keyword:mult - cp's OEIS frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

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

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

A010052 Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Also parity of the divisor function A000005 if n >= 1. - Omar E. Pol, Jan 14 2012
This sequence can be considered as k=1 analog of A025426 (k=2), A025427 (k=3), A025428 (k=4); see also A000161. - M. F. Hasler, Jan 25 2013
Also, the decimal expansion of Sum_{n >= 0} 1/(10^n)^n. -  Eric Desbiaux, Mar 15 2009, rephrased and simplified by M. F. Hasler, Jan 26 2013
Run lengths of zeros gives A005843, the nonnegative even numbers. - Jeremy Gardiner, Jan 14 2018
Inverse Möbius transform of Liouville's lambda function (A008836), n >= 1. - Wesley Ivan Hurt, Jun 22 2024

			G.f. = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 + x^81 + ...

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 3-4, also p. 166, Ex. 5.5.1.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, Problem 20.
Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (11.14).
Michael D. Hirschhorn, The Power of q, Springer, 2017. See phi(q) page 8.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 55.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
David Christopher and Tamil Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), Article 15.11.5.
Robert Price, Comments on A010052 concerning Elementary Cellular Automata, Jan 29 2016.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Stephen Wolfram, A New Kind of Science.
Index entries for characteristic functions
Index to Elementary Cellular Automata.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to cellular automata.

Column k=1 of A243148, A337165, A341040 (for n>0).
Cf. A000005, A000122, A005369, A007913, A008836 (Mobius transf.), A037011, A063524, A258998, A271102 (Dirichlet inv), A046951 (inv. Mobius trans.).
First differences of A000196.

a010052 n = fromEnum $ a000196 n ^ 2 == n
-- Reinhard Zumkeller, Jan 26 2012, Feb 20 2011
a010052_list = concat (iterate (\xs -> xs ++ [0,0]) [1])
-- Reinhard Zumkeller, Apr 27 2012

readlib(issqr): f := i->if issqr(i) then 1 else 0; fi; [ seq(f(i),i=0..100) ];

lst = {}; Do[AppendTo[lst, 2*Sum[Floor[n/k] - Floor[(n - 1)/k], {k, Floor[Sqrt[n]]}] - DivisorSigma[0, n]], {n, 93}]; Prepend[lst, 1] (* Eric Desbiaux, Jan 29 2012 *)
Table[If[IntegerQ[Sqrt[n]],1,0],{n,0,100}] (* Harvey P. Dale, Jul 19 2014 *)
a[n_] := SeriesCoefficient[1/(1 - q)* QHypergeometricPFQ[{-q, -q}, {-(q^2)}, -q, -q], {q, 0, Abs@n}] (* Mats Granvik, Jan 01 2016 *)
Range[0, 120] /. {n_ /; IntegerQ@ Sqrt@ n -> 1, n_ /; n != 1 -> 0} (* Michael De Vlieger, Jan 02 2016 *)
a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[LiouvilleLambda[k]]*Sqrt[LiouvilleLambda[n/k]]], 0], {k, 1, n}] (* Mats Granvik, Aug 10 2018 *)

PARI
```
{a(n) = issquare(n)};
```

a(n)=if(n<1,1,sumdiv(n,d,(-1)^bigomega(d))) \\ Benoit Cloitre, Oct 25 2009

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

def A010052(n): return int(math.isqrt(n)**2==n) ##  appears to be faster than sympy.ntheory.primetest.is_square, up to 10^8 at least.
# M. F. Hasler, Mar 21 2022

(define (A010052 n) (if (zero? n) 1 (- (A000196 n) (A000196 (- n 1))))) ;; (For the definition of A000196, see under that entry). - Antti Karttunen, Nov 03 2017

