibeas, Mar 19 2015
a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - Zak Seidov, Jan 30 2016
All solutions of the Diophantine equation n*x=y^2 or, equivalently, G(n,x)=y, with G being the geometric mean, are of the form x=k^2*a(n), y=k*sqrt(n*a(n)), where k is a positive integer. - Stanislav Sykora, Feb 03 2016
If f is a multiplicative function then Sum_{d divides n} f(a(d)) is also multiplicative. For example, A010052(n) = Sum_{d divides n} mu(a(d)) and A046951(n) = Sum_{d divides n} mu(a(d)^2). - Peter Bala, Jan 24 2024

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Krassimir T. Atanassov, On the 22nd, 23rd, and the 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.
Andrew Booker, Ghaith Hiary, and Jon Keating, Detecting squarefree numbers, CNTA XII (2012).
Henry Bottomley, Some Smarandache-type multiplicative sequences.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Vlad Copil and Laurenţiu Panaitopol, Properties of a sequence generated by positive integers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 2 (2007), pp. 131-137; alternative link.
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Eric Weisstein's World of Mathematics, Squarefree Part.

See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.
See A027746, A027748, A124010 for factorization data for n.
Analogous sequences: A050985, A053165, A055231.
Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums), A090699, A350389.
Related to A006519 via A225546.

a007913 n = product $
            zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)
-- Reinhard Zumkeller, Jul 06 2012

[ Squarefree(n) : n in [1..256] ]; // N. J. A. Sloane, Dec 23 2006

A007913 := proc(n) local f,a,d; f := ifactors(n)[2] ; a := 1 ; for d in f do if type(op(2,d),'odd') then a := a*op(1,d) ; end if; end do: a; end proc: # R. J. Mathar, Mar 18 2011
# second Maple program:
a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 20 2015
seq(n / expand(numtheory:-nthpow(n, 2)), n=1..77);  # Peter Luschny, Jul 12 2022

data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[] -> 1; sfp = data/sp /. Sqrt[x] -> x (* Artur Jasinski, Nov 03 2008 *)
Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,100}] (* Zak Seidov, Apr 08 2009 *)
Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* T. D. Noe, May 20 2013 *)
Sqrt[#] /. (c_:1)*a_^(b_:0) -> (c*a^b)^2& /@ Range@100 (* Bill Gosper, Jul 18 2015 *)

PARI
```
a(n)=core(n)
```

from sympy import factorint, prod
def A007913(n):
    return prod(p for p, e in factorint(n).items() if e % 2)
# Chai Wah Wu, Feb 03 2015

[squarefree_part(n) for n in (1..77)] # Peter Luschny, Feb 04 2015

Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson, Aug 01 2001
a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - Philippe Deléham, Mar 28 2004
Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - Anthony Browne, Jun 06 2016
a(n) = rad(n)/a(n/rad(n)), where rad = A007947. This recurrence relation together with a(1) = 1 generate the sequence. - Velin Yanev, Sep 19 2017
From Peter Munn, Nov 18 2019: (Start)
a(k*m) = A059897(a(k), a(m)).
a(n) = n / A008833(n).
(End)
a(A225546(n)) = A225546(A006519(n)). - Peter Munn, Jan 04 2020
From Amiram Eldar, Mar 14 2021: (Start)
Theorems proven by Copil and Panaitopol (2007):
Lim sup_{n->oo} a(n+1)-a(n) = oo.
Lim inf_{n->oo} a(n+1)-a(n) = -oo.
Sum_{k=1..n} 1/a(k) ~ c*sqrt(n) + O(log(n)), where c = zeta(3/2)/zeta(3) (A090699). (End)
a(n) = A019554(n)^2/n. - Jianing Song, May 08 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Oct 25 2022
a(n) = A007947(A350389(n)). - Amiram Eldar, Jan 20 2024

More terms from Michael Somos, Nov 24 2001

Definition reformulated by Daniel Forgues, Mar 24 2009

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520

Offset: 1

Paul Boddington, Jul 21 2005

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)
Index to divisibility sequences.
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to primorial numbers.

