A077610 -id:A077610 - cp's OEIS frontend

A049599 Number of (1+e)-divisors of n: if n = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of n.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 6, 2, 12, 4, 6, 4, 4, 4, 6, 2, 6, 6, 9, 2, 8, 2

Offset: 1

Yasutoshi Kohmoto

A divisor of n is a (1+e)-divisor if and only if it is a unitary divisor of an exponential divisor of n (see A077610 and A322791). - Amiram Eldar, Feb 26 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A049603, A051378.

Cf. A124010, A000005, A049419, A077610, A322791.

a049599 = product . map ((+ 1) . a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

a[n_] := Times @@ (DivisorSigma[0, #] + 1 &)  /@ FactorInteger[n][[All, 2]]; a[1] = 1; Table[a[n], {n, 1, 103}] (* Jean-François Alcover, Oct 10 2011 *)

a(n) = vecprod(apply(x->numdiv(x)+1, factor(n)[, 2])); \\ Amiram Eldar, Aug 13 2023

If n = Product p(i)^r(i) then a(n) = Product (tau(r(i))+1), where tau(n) = number of divisors of n, cf. A000005. - Vladeta Jovovic, Apr 29 2001

More terms from Naohiro Nomoto, Apr 12 2001

A165430 Table T(n,m) read by rows: the greatest common unitary divisor of n and m, n>=1, 1<=m<=n.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 18 2009

The maximum number which appears in row n and also in row m of A077610. The sequence of the counts of 1 in row n=1,2,3,... is 1, 1, 2, 3, 4, 3, 6, 7, 8, 6, 10, 8, 12, 9, 9,...

			The table starts
1;
1,2
1,1,3
1,1,1,4
1,1,1,1,5
1,2,3,1,1,6
1,1,1,1,1,1,7
1,1,1,1,1,1,1,8
1,1,1,1,1,1,1,1,9
1,2,1,1,5,2,1,1,1,10

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Pentti Haukkanen, On a gcd-sum function, Aequat. Math. 76 (1-2) (2008) 168-178.
L. Toth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2, function (k,n)**.

Cf. A034444, A275254 (row sums)

import Data.List (intersect)
a165430 n k = last (a077610_row n `intersect` a077610_row k)
a165430_row n = map (a165430 n) [1..n]
a165430_tabl = map a165430_row [1..]
-- Reinhard Zumkeller, Mar 04 2013

A077610 := proc(n) local a; a := {} ; for d in numtheory[divisors](n) do if gcd(d,n/d) = 1 then a := a union {d} ; fi; od: a; end:
  A165430 := proc(n,m) local cud ; cud := A077610(n) intersect A077610(m) ; max(op(cud)) ; end:
seq(seq(A165430(n,m),m=1..n),n=1..20) ;

A077610[n_] := Module[{a = {}}, Do[If[GCD[d, n/d] == 1, a = a ~Union~ {d}], {d, Divisors[n]}]; a]; A165430[n_, m_] := Module[{cud = A077610[n] ~Intersection~ A077610[m]}, Max[cud]]; Table[Table[A165430[n, m], {m, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d);}
T(n,m) = vecmax(setintersect(udivs(n), udivs(m))); \\ Michel Marcus, Oct 11 2015

A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 18 2025

			Triangle begins:
  1
  1, 2
  1, 1, 3
  1, 1, 1, 4
  1, 1, 1, 1, 5
  1, 2, 3, 2, 1, 6
  1, 1, 1, 1, 1, 1, 7
  1, 1, 1, 1, 1, 1, 1, 8
  1, 1, 1, 1, 1, 1, 1, 1, 9
  1, 2, 1, 2, 5, 2, 1, 2, 1, 10

Amiram Eldar, Table of n, a(n) for n = 1..10585 (first 145 rows flattened)

Cf. A005117, A050873, A077610, A145388 (row sums), A165430, A225174, A384046.

Upper right triangle of A322482.

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; T[n_, k_] := Max[Intersection[udiv[n], Divisors[k]]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
T(n, k) = vecmax(setintersect(udiv(n), divisors(k)));

T(n, 1) = 1.
T(n, n) = n.
T(n, k) <= A050873(n, k) = gcd(n, k), with equality if n is squarefree (A005117).