a(n) = floor(sqrt(n)) - floor(sqrt(n-1)), for n > 0.
a(n) = A000005(n) mod 2, n > 0. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-w)^2 - (v-w)*(v+w-1) - Michael Somos, Jul 19 2004
Dirichlet g.f.: zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
G.f.: (theta_3(0,x) + 1)/2, where theta_3 is a Jacobi theta function. - Franklin T. Adams-Watters, Jun 19 2006 [See A000122 for theta_3.]
a(n) = f(n,0) with f(x,y) = f(x-2*y-1,y+1) if x > 0, otherwise 0^(-x). - Reinhard Zumkeller, Sep 26 2008
a(n) = Sum_{d|n} (-1)^bigomega(d), for n >= 1. - Benoit Cloitre, Oct 25 2009
a(n) <= A093709(n). - Reinhard Zumkeller, Nov 14 2009
a(A000290(n)) = 1; a(A000037(n)) = 0. - Reinhard Zumkeller, Jun 20 2011
a(n) = 0 ^ A053186(n). - Reinhard Zumkeller, Feb 12 2012
a(n) = A063524(A007913(n)), for n > 0. - Reinhard Zumkeller, Jul 09 2014
a(n) = -(-1)^n * A258998(n) unless n = 0. 2 * a(n) = A000122(n) unless n = 0. - Michael Somos, Jun 16 2015
a(n) = A037011(A156552(n)), provided that A037011(n) = A000035(A106737(n)). [See A037011.] - Antti Karttunen, Nov 03 2017
a(n*m) = a(n/gcd(n,m))*a(m/gcd(n,m)) for all n and m > 0 (conjectured). - Velin Yanev, Feb 13 2019 [Proof from Michael B. Porter, Feb 16 2019: If nm is a square, nm = product_i (p_i^2), where p_i are prime, not necessarily distinct. Each p_i either appears twice in n, twice in m, or one time in each and therefore in the gcd. So n/gcd(n,m) and m/gcd(n,m) are both squares. If nm is not a square, there is a q_j that appears in one of n or m but not in the gcd. So either n/gcd(n,m) or m/gcd(n,m) is not a square.]
a(n) = Sum_{d|n} A008836(d), n >= 1, a(0) = 1. - Jinyuan Wang, Apr 20 2019
G.f.: A(q) = Sum_{n >= 0}  q^(2*n)*Product_{k >= 2*n+1} 1 - (-q)^k. - Peter Bala, Feb 22 2021
Multiplicative with a(p^e) = 1 if e is even, and 0 otherwise. - Amiram Eldar, Dec 29 2022
a(n) = Sum_{d|n} mobius(core(n)), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

More terms from Franklin T. Adams-Watters, Jun 19 2006

A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

If n = Product p_i^a_i, d = Product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.
Also the number of squarefree divisors of n. - Labos Elemer
Also number of divisors of the squarefree kernel of n: a(n) = A000005(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
Also shadow transform of pronic numbers A002378.
For n >= 1 define an n X n (0,1) matrix A by A[i,j] = 1 if lcm(i,j) = n, A[i,j] = 0 if lcm(i,j) <> n for 1 <= i,j <= n. a(n) is the rank of A. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 11 2003
a(n) is also the number of solutions to x^2 - x == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
a(n) is the number of squarefree divisors of n, but in general the set of unitary divisors of n is not the set of squarefree divisors (compare the rows of A077610 and A206778). - Jaroslav Krizek, May 04 2009
Row lengths of the triangles in A077610 and in A206778. - Reinhard Zumkeller, Feb 12 2012
a(n) is also the number of distinct residues of k^phi(n) (mod n), k=0..n-1. - Michel Lagneau, Nov 15 2012
a(n) is the number of irreducible fractions y/x that satisfy x*y=n (and gcd(x,y)=1), x and y positive integers. - Luc Rousseau, Jul 09 2017
a(n) is the number of (x,y) lattice points satisfying both x*y=n and (x,y) is visible from (0,0), x and y positive integers. - Luc Rousseau, Jul 10 2017
Conjecture: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the unitary divisors of n is divisible by the sum of the k-th powers of the odd unitary divisors of n (note that this sequence lists the sum of the 0th powers of the unitary divisors of n). - Ivan N. Ianakiev, Feb 18 2018
a(n) is the number of one-digit numbers, k, when written in base n such that k and k^2 end in the same digit. - Matthew Scroggs, Jun 01 2018
Dirichlet convolution of A271102 and A000005. - Vaclav Kotesovec, Apr 08 2019
Conjecture: Let b(i; n), n > 0, be multiplicative sequences for some fixed integer i >= 0 with b(i; p^e) = (Sum_{k=1..i+1} A164652(i, k) * e^(k-1)) * (i+2) / (i!) for prime p and e > 0. Then we have Dirichlet generating functions: Sum_{n > 0} b(i; n) / n^s = (zeta(s))^(i+2) / zeta((i+2) * s). Examples for i=0 this sequence, for i=1 A226602, and for i=2 A286779. - Werner Schulte, Feb 17 2022
The smallest integer with 2^m unitary divisors, or equivalently, the smallest integer with 2^m squarefree divisors, is A002110(m). - Bernard Schott, Oct 04 2022

			a(12) = 4 because the four unitary divisors of 12 are 1, 3, 4, 12, and also because the four squarefree divisors of 12 are 1, 2, 3, 6.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.

T. D. Noe, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 49-50.
Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
Jon Maiga, Upper bound of Fibonacci entry points, 2019.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.
Index entries for sequences computed from exponents in factorization of n

Cf. A077610, A048105, A000173, A013928, A000079, A001221, A002110, A034448, A047994, A061142, A277561.
Sum of the k-th powers of the squarefree divisors of n for k=0..10: this sequence (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: this sequence (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), this sequence (k=10).
Cf. A020821 (Dgf at s=2), A177057 (Dgf at s=4).

a034444 = length . a077610_row  -- Reinhard Zumkeller, Feb 12 2012

[#[d:d in Divisors(n)|Gcd(d,n div d) eq 1]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