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390, 19835652870, 2333606220

Offset: 0

Labos Elemer, Jul 25 2000

nonn

a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm(A001405(n-1), A001405(n)) (for n>0).
A055773(n) divides a(n), A001316(floor(n/2)) divides a(n).
Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012
a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]

			a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Didier Guillet, On swinging factorials and the lonely runner conjecture (Text in French).
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Orbitals.
Peter Luschny, Swinging Factorial.

Bisections are A000984 and A002457.

Cf. A000142, A001405, A000188, A055772, A056042, A211226, A162246, A189231, A212303.

[(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011

SeriesCoeff := proc(s,n) series(s(w,n),w,n+2);
convert(%,polynom); coeff(%,w,n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)),k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0,1,n^irem(n,2)*(4/n)^irem(n+1,2)*a2(n-1)) end;
a3 := n -> n!/iquo(n,2)!^2;
g4 := z -> BesselI(0,2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4,n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5,n);
g6 := (z,n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6,n);
a7 := n -> combinat[multinomial](n,floor(n/2),n mod 2,floor(n/2));
h := n -> binomial(n,floor(n/2)); # A001405
a8 := n -> ilcm(h(n-1),h(n));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 02 2010 *)
f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 10 2010 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)

a(n)=n!/(n\2)!^2 \\ Charles R Greathouse IV, May 02 2011

def A056040():
    r, n = 1, 0
    while True:
        yield r
        n += 1
        r *= 4/n if is_even(n) else n
a = A056040(); [next(a) for i in range(36)]  # Peter Luschny, Oct 24 2013

a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
E.g.f.: (1+x)*BesselI(0, 2*x). - Vladeta Jovovic, Jan 19 2004
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
a(2n+1) = A046212(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012
D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012
Sum_{n>=0} 1/a(n) = 4/3 + 8*Pi/(9*sqrt(3)). - Alexander R. Povolotsky, Aug 18 2012
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013
a(n) = Sum_{k=0..n} A189231(n, 2*k). (Cf. A212303 for the odd case.) - Peter Luschny, Oct 30 2013
a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014
a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015
Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022

Extended and edited by Peter Luschny, Jun 28 2009

A046951 a(n) is the number of squares dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Rediscovered by the HR automatic theory formation program.
a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001
We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009
Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]
Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015
The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020
The number of unordered factorizations of n into cubefree powers of primes (1, primes and squares of primes, A166684). - Amiram Eldar, Jun 12 2025

			a(16) = 3 because the squares 1, 4, and 16 divide 16.
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Antonio Amariti, Claudius Klare, Domenico Orlando and Susanne Reffert, The M-theory origin of global properties of gauge theories, Nuclear Physics B, Vol. 901 (2015), pp. 318-337, arXiv preprint, arXiv:1507.04743 [hep-th], 2015 (see (A.13)).
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics
Ian G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).
Andrew V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.
Werner Georg Nowak and László Tóth, On the average number of subgroups of the group Z_m X Z_n, International Journal of Number Theory, Vol. 10, No. 2 (2014), pp. 363-374, arXiv preprint, arXiv:1307.1414 [math.NT], 2013.
N. J. A. Sloane, Transforms.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000188, A004101, A005117 (positions of 1's), A008619, A008833, A013936 (partial sums), A038538, A046952, A052304, A056595, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A130279, A156552, A166684, A278161, A307445 (Dirichlet inverse).
One more than A071325.
Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).
Sum of the k-th powers of the square divisors of n for k=0..10: this sequence (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: this sequence (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
Cf. A082293 (a(n)==2), A082294 (a(n)==3).

a046951 = sum . map a010052 . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[#[d: d in Divisors(n)|IsSquare(d)]:n in [1..120]]; // Marius A. Burtea, Jan 21 2020

A046951 := proc(n)
    local a,s;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*(1+floor(op(2,p)/2)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 17 2012
# Alternatively:
isbidivisible := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d:
a := n -> nops(select(k -> isbidivisible(n, k), [seq(1..n)])): # Peter Luschny, Jun 13 2025

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)
Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)
f[p_, e_] := 1 + Floor[e/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n)=my(f=factor(n));for(i=1,#f[,1],f[i,2]\=2);numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

a(n)=factorback(apply(e->e\2+1, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 17 2015

from math import prod
from sympy import factorint
def A046951(n): return prod((e>>1)+1 for e in factorint(n).values()) # Chai Wah Wu, Aug 04 2024

def is_bidivisible(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == d
def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_bidivisible(n, k)]
print([len(aList(n)) for n in range(1, 126)])  # Peter Luschny, Jun 13 2025