A068068 Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d,n/d)=1.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 19 2002

Shadow transform of triangular numbers.
a(n) is the number of primitive Pythagorean triangles with inradius n. For the smallest inradius of exactly 2^n primitive Pythagorean triangles see A070826.
Number of primitive Pythagorean triangles with leg 4n. For smallest (even) leg of exactly 2^n PPTs, see A088860. - Lekraj Beedassy, Jul 12 2006
As shown by Chi and Killgrove, a(n) is the total number of primitive Pythagorean triples satisfying area = n * perimeter, or equivalently 2 raised to the power of the number of distinct, odd primes contained in n. - Ant King, Mar 15 2011
This is the case k=0 of the sum over the k-th powers of the odd unitary divisors of n, which is multiplicative with a(2^e)=1 and a(p^e)=1+p^(e*k), p>2, and has Dirichlet g.f. zeta(s)*zeta(s-k)*(1-2^(k-s))/( zeta(2s-k)*(1-2^(k-2*s)) ). - R. J. Mathar, Jun 20 2011
Also the number of odd squarefree divisors of n: a(n) = Sum_{k = 1..A034444(k)} (A077610(n,k) mod 2) = Sum_{k = 1..A034444(k)} (A206778(n,k) mod 2). - Reinhard Zumkeller, Feb 12 2012
a(n) is also the number of even unitary divisors of 2*n. - Amiram Eldar, Jan 28 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henjin Chi and Raymond Killgrove, Problem 1447, Crux Math 15(5), May 1989.
Henjin Chi and Raymond Killgrove, Solution to Problem 1447, Crux Math 16(7), September 1990.
L. J. Gerstein, Pythagorean triples and inner products, Math. Mag. 78 (2005), 205-213.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012.
OEIS Wiki, Shadow transform.
Neville Robbins, On the number of primitive Pythagorean triangles with a given inradius, Fibonacci Quart. 44(4) (2006), 368-369.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A001221, A001620, A024361, A034444, A056901, A068067, A008683, A323239.

a068068 = length . filter odd . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

A068068 := proc(n) local a,f; a :=1 ; for f in ifactors(n)[2] do if op(1,f) > 2 then a := a*2 ; end if; end do: a ; end proc: # R. J. Mathar, Apr 16 2011

a[n_] := Length[Select[Divisors[n], OddQ[ # ]&&GCD[ #, n/# ]==1&]]
a[n_] := 2^(PrimeNu[n]+Mod[n, 2]-1); Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := If[p == 2, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = sumdiv(n, d, (d%2)*(gcd(d, n/d)==1)); \\ Michel Marcus, May 13 2014

a(n) = 2^omega(n>>valuation(n,2)) \\ Charles R Greathouse IV, May 14 2014

a(n) = A034444(2n)/2. If n is even, a(n) = 2^(omega(n)-1); if n is odd, a(n) = 2^omega(n). Here omega(n) = A001221(n) is the number of distinct prime divisors of n.
Multiplicative with a(2^e) = 1, a(p^e) = 2, p>2. - Christian G. Bower May 18 2005
a(n) = A024361(4n). - Lekraj Beedassy, Jul 12 2006
Dirichlet g.f.: zeta^2(s)/ ( zeta(2*s)*(1+2^(-s)) ). Dirichlet convolution of A034444 and A154269. - R. J. Mathar, Apr 16 2011
a(n) = Sum_{d|n} mu(2*d)^2. - Ridouane Oudra, Aug 11 2019
Sum_{k=1..n} a(k) ~ 4*n*((log(n) + 2*gamma - 1 + log(2)/3) / Pi^2 - 12*zeta'(2) / Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 18 2020
a(n) = Sum_{d divides n, d odd} mu(d)^2. - Peter Bala, Feb 01 2024

Edited by Dean Hickerson, Jun 08 2002

A206778 Irregular triangle in which n-th row lists squarefree divisors (A005117) of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 1, 3, 1, 2, 5, 10, 1, 11, 1, 2, 3, 6, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 1, 17, 1, 2, 3, 6, 1, 19, 1, 2, 5, 10, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 6, 1, 5, 1, 2, 13, 26, 1, 3, 1, 2, 7, 14, 1, 29

Offset: 1

Reinhard Zumkeller, Feb 12 2012

			Triangle begins:
.   1: [1]
.   2: [1, 2]
.   3: [1, 3]
.   4: [1, 2]
.   5: [1, 5]
.   6: [1, 2, 3, 6]
.   7: [1, 7]
.   8: [1, 2]
.   9: [1, 3]
.  10: [1, 2, 5, 10]
.  11: [1, 11]
.  12: [1, 2, 3, 6].

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

Cf. A008966, A034444 (row lengths), A048250 (row sums), A206787; A077610.