[&+[Abs(MoebiusMu(d)):d in Divisors(n)]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

with(numtheory): for n from 1 to 200 do printf(`%d,`,2^nops(ifactors(n)[2])) od:
with(numtheory);
# returns the number of unitary divisors of n and a list of them
f:=proc(n)
local ct,i,t1,ans;
ct:=0; ans:=[];
t1:=divisors(n);
for i from 1 to nops(t1) do
d:=t1[i];
if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi;
od:
RETURN([ct,ans]);
end;
# N. J. A. Sloane, May 01 2013
# alternative Maple program:
a:= n-> 2^nops(ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Jan 23 2024
a := n -> 2^NumberTheory:-NumberOfPrimeFactors(n, distinct):  # Peter Luschny, May 13 2025

a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* Jean-François Alcover, Apr 05 2011 *)
Table[2^PrimeNu[n],{n,110}] (* Harvey P. Dale, Jul 14 2011 *)

a(n)=1<Charles R Greathouse IV, Feb 11 2011

for(n=1, 100, print1(direuler(p=2, n, (1+X)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

from sympy import divisors, gcd
def a(n):
    return sum(1 for d in divisors(n) if gcd(d, n//d)==1)
# Indranil Ghosh, Apr 16 2017

from sympy import primefactors
def a(n): return 2**len(primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 16 2017

(define (A034444 n) (if (= 1 n) n (* 2 (A034444 (A028234 n))))) ;; Antti Karttunen, May 29 2017

a(n) = Sum_{d|n} abs(mu(n)) = 2^(number of different primes dividing n) = 2^A001221(n), with mu(n) = A008683(n). [Added Möbius formula. - Wolfdieter Lang, Jan 11 2020]
a(n) = Product_{ primes p|n } (1 + Legendre(1, p)).
Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley, Oct 25 2001
a(n) = Sum_{d|n} tau(d^2)*mu(n/d), Dirichlet convolution of A048691 and A008683. - Benoit Cloitre, Oct 03 2002
Dirichlet generating function: zeta(s)^2/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
Inverse Mobius transform of A008966. - Franklin T. Adams-Watters, Sep 11 2005
Asymptotically [Finch] the cumulative sum of a(n) = Sum_{n=1..N} a(n) ~ (6/(Pi^2))*N*log(N) + (6/(Pi^2))*(2*gamma - 1 - (12/(Pi^2))*zeta'(2))*N + O(sqrt(N)). - Jonathan Vos Post, May 08 2005 [typo corrected by Vaclav Kotesovec, Sep 13 2018]
a(n) = Sum_{d|n} floor(rad(d)/d), where rad is A007947 and floor(rad(n)/n) = A008966(n). - Enrique Pérez Herrero, Nov 13 2009
a(n) = A000005(n) - A048105(n); number of nonzero terms in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013
G.f.: Sum_{n>0} A008966(n)*x^n/(1-x^n). - Mircea Merca, Feb 25 2014
a(n) = Sum_{d|n} lambda(d)*mu(d), where lambda is A008836. - Enrique Pérez Herrero, Apr 27 2014
a(n) = A277561(A156552(n)). - Antti Karttunen, May 29 2017
a(n) = A005361(n^2)/A005361(n). - Velin Yanev, Jul 26 2017
L.g.f.: -log(Product_{k>=1} (1 - mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A001615(d) * A023900(n/d). - Torlach Rush, Jan 20 2020
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = 1. - Amiram Eldar, May 29 2020
a(n) = lim_{k->oo} A000005(n^(2*k))/A000005(n^k). - Velin Yanev and Amiram Eldar, Jan 10 2025

More terms from James Sellers, Jun 20 2000

A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144

Offset: 1

N. J. A. Sloane, Dec 11 1999

Row sums of the triangle in A077610. - Reinhard Zumkeller, Feb 12 2002

Multiplicative with a(p^e) = p^e+1 for e>0. - Franklin T. Adams-Watters, Sep 11 2005

			Unitary divisors of 12 are 1, 3, 4, 12. Or, 12=3*2^2 hence usigma(12)=(3+1)*(2^2+1)=20.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

T. D. Noe, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Prime injections and quasipolarities, Matematiche 69 (2014) 159-168
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique 2015 Vol. 97, Issue 111, pp. 175-180.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Cf. A000203, A034444, A034460, A047994, A048250, A064000, A064609, A191414.
Cf. A063937 (squares > 1).
Cf. A188999, A301981, A301982.

a034448 = sum . a077610_row  -- Reinhard Zumkeller, Feb 12 2012
(Python 3.8+)
from math import prod
from sympy import factorint
def A034448(n): return prod(p**e+1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 20 2021

A034448 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: RETURN(ans) end:
a := proc(n) local i; numtheory[divisors](n); select(d -> igcd(d,n/d)=1, %); add(i,i=%) end; # Peter Luschny, May 03 2009

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Table[ usigma[n], {n, 71}] (* Robert G. Wilson v, Aug 28 2004 *)
Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 70}] (* Michael De Vlieger, Mar 01 2017 *)
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Array[usigma, 100] (* faster since avoids generating divisors, Giovanni Resta, Apr 23 2017 *)