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))
(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))
;; Antti Karttunen, Nov 14 2016

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016
a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.
Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007
a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007
Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).
G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013
a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014
From Antti Karttunen, Nov 14 2016: (Start)
a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).
a(n) = A278161(A156552(n)). (End)
G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016
From  Antti Karttunen, Nov 12 2017: (Start)
a(n) = A000005(n) - A056595(n).
a(n) = 1 + A071325(n).
a(n) = 1 + A001222(A293515(n)). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A000005(d) * A008836(n/d). - Torlach Rush, Jan 21 2020
a(n) = A000005(sqrt(A008833(n))). - Amiram Eldar, Jul 07 2020
a(n) = Sum_{d divides n} mu(core(d)^2), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

A008833 Largest square dividing n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 1

Offset: 1

N. J. A. Sloane

The Dirichlet generating function of the arithmetic function of the largest t-th power dividing n is zeta(s)*zeta(t*s-t)/zeta(s*t), here with t=2 and in A008834 and A008835 with t=3 and t=4, respectively. - R. J. Mathar, Feb 19 2011

Daniel Forgues, Table of n, a(n) for n = 1..100000
Henry Bottomley, Some Smarandache-type multiplicative sequences
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions arXiv:1106.4038 [math.NT], 2011-2012, Remark 16.
Andrew Reiter, On (mod n) spirals, 2014, see also posting to Number Theory Mailing List, Mar 23 2014.
Eric Weisstein's World of Mathematics, Square part

Cf. A000188, A005563, A007913, A019554, A059897, A068310.

Cf. A008836, A358272, A001615.

a008833 n = head $ filter ((== 0) . (mod n)) $
   reverse $ takeWhile (<= n) $ tail a000290_list
-- Reinhard Zumkeller, Nov 13 2011

A008833 := proc(n)
    expand(numtheory:-nthpow(n,2)) ;
end proc:
seq(A008833(n), n=1..100) ;

a[n_] := First[ Select[ Reverse[ Divisors[n]], IntegerQ[Sqrt[#]]&, 1]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 12 2011 *)
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jul 07 2020 *)

A008833(n)=n/core(n) \\ Michael B. Porter, Oct 17 2009

from sympy.ntheory.factor_ import core
def A008833(n): return n//core(n) # Chai Wah Wu, Dec 30 2021

a(n) = A000188(n)^2 = n/A007913(n). Cf. A019554.
Multiplicative with a(p^e) = p^(2[e/2]). - David W. Wilson, Aug 01 2001
Dirichlet g.f.: zeta(s)*zeta(2s-2)/zeta(2s). - R. J. Mathar, Oct 31 2011
a(n) = A005563(n-1) / A068310(n) for n > 1. - Reinhard Zumkeller, Nov 26 2011
Sum_{k=1..n} a(k) ~ Zeta(3/2) * n^(3/2) / (3*Zeta(3)). - Vaclav Kotesovec, Feb 01 2019
a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Nov 30 2019
From Ridouane Oudra, May 11 2025: (Start)
a(n) = Sum_{d|n} lambda(d)*d*psi(n/d), where lambda = A008836 and psi = A001615.
a(n) = lambda(n) * Sum_{d|n} lambda(d)*d*phi(n/d).
a(n) = A008836(n) * A358272(n). (End)

A059897 Symmetric square array read by antidiagonals: A(n,k) is the product of all factors that occur in one, but not both, of the Fermi-Dirac factorizations of n and k.