Cf. A027750, A050320, A050326, A050328.

a206778 n k = a206778_row n !! k
a206778_row = filter ((== 1) . a008966) . a027750_row
a206778_tabf = map a206778_row [1..]
-- Reinhard Zumkeller, May 03 2013, Feb 12 2012

A206778 := proc(n)
    local sqdvs ,nfac,d;
    sqdvs := {} ;
    nfac := ifactors(n)[2] ;
    for d in numtheory[divisors](n) do
        if issqrfree(d) then
            sqdvs := sqdvs union {d} ;
        end if;
    end do:
    sort(sqdvs) ;
end proc:
seq(op(A206778(n)),n=1..10) ; # R. J. Mathar, Mar 06 2023

Flatten[Table[Select[Divisors[n],SquareFreeQ],{n,30}]] (* Harvey P. Dale, Apr 11 2012 *)

row(n) = select(x -> issquarefree(x), divisors(n)); \\ Amiram Eldar, May 02 2025

A384046 Triangle in which the n-th row gives the numbers from 1 to n whose largest divisor that is a unitary divisor of n is 1.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 18 2025

			Triangle begins:
  1,
  1,
  1, 2,
  1, 2, 3,
  1, 2, 3, 4,
  1, 5,
  1, 2, 3, 4, 5, 6,
  1, 2, 3, 4, 5, 6, 7,
  1, 2, 3, 4, 5, 6, 7, 8,
  1, 3, 7, 9

Amiram Eldar, Table of n, a(n) for n = 1..10204 (first 170 rows flattened)
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.

The unitary analog of A038566.

Cf. A047994 (row lengths), A333576 (row sums), A077610, A225174, A384047.

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; uGCD[n_, k_] := Max[Intersection[udiv[n], Divisors[k]]]; row[n_] := Select[Range[n], uGCD[n, #] == 1 &]; Array[row, 10] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
ugcd(n, k) = vecmax(setintersect(udiv(n), divisors(k)));
row(n) = select(x -> ugcd(n, x) == 1, vector(n, i, i));

T(n, 1) = 1.

A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jun 13 2000

Number of distinct values of gcd(d, n!/d) if d runs over divisors of n! seems to be A046951(n).
a(n) = 1 iff n is a square. - Bernard Schott, Oct 22 2019
a(n) is the number of the unitary divisors (cf. A077610) of n that are exponentially odd (A268335). - Amiram Eldar, Nov 11 2022
The number of infinitary divisors of n that are squarefree (A005117). - Amiram Eldar, Jan 09 2024

			n=120, the set of gcd(d, 120/d) values for the 16 divisors of 120 is {1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1}. The max is 2 and it occurs 8 times, so a(120)=8. This sequence seems to consist of powers of 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A005117, A007913, A034444, A077610, A162642, A268335, A295316, A350389.

with(numtheory):
a:= n->(p->coeff(p, x, degree(p)))(add(x^igcd(d, n/d), d=divisors(n))):
seq(a(n), n=1..105);  # Alois P. Heinz, Jul 21 2015

a[n_] := With[{g = GCD[#, n/#]& /@ Divisors[n]}, Count[g, Max[g]]];
Array[a, 105] (* Jean-François Alcover, Mar 28 2017 *)
f[p_, e_] := 2^Mod[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 11 2022 *)

A055076(n) = if(1==n,n,my(es=factor(n)[,2]~); prod(i=1,#es,2^(es[i]%2))); \\ Antti Karttunen, Apr 05 2021

;; With memoization-macro definec.
(definec (A055076 n) (if (= 1 n) n (* (+ 1 (A000035 (A067029 n))) (A055076 (A028234 n))))) ;; Antti Karttunen, Dec 02 2017

Multiplicative with a(p^e) = 2^(e mod 2). - Vladeta Jovovic, Dec 13 2002
a(n) = 2^A162642(n). - Antti Karttunen, Dec 02 2017
a(n) = A034444(A007913(n)). [Found by LODA miner, see C. Krause link. Essentially the same formula as the above ones] - Antti Karttunen, Apr 05 2021
From Amiram Eldar, Sep 09 2023: (Start)
a(n) = A034444(A350389(n)).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 2/p^s). (End)
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - 3/p^(2*s) + 2/p^(3*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.286747428434478734107892712789838446434331844097056995641477859336652243...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = f(1) * 2.798014228561519243358371276385174449737670294137200281334256087932048625...
and gamma is the Euler-Mascheroni constant A001620. (End)

A358347 a(n) is the sum of the unitary divisors of n that are squares.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 1, 26, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 50, 1, 1, 1, 1, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1

Offset: 1

Amiram Eldar, Nov 11 2022

The number of unitary divisors of n that are squares is A056624(n).

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

Cf. A056624, A077610, A078434, A247041, A268335, A350388, A351568.

Similar sequences: A033634, A034448, A035316, A358346.

f[p_, e_] := If[OddQ[e], 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, f[i,1]^f[i,2] + 1));}

a(n) >= 1 with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = p^e + 1 if e is even, and 1 otherwise.
a(n) = A034448(n)/A358346(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)) = 0.6491241554... .
Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(3*s-2). - Amiram Eldar, Jan 29 2023
a(n) = A034448(A350388(n)). - Amiram Eldar, Sep 09 2023

A071974 Numerator of rational number i/j such that Sagher map sends i/j to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jun 19 2002

The Sagher map sends Product p_i^e_i / Product q_i^f_i (p_i and q_i being distinct primes) to Product p_i^(2e_i) * Product q_i^(2f_i-1). This is multiplicative.