A034448(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d)) \\ Rick L. Shepherd

A034448(n) = {my(f=factorint(n)); prod(k=1, #f[,2], f[k,1]^f[k,2]+1)} \\ Andrew Lelechenko, Apr 22 2014

a(n)=sumdivmult(n,d,if(gcd(d,n/d)==1,d)) \\ Charles R Greathouse IV, Sep 09 2014

If n = Product p_i^e_i, usigma(n) = Product (p_i^e_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet generating function: zeta(s)*zeta(s-1)/zeta(2s-1). - Franklin T. Adams-Watters, Sep 11 2005
Conjecture: a(n) = sigma(n^2/rad(n))/sigma(n/rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
This conjecture is easily verified since all the functions involved are multiplicative and proving it for prime powers is straightforward. - Juan José Alba González, Mar 19 2021
From Amiram Eldar, May 29 2020: (Start)
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = n.
a(n) <= sigma(n) = A000203(n), with equality if and only if n is squarefree (A005117). (End)
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 20 2021
a(n) = uphi(n^2)/uphi(n) = A191414(n)/uphi(n), where uphi(n) = A047994(n). - Amiram Eldar, Sep 21 2024

More terms from Erich Friedman

A000004 The zero sequence.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, Table of n, a(n) for n = 0..1000 [Useful when plotting one sequence against another.]
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 10.
Index entries for characteristic functions
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (1).
Index entries for sequences that are fixed points of mappings

Cf. A000012 (all 1's), A007395 (all 2's), A010701 (all 3's).

Cf. A000007(n) = 0^n: characteristic function of {0}.

a000004 = const 0
a000004_list = repeat 0  -- Reinhard Zumkeller, May 07 2012

Magma
```
[ 0 : n in [0..100]];
```
Maple
```
A000004 := n->0;
```

a[ n_] := 0;
Table[0, {n, 100}] (* Matthew House, Jul 14 2015 *)
LinearRecurrence[{1},{0},102] (* Ray Chandler, Jul 15 2015 *)

PARI
```
vector(100,n,0)
```

print([0 for n in range(102)]) # Michael S. Branicky, Apr 04 2022

R
```
rep(0,100)
```

a(n) = 0 for all integer n.

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

A006519 Highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 64, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2

Offset: 1

N. J. A. Sloane, Simon Plouffe

Least positive k such that m^k + 1 divides m^n + 1 (with fixed base m). - Vladimir Baltic, Mar 25 2002
To construct the sequence: start with 1, concatenate 1, 1 and double last term gives 1, 2. Concatenate those 2 terms, 1, 2, 1, 2 and double last term 1, 2, 1, 2 -> 1, 2, 1, 4. Concatenate those 4 terms: 1, 2, 1, 4, 1, 2, 1, 4 and double last term -> 1, 2, 1, 4, 1, 2, 1, 8, etc. - Benoit Cloitre, Dec 17 2002
a(n) = gcd(seq(binomial(2*n, 2*m+1)/2, m = 0 .. n - 1)) (odd numbered entries of even numbered rows of Pascal's triangle A007318 divided by 2), where gcd() denotes the greatest common divisor of a set of numbers. Due to the symmetry of the rows it suffices to consider m = 0 .. floor((n-1)/2). - Wolfdieter Lang, Jan 23 2004
Equals the continued fraction expansion of a constant x (cf. A100338) such that the continued fraction expansion of 2*x interleaves this sequence with 2's: contfrac(2*x) = [2; 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 8, 2, ...].
Simon Plouffe observes that this sequence and A003484 (Radon function) are very similar, the difference being all zeros except for every 16th term (see A101119 for nonzero differences). Dec 02 2004
This sequence arises when calculating the next odd number in a Collatz sequence: Next(x) = (3*x + 1) / A006519, or simply (3*x + 1) / BitAnd(3*x + 1, -3*x - 1). - Jim Caprioli, Feb 04 2005
a(n) = n if and only if n = 2^k. This sequence can be obtained by taking a(2^n) = 2^n in place of a(2^n) = n and using the same sequence building approach as in A001511. - Amarnath Murthy, Jul 08 2005
Also smallest m such that m + n - 1 = m XOR (n - 1); A086799(n) = a(n) + n - 1. - Reinhard Zumkeller, Feb 02 2007
Number of 1's between successive 0's in A159689. - Philippe Deléham, Apr 22 2009
Least number k such that all coefficients of k*E(n, x), the n-th Euler polynomial, are integers (cf. A144845). - Peter Luschny, Nov 13 2009
In the binary expansion of n, delete everything left of the rightmost 1 bit. - Ralf Stephan, Aug 22 2013
The equivalent sequence for partitions is A194446. - Omar E. Pol, Aug 22 2013
Also the 2-adic value of 1/n, n >= 1. See the Mahler reference, definition on p. 7. This is a non-archimedean valuation. See Mahler, p. 10. Sometimes called 2-adic absolute value of 1/n. - Wolfdieter Lang, Jun 28 2014
First 2^(k-1) - 1 terms are also the heights of the successive rectangles and squares of width 2 that are adjacent to any of the four sides of the toothpick structure of A139250 after 2^k stages, with k >= 2. For example: if k = 5 the heights after 32 stages are [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1] respectively, the same as the first 15 terms of this sequence. - Omar E. Pol, Dec 29 2020