Original entry on oeis.org

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

Offset: 1

Marc LeBrun, Feb 06 2001

Old name: Square array read by antidiagonals: T(i,j) = product prime(k)^(Ei(k) XOR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; XOR is the bitwise operation on binary representation of the exponents.
Analogous to multiplication, with XOR replacing +.
From Peter Munn, Apr 01 2019: (Start)
(1) Defines an abelian group whose underlying set is the positive integers. (2) Every element is self-inverse. (3) For all n and k, A(n,k) is a divisor of n*k. (4) The terms of A050376, sometimes called Fermi-Dirac primes, form a minimal set of generators. In ordered form, it is the lexicographically earliest such set.
The unique factorization of positive integers into products of distinct terms of the group's lexicographically earliest minimal set of generators seems to follow from (1) (2) and (3).
From (1) and (2), every row and every column of the table is a self-inverse permutation of the positive integers. Rows/columns numbered by nonmembers of A050376 are compositions of earlier rows/columns.
It is a subgroup of the equivalent group over the nonzero integers, which has -1 as an additional generator.
As generated by A050376, the subgroup of even length words is A000379. The complementary set of odd length words is A000028.
The subgroup generated by A000040 (the primes) is A005117 (the squarefree numbers).
(End)
Considered as a binary operation, the result is (the squarefree part of the product of its operands) times the square of (the operation's result when applied to the square roots of the square parts of its operands). - Peter Munn, Mar 21 2022

			A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 XOR 3) * 3^(3 XOR 5) = 2^6 * 3^6 = 46656.
The top left 12 X 12 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12
   2,  1,  6,  8, 10,  3, 14,  4,  18,   5,  22,  24
   3,  6,  1, 12, 15,  2, 21, 24,  27,  30,  33,   4
   4,  8, 12,  1, 20, 24, 28,  2,  36,  40,  44,   3
   5, 10, 15, 20,  1, 30, 35, 40,  45,   2,  55,  60
   6,  3,  2, 24, 30,  1, 42, 12,  54,  15,  66,   8
   7, 14, 21, 28, 35, 42,  1, 56,  63,  70,  77,  84
   8,  4, 24,  2, 40, 12, 56,  1,  72,  20,  88,   6
   9, 18, 27, 36, 45, 54, 63, 72,   1,  90,  99, 108
  10,  5, 30, 40,  2, 15, 70, 20,  90,   1, 110, 120
  11, 22, 33, 44, 55, 66, 77, 88,  99, 110,   1, 132
  12, 24,  4,  3, 60,  8, 84,  6, 108, 120, 132,   1
From _Peter Munn_, Apr 04 2019: (Start)
The subgroup generated by {6,8,10}, the first three integers > 1 not in A050376, has the following table:
    1     6     8    10    12    15    20   120
    6     1    12    15     8    10   120    20
    8    12     1    20     6   120    10    15
   10    15    20     1   120     6     8    12
   12     8     6   120     1    20    15    10
   15    10   120     6    20     1    12     8
   20   120    10     8    15    12     1     6
  120    20    15    12    10     8     6     1
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array
Eric Weisstein's World of Mathematics, Group, Square Part, Squarefree Part.

Cf. A000040, A003987, A003991, A028233, A028234, A050376, A059896, A089913, A207901, A268387, A284577, A302033.
Cf. A284567 (A000142 or A003418-analog for this operation).
Rows/columns: A073675 (2), A120229 (3), A120230 (4), A307151 (5), A307150 (6), A307266 (8), A307267 (24).
Particularly significant subgroups or cosets: A000028, A000379, A003159, A005117, A030229, A252895. See also the lists in A329050, A352273.
Sequences that relate this sequence to multiplication: A000188, A007913, A059895.

a[i_, i_] = 1;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, e1[] = 0; Scan[(e1[#[[1]]] = #[[2]])&, f1]; e2[] = 0; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitXor[e1[#], e2[#]]& /@ Union[f1[[All, 1]], f2[[All, 1]]])];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

T(n,k) = {if (n==1, return (k)); if (k==1, return (n)); my(fn=factor(n), fk=factor(k)); vp = setunion(fn[,1]~, fk[,1]~); prod(i=1, #vp, vp[i]^(bitxor(valuation(n, vp[i]), valuation(k, vp[i]))));} \\ Michel Marcus, Apr 03 2019

T(i, j) = {if(gcd(i, j) == 1, return(i * j)); if(i == j, return(1)); my(f = vecsort(concat(factor(i)~, factor(j)~)), t = 1, res = 1); while(t + 1 <= #f, if(f[1, t] == f[1, t+1], res *= f[1, t] ^ bitxor(f[2, t] , f[2, t+1]); t+=2; , res*= f[1, t]^f[2, t]; t++; ) ); if(t == #f, res *= f[1, #f] ^ f[2, #f]); res } \\ David A. Corneth, Apr 03 2019