			The Sagher map sends the following fractions to 1, 2, 3, 4, ...: 1/1, 1/2, 1/3, 2/1, 1/5, 1/6, 1/7, 1/4, 3/1, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David M. Bradley, Counting the Positive Rationals: A Brief Survey, arXiv:math/0509025 [math.HO], 2005.
Gerald Freilich, A denumerability formula for the rationals, Amer. Math. Monthly, Nov 1965, pp. 1013-1014.
Kevin McCrimmon, Enumeration of the positive rationals, Amer. Math. Monthly, Nov 1960, p. 868.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
Index entries for sequences related to enumerating the rationals

Cf. A071975. Differs from A056622 at a(32).
Cf. A000290, A027748, A077591, A124010, A350388.
For other bijective mappings from integers to positive rationals see A002487, A020652/A020653, A038568/A038569, A229994/A077610, A295515.
Cf. A307868.

a071974 n = product $ zipWith (^) (a027748_row n) $
   map (\e -> (1 - e `mod` 2) * e `div` 2) $ a124010_row n
-- Reinhard Zumkeller, Jun 15 2012

f[{p_, a_}] := If[EvenQ[a], p^(a/2), 1]; a[n_] := Times@@(f/@FactorInteger[n])
Table[Sqrt@ SelectFirst[Reverse@ Divisors@ n, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 104}] (* Michael De Vlieger, Dec 06 2018 *)

a(n)=local(v=factor(n)~); prod(k=1,length(v),if(v[2,k]%2,1,v[1,k]^(v[2,k]/2)))

from math import prod
from sympy import factorint
def A071974(n): return prod(p**(e>>1) for p, e in factorint(n).items() if e&1^1) # Chai Wah Wu, Jul 27 2024

If n=Product p_i^e_i, then a_n=Product p_i^f(e_i), where f(n)=n/2 if n is even and f(n)=0 if n is odd. - Reiner Martin, Jul 08 2002
a(n^2) = n, A071975(n^2) = 1, cf. A000290; a(2*(2*n-1)^2) = 2*n+1, A071975(2*(2*n-1)^2) = 2, cf. A077591. - Reinhard Zumkeller, Jul 10 2011
From Amiram Eldar, Nov 02 2023, Jul 26 2024: (Start)
a(n) = sqrt(A350388(n)) (square root of largest unitary divisor of n that is a square).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-1)). (End)
From Vaclav Kotesovec, May 05 2025: (Start)
Let f(s) = Product_{p prime} (1 -  (p^s + p)/((p^s + 1)*p^(2*s))).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A307868 = Product_{p prime} (1 - 2/(p*(1+p))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321...
f'(1) = f(1) * Sum_{p prime} (5*p+3)*log(p) / ((p+1)*(p^2+p-2)) = f(1) * 2.1244279471327068377850377690765768532203174482128717024402373817115555...
and gamma is the Euler-Mascheroni constant A001620. (End)

More terms from Reiner Martin, Jul 08 2002

Additional references supplied by Kevin Ryde added by N. J. A. Sloane, May 31 2012

A322857 a(1) = 1; a(n) = sum of exponential unitary divisors of n for n > 1.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 18, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67

Offset: 1

Amiram Eldar, Dec 29 2018

The exponential unitary (or e-unitary) divisors of n = Product p(i)^a(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of a(i).

Ivan Neretin, Table of n, a(n) for n = 1..10000
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Cf. A361255, A051377, A077610, A278908 (number of exponential unitary divisors).

f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ f @@@ FactorInteger[n]; Array[eusigma, 100]

ff(p, e) = sumdiv(e, d, if (gcd(d, e/d)==1, p^d));
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = ff(f[k,1], f[k,2]); f[k,2] = 1); factorback(f); \\ Michel Marcus, Dec 29 2018

Multiplicative with a(p^e) = Sum_{d|e, gcd(d, e/d)==1} p^d.

cp's OEIS Frontend

A049599 Number of (1+e)-divisors of n: if n = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of n.

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A165430 Table T(n,m) read by rows: the greatest common unitary divisor of n and m, n>=1, 1<=m<=n.

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A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

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A068068 Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d,n/d)=1.

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A206778 Irregular triangle in which n-th row lists squarefree divisors (A005117) of n.

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A384046 Triangle in which the n-th row gives the numbers from 1 to n whose largest divisor that is a unitary divisor of n is 1.

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A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

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