			2^3 divides 24, but 2^4 does not divide 24, so a(24) = 8.
2^0 divides 25, but 2^1 does not divide 25, so a(25) = 1.
2^1 divides 26, but 2^2 does not divide 26, so a(26) = 2.
Per _Marc LeBrun_'s 2000 comment, a(n) can also be determined with bitwise operations in two's complement. For example, given n = 48, we see that n in binary in an 8-bit byte is 00110000 while -n is 11010000. Then 00110000 AND 11010000 = 00010000, which is 16 in decimal, and therefore a(48) = 16.
G.f. = x + 2*x^2 + x^3 + 4*x^4 + x^5 + 2*x^6 + x^7 + 8*x^8 + x^9 + ...

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Dzmitry Badziahin and Jeffrey Shallit, An Unusual Continued Fraction, arXiv:1505.00667 [math.NT], 2015.
Dzmitry Badziahin and Jeffrey Shallit, An unusual continued fraction, Proc. Amer. Math. Soc. 144 (2016), 1887-1896.
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
M. Beeler, R. W. Gosper and R. Schroeppel, Item 175, in Beeler, M., Gosper, R. W. and Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb 29 1972.
Ron Brown and Jonathan L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.
Daniel Bruns, Wojciech Mostowski, and Mattias Ulbrich, Implementation-level verification of algorithms with KeY, International Journal on Software Tools for Technology Transfer, November 2013.
Victor Meally, Letter to N. J. A. Sloane, May 1975.
Laurent Orseau, Levi H. S. Lelis, and Tor Lattimore, Zooming Cautiously: Linear-Memory Heuristic Search With Node Expansion Guarantees, arXiv:1906.03242 [cs.AI], 2019.
Laurent Orseau, Levi H. S. Lelis, Tor Lattimore, and Théophane Weber, Single-Agent Policy Tree Search With Guarantees, arXiv:1811.10928 [cs.AI], 2018, also in Advances in Neural Information Processing Systems, 32nd Conference on Neural Information Processing Systems (NIPS 2018), Montréal, Canada.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions.
Ralf Stephan, Table of generating functions.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Eric Weisstein's World of Mathematics, Even Part.
Wikipedia, Converse nonimplication.
Index entries for sequences related to binary expansion of n.

Partial sums are in A006520, second partial sums in A022560.
Sequences used in definitions of this sequence: A000079, A001511, A004198, A007814.
Cf. A000265, A100338, A003484, A001620, A101119, A002487, A139250.
Sequences with related definitions: A038712, A171977, A135481 (GS(1, 6)).
This is Guy Steele's sequence GS(5, 2) (see A135416).
Related to A007913 via A225546.
A059897 is used to express relationship between sequence terms.
Cf. A091476 (Dgf at s=2).

import Data.Bits ((.&.))
a006519 n = n .&. (-n) :: Integer
-- Reinhard Zumkeller, Mar 11 2012, Dec 29 2011

using IntegerSequences
[EvenPart(n) for n in 1:102] |> println  # Peter Luschny, Sep 25 2021

[2^Valuation(n, 2): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015

with(numtheory): for n from 1 to 200 do if n mod 2 = 1 then printf(`%d,`,1) else printf(`%d,`,2^ifactors(n)[2][1][2]) fi; od:
A006519 := proc(n) if type(n,'odd') then 1 ; else for f in ifactors(n)[2] do if op(1,f) = 2 then return 2^op(2,f) ; end if; end do: end if; end proc: # R. J. Mathar, Oct 25 2010
A006519 := n -> 2^padic[ordp](n,2): # Peter Luschny, Nov 26 2010

lowestOneBit[n_] := Block[{k = 0}, While[Mod[n, 2^k] == 0, k++]; 2^(k - 1)]; Table[lowestOneBit[n], {n, 102}] (* Robert G. Wilson v Nov 17 2004 *)
Table[2^IntegerExponent[n, 2], {n, 128}] (* Jean-François Alcover, Feb 10 2012 *)
Table[BitAnd[BitNot[i - 1], i], {i, 1, 102}] (* Peter Luschny, Oct 10 2019 *)