A059897(n,k) = if(n==k, 1, core(n*k) * A059897(core(n,1)[2],core(k,1)[2])^2) \\ Peter Munn, Mar 21 2022

(define (A059897 n) (A059897bi (A002260 n) (A004736 n)))
(define (A059897bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) (* m b)) ((= 1 b) (* m a)) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A003987bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (/ a (A028233 a)) b (* m (A028233 a)))) (else (loop a (/ b (A028233 b)) (* m (A028233 b)))))))
;; Antti Karttunen, Apr 11 2017

For all x, y >= 1, A(x,y) * A059895(x,y)^2 = x*y. - Antti Karttunen, Apr 11 2017
From Peter Munn, Apr 01 2019: (Start)
A(n,1) = A(1,n) = n
A(n, A(m,k)) = A(A(n,m), k)
A(n,n) = 1
A(n,k) = A(k,n)
if i_1 <> i_2 then A(A050376(i_1), A050376(i_2)) = A050376(i_1) * A050376(i_2)
if A(n,k_1) = n * k_1 and A(n,k_2) = n * k_2 then A(n, A(k_1,k_2)) = n * A(k_1,k_2)
(End)
T(k, m) = k*m for coprime k and m. - David A. Corneth, Apr 03 2019
if A(n*m,m) = n, A(n*m,k) = A(n,k) * A(m,k) / k. - Peter Munn, Apr 04 2019
A(n,k) = A007913(n*k) * A(A000188(n), A000188(k))^2. - Peter Munn, Mar 21 2022

New name from Peter Munn, Mar 21 2022

A056170 Number of non-unitary prime divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0

Offset: 1

Labos Elemer, Jul 27 2000

A prime factor of n is unitary iff its exponent is 1 in the prime factorization of n. (Of course for any prime p, GCD(p, n/p) is either 1 or p. For a unitary prime factor it must be 1.)
Number of squared primes dividing n. - Reinhard Zumkeller, May 18 2002
a(A005117(n)) = 0; a(A013929(n)) > 0; a(A190641(n)) = 1. - Reinhard Zumkeller, Dec 29 2012
First differences of A013940. - Jason Kimberley, Feb 01 2017
Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A001221, A003557, A013940, A034444, A046660, A048105, A056169, A085548, A124010, A162641, A212177, A275812, A295659, A295666.

Cf. A057427(a(n)) = 1 - A008966(n).

a056170 = length . filter (> 1) . a124010_row
-- Reinhard Zumkeller, Dec 29 2012

A056170:=func;
[A056170(n):n in[1..105]];
// Jason Kimberley, Jan 22 2017

A056170 := n -> nops(select(t -> (t[2]>1), ifactors(n)[2]));
seq(A056170(n),n=1..100); # Robert Israel, Jun 03 2014

a[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Table[a[n], {n, 105}]  (* Jean-François Alcover, Mar 23 2011 *)
Table[Count[FactorInteger[n][[All,2]],?(#>1&)],{n,110}] (* _Harvey P. Dale, Jul 08 2019 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]>1) \\ Charles R Greathouse IV, May 18 2015

from sympy import factorint
def a(n):
    f = factorint(n)
    return sum([1 for i in f if f[i]!=1]) # Indranil Ghosh, Apr 24 2017

Additive with a(p^e) = 0 if e = 1, 1 otherwise.
G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Jan 01 2017
a(n) = log_2(A000005(A071773(n))). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A001221(n) - A056169(n).
a(n) = omega(A000188(n)) = omega(A003557(n)) = omega(A057521(n)) = omega(A295666(n)), where omega = A001221.
For all n >= 1 it holds that:
a(A003557(n)) = A295659(n).
a(n) >= A162641(n).
(End)
Dirichlet g.f.: primezeta(2s)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^2 = 0.452247... (A085548). - Amiram Eldar, Nov 01 2020
a(n) = A275812(n) - A046660(n). - Amiram Eldar, Jan 09 2024

Minor edits by Franklin T. Adams-Watters, Mar 23 2011

cp's OEIS Frontend

A355264 a(n) = n * largest-nth-power(n, 2) = n * A000188(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.

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A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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A003557 n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.

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A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

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A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

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