PARI
```
{a(n) = 2^valuation(n, 2)};
```
PARI
```
a(n)=1<Joerg Arndt, Jun 10 2011
```

a(n)=bitand(n,-n); \\ Joerg Arndt, Jun 10 2011

a(n)=direuler(p=2,n,if(p==2,1/(1-2*X),1/(1-X)))[n] \\ Ralf Stephan, Mar 27 2015

def A006519(n): return n&-n # Chai Wah Wu, Jul 06 2022

(1 to 128).map(Integer.lowestOneBit()) // _Alonso del Arte, Mar 04 2020

a(n) = n AND -n (where "AND" is bitwise, and negative numbers are represented in two's complement in a suitable bit width). - Marc LeBrun, Sep 25 2000, clarified by Alonso del Arte, Mar 16 2020
Also: a(n) = gcd(2^n, n). - Labos Elemer, Apr 22 2003
Multiplicative with a(p^e) = p^e if p = 2; 1 if p > 2. - David W. Wilson, Aug 01 2001
G.f.: Sum_{k>=0} 2^k*x^2^k/(1 - x^2^(k+1)). - Ralf Stephan, May 06 2003
Dirichlet g.f.: zeta(s)*(2^s - 1)/(2^s - 2) = zeta(s)*(1 - 2^(-s))/(1 - 2*2^(-s)). - Ralf Stephan, Jun 17 2007
a(n) = 2^floor(A002487(n - 1) / A002487(n)). - Reikku Kulon, Oct 05 2008
a(n) = 2^A007814(n). - R. J. Mathar, Oct 25 2010
a((2*k - 1)*2^e) = 2^e, k >= 1, e >= 0. - Johannes W. Meijer, Jun 07 2011
a(n) = denominator of Euler(n-1, 1). - Arkadiusz Wesolowski, Jul 12 2012
a(n) = A011782(A001511(n)). - Omar E. Pol, Sep 13 2013
a(n) = (n XOR floor(n/2)) XOR (n-1 XOR floor((n-1)/2)) = n - (n AND n-1) (where "AND" is bitwise). - Gary Detlefs, Jun 12 2014
a(n) = ((n XOR n-1)+1)/2. - Gary Detlefs, Jul 02 2014
a(n) = A171977(n)/2. - Peter Kern, Jan 04 2017
a(n) = 2^(A001511(n)-1). - Doug Bell, Jun 02 2017
a(n) = abs(A003188(n-1) - A003188(n)). - Doug Bell, Jun 02 2017
Conjecture: a(n) = (1/(A000203(2*n)/A000203(n)-2)+1)/2. - Velin Yanev, Jun 30 2017
a(n) = (n-1) o n where 'o' is the bitwise converse nonimplication. 'o' is not commutative. n o (n+1) = A135481(n). - Peter Luschny, Oct 10 2019
From Peter Munn, Dec 13 2019: (Start)
a(A225546(n)) = A225546(A007913(n)).
a(A059897(n,k)) = A059897(a(n), a(k)). (End)
Sum_{k=1..n} a(k) ~ (1/(2*log(2)))*n*log(n) + (3/4 + (gamma-1)/(2*log(2)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022
a(n) = n / A000265(n). - Amiram Eldar, May 22 2025

More terms from James Sellers, Jun 20 2000

A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, 90, 60, 144, 62, 96, 96, 96, 84, 144, 68, 108, 96

Offset: 1

N. J. A. Sloane

Number of primitive sublattices of index n in generic 2-dimensional lattice; also index of Gamma_0(n) in SL_2(Z).
A generic 2-dimensional lattice L =  consists of all vectors of the form mV + nW, (m,n integers). A sublattice S =  has index |ad-bc| and is primitive if gcd(a,b,c,d) = 1. The generic lattice L has precisely a(2) = 3 sublattices of index 2, namely <2V,W>,  and  (which = ) and so on for other indices.
The sublattices of index n are in 1-to-1 correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is the present sequence.
SL_2(Z) = Gamma is the group of all 2 X 2 matrices [a b; c d] where a,b,c,d are integers with ad-bc = 1 and Gamma_0(N) is usually defined as the subgroup of this for which N|c. But conceptually Gamma is best thought of as the group of (positive) automorphisms of a lattice , its typical element taking V -> aV + bW, W -> cV + dW and then Gamma_0(N) can be defined as the subgroup consisting of the automorphisms that fix the sublattice  of index N. - J. H. Conway, May 05 2001
Dedekind proved that if n = k_i*j_i for i in I represents all the ways to write n as a product, and e_i=gcd(k_i,j_i), then a(n)= sum(k_i / (e_i * phi(e_i)), i in I ) [cf. Dickson, History of the Theory of Numbers, Vol. 1, p. 123].
Also a(n)= number of cyclic subgroups of order n in an Abelian group of order n^2 and type (1,1) (Fricke). - Len Smiley, Dec 04 2001
The polynomial degree of the classical modular equation of degree n relating j(z) and j(nz) is psi(n) (Fricke). - Michael Somos, Nov 10 2006; clarified by Katherine E. Stange, Mar 11 2022
The Mobius transform of this sequence is A063659. - Gary W. Adamson, May 23 2008
The inverse Mobius transform of this sequence is A060648. - Vladeta Jovovic, Apr 05 2009
The Dirichlet inverse of this sequence is A008836(n) * A048250(n). - Álvar Ibeas, Mar 18 2015
The Riemann Hypothesis is true if and only if a(n)/n - e^gamma*log(log(n)) < 0 for any n > 30. - Enrique Pérez Herrero, Jul 12 2011
The Riemann Hypothesis is also equivalent to another inequality, see the Sole and Planat link. - Thomas Ordowski, May 28 2017
An infinitary analog of this sequence is the sum of the infinitary divisors of n (see A049417). - Vladimir Shevelev, Apr 01 2014
Problem: are there composite numbers n such that n+1 divides psi(n)? - Thomas Ordowski, May 21 2017
The sum of divisors d of n such that n/d is squarefree. - Amiram Eldar, Jan 11 2019
Psi(n)/n is a new maximum for each primorial (A002110) [proof in link: Patrick Sole and Michel Planat, Proposition 1 page 2]. - Bernard Schott, May 21 2020
From Jianing Song, Nov 05 2022: (Start)
a(n) is the number of subgroups of C_n X C_n that are isomorphic to C_n, where C_n is the cyclic group of order n. Proof: the number of elements of order n in C_n X C_n is A007434(n) (they are the elements of the form (a,b) in C_n X C_n where gcd(a,b,n) = 1), and each subgroup isomorphic to C_n contains phi(n) generators, so the number of such subgroups is A007434(n)/phi(n) = a(n).
The total number of order-n subgroups of C_n X C_n is A000203(n). (End)

			Let L = <V,W> be a 2-dimensional lattice. The 6 primitive sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V+W,2W>, <2V,2W+V>. Compare A000203.
G.f. = x + 3*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 12*x^8 + 12*x^9 + ...

Tom Apostol, Intro. to Analyt. Number Theory, page 71, Problem 11, where this is called phi_1(n).
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 228.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 220.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (1).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
O. Bordelles and B. Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, Representing and counting the subgroups of the group Z_m X Z_n, arXiv:1211.1797 [math.GR], 2012.
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications 14:2 (2015), 73-88.
F. A. Lewis et al., Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
E. Pérez Herrero, Recycling Hardy & Wright, Average Order of Dedekind Psi Function, Psychedelic Geometry Blogspot.
Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, to appear in Journal of Combinatorics and Number Theory, arXiv:1011.1825 [math.NT], 2010-2011.
Eric Weisstein's World of Mathematics, Dedekind Function
Wikipedia, Dedekind psi function
Index entries for "core" sequences
Index entries for sequences related to sublattices

Other sequences that count lattices/sublattices: A000203 (with primitive condition removed), A003050 (hexagonal lattice instead), A003051, A054345, A160889, A160891.
Cf. A002110, A027748, A124010, A019269.
Cf. A301594.
Cf. A063659 (Möbius transform), A082020 (average order), A156303 (Euler transform), A173290 (partial sums), A175836 (partial products), A203444 (range).
Cf. A210523 (record values).
Algebraic combinations with other core sequences: A000082, A033196, A175732, A291784, A344695.
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), this sequence (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).
Cf. A082695 (Dgf at s=3), A339925 (Dgf at s=4).

import Data.Ratio (numerator)
a001615 n = numerator (fromIntegral n * (product $
            map ((+ 1) . recip . fromIntegral) $ a027748_row n))
-- Reinhard Zumkeller, Jun 03 2013, Apr 12 2012

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[MoebiusMu(k)^2*x^k/(1-x^k)^2: k in [1..2*m]]) )); // G. C. Greubel, Nov 23 2018

A001615 := proc(n) n*mul((1+1/i[1]),i=ifactors(n)[2]) end; # Mark van Hoeij, Apr 18 2012

Join[{1}, Table[n Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] (* T. D. Noe, Jun 11 2006 *)
Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 100}] (* Jan Mangaldan, Aug 22 2013 *)
a[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}]; (* Michael Somos, Jan 10 2015 *)
Table[n Product[1 + 1/p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 08 2021 *)
Table[n DivisorSum[n, MoebiusMu[#]^2/# &], {n, 20}] (* Eric W. Weisstein, Mar 09 2025 *)

{a(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n])};

{a(n) = if( n<1, 0, n * sumdiv( n, d, moebius(d)^2 / d))}; /* Michael Somos, Nov 10 2006 */

a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^f[i,2] + f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Aug 22 2013

a(n) = n * sumdivmult(n, d, issquarefree(d)/d) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import primefactors
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist) # Chai Wah Wu, Jun 03 2021

def A001615(n) : return n*mul(1+1/p for p in prime_divisors(n))
[A001615(n) for n in (1..69)] # Peter Luschny, Jun 10 2012

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s). - Michael Somos, May 19 2000
Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson, Aug 01 2001
a(n) = A003557(n)*A048250(n) = n*A000203(A007947(n))/A007947(n). - Labos Elemer, Dec 04 2001
a(n) = n*Sum_{d|n} mu(d)^2/d, Dirichlet convolution of A008966 and A000027. - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(n/d)^2 * d. - Joerg Arndt, Jul 06 2011
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_2(n)/J_1(n) = J_2(n)/phi(n) = A007434(n)/A000010(n), where J_k is the k-th Jordan Totient Function.
a(n) = (1/phi(n))*Sum_{d|n} mu(n/d)*d^(b-1), for b=3. (End)
a(n) = n / Sum_{d|n} mu(d)/a(d). - Enrique Pérez Herrero, Jun 06 2012
a(n^k)= n^(k-1) * a(n). - Enrique Pérez Herrero, Jan 05 2013
If n is squarefree, then a(n) = A049417(n) = A000203(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{d^2 | n} mu(d) * A000203(n/d^2). - Álvar Ibeas, Dec 20 2014
The average order of a(n) is 15*n/Pi^2. - Enrique Pérez Herrero, Jan 14 2012. See Apostol. - N. J. A. Sloane, Sep 04 2017
G.f.: Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 25 2018
a(n) = Sum_{d|n} 2^omega(d) * phi(n/d), Dirichlet convolution of A034444 and A000010. - Daniel Suteu, Mar 09 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} 2^omega(gcd(n,k)).
a(n) = Sum_{k=1..n} 2^omega(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = abs(A158523(n)) = A158523(n) * A008836(n). - Enrique Pérez Herrero, Nov 07 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^2). - Ridouane Oudra, Mar 26 2025

More terms from Olivier Gérard, Aug 15 1997

A008966 a(n) = 1 if n is squarefree, otherwise 0.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0

Offset: 1

N. J. A. Sloane

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

The infinite lower triangular matrix with A008966 on the main diagonal and the rest zeros is the square of triangle A143255. - Gary W. Adamson, Aug 02 2008

Daniel Forgues, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Moebius Function
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A005117, A008836 (Dirichlet inverse), A013928 (partial sums).
Cf. A179211, A179212, A179213, A179214, A179215.
Cf. A020639, A073576, A087188.
Cf. A124010, A212793, A085357, A156552.
Parity of A002033.
Cf. A047999, A036987
Cf. A082020 (Dgf at s=2), A157289 (Dgf at s=3), A157290 (Dgf at s=4).

a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012

Magma
```
[ Abs(MoebiusMu(n)) : n in [1..100]];
```

A008966 := proc(n) if numtheory[issqrfree](n) then 1 ; else 0 ; end if; end proc: # R. J. Mathar, Mar 14 2011

A008966[n_] := Abs[MoebiusMu[n]]; Table[A008966[n], {n, 100}] (* Enrique Pérez Herrero, Apr 15 2010 *)
Table[If[SquareFreeQ[n],1,0],{n,100}] (* or *) Boole[SquareFreeQ/@ Range[ 100]] (* Harvey P. Dale, Feb 28 2015 *)

MuPAD
```
func(abs(numlib::moebius(n)), n):
```
PARI
```
a(n)=if(n<1,0,direuler(p=2,n,1+X))[n]
```

a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015

from sympy import factorint
def A008966(n): return int(max(factorint(n).values(),default=1)==1) # Chai Wah Wu, Apr 05 2023

Dirichlet g.f.: zeta(s)/zeta(2s).
a(n) = abs(mu(n)), where mu is the Moebius function (A008683).
a(n) = 0^(bigomega(n) - omega(n)), where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Multiplicative with p^e -> 0^(e - 1), p prime and e > 0. - Reinhard Zumkeller, Jul 15 2003
a(n) = 0^(A046951(n) - 1). - Reinhard Zumkeller, May 20 2007
a(n) = 1 - A107078(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = floor(rad(n)/n), where rad() is A007947. - Enrique Pérez Herrero, Nov 13 2009
A175046(n) = a(n)*A073311(n). - Reinhard Zumkeller, Apr 05 2010
a(n) = floor(A000005(n^2)/A007425(n)). - Enrique Pérez Herrero, Apr 15 2010
a(A005117(n)) = 1; a(A013929(n)) = 0; a(n) = A013928(n + 1) - A013928(n). - Reinhard Zumkeller, Jul 05 2010
a(n) * A112526(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = mu(n) * lambda(n) = A008836(n) * A008683(n). - Enrique Pérez Herrero, Nov 29 2013
a(n) = Sum_{d|n} 2^omega(d)*mu(n/d). - Geoffrey Critzer, Feb 22 2015
a(n) = A085357(A156552(n)). - Antti Karttunen, Mar 06 2017
Limit_{n->oo} (1/n)*Sum_{j=1..n} a(j) = 6/Pi^2. - Andres Cicuttin, Aug 13 2017
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^bigomega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021

Deleted an unclear comment. - N. J. A. Sloane, May 30 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

cp's OEIS Frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

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A010052 Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.

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A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

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A000004 The zero sequence.

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