A306016 -id:A306016 - cp's OEIS frontend

A092089 Number of odd-length palindromes among the k-tuples of partial quotients of the continued fraction expansions of n/r, r = 1, ..., n.

1, 2, 3, 4, 3, 6, 3, 8, 5, 6, 3, 12, 3, 6, 9, 12, 3, 10, 3, 12, 9, 6, 3, 24, 5, 6, 7, 12, 3, 18, 3, 16, 9, 6, 9, 20, 3, 6, 9, 24, 3, 18, 3, 12, 15, 6, 3, 36, 5, 10, 9, 12, 3, 14, 9, 24, 9, 6, 3, 36, 3, 6, 15, 20, 9, 18, 3, 12, 9, 18, 3, 40, 3, 6, 15, 12, 9, 18, 3, 36, 9, 6, 3, 36, 9, 6, 9, 24, 3

Offset: 1

John W. Layman, Mar 29 2004

Suggested by R. K. Guy, Mar 26 2004
From Jianing Song, Mar 24 2019: (Start)
a(n) is also the number of inequivalent residue classes modulo n where the equivalence relation is defined as [a] ~ [b] (mod n) if and only if there exists some k such that gcd(k, n) = 1 and that a*k^2 == b (mod n). For example, for n = 16, the inequivalent residue classes are {[0], [1], [2], [3], [4], [5], [6], [7], [8], [10], [12], [14]}, so a(16) = 14.
Proof: let S(n) be the set of inequivalent residue classes modulo n, so our goal is to show that |S(n)| = a(n) for all n. By the Chinese Remainder Theorem, if gcd(s, t) = 1, then [a] ~ [b] (mod s*t) if and only if [a] ~ [b] (mod s) and [a] ~ [b] (mod t), so there is a one-to-one correspondence between S(s*t) and S(s) X S(t), that is, |S(n)| is multiplicative. It is obvious that |S(p^e)| = a(p^e), so |S(n)| = a(n) for all n. (End)

			[1, 2, 1, 2, 1] <-> 1+1/(2+1/(1+1/(2+1/1))) = 15/11 is one of the nine palindromes {[15], [5], [3, 1, 3], [3], [1, 1, 1], [1, 2, 1, 2, 1], [1, 3, 1], [1, 13, 1], [1]} among the continued fraction expansions of 15/r for r = 1..15. Thus a(15)=9.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Agustina Czenky, Diagramatics for cyclic pointed fusion categories, arXiv:2404.08084 [math.QA], 2024. See p. 15.
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110 (see (37) in Corollary 15, p. 108).
Eric Weisstein's World of Mathematics, Partial quotient.

Cf. A001620, A201994, A306016.

Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, OddQ@ p, 2 e + 1, And[p == 2, e == 1], 2, True, 4 (e - 1)]], {n, 89}] (* Michael De Vlieger, Sep 11 2017 *)

a(n) = if (n % 2, numdiv(n^2), if (n/2 % 2, 2*numdiv((n/2)^2), val = valuation(n, 2); 4*(val-1)*numdiv((n/2^val)^2))); \\ Michel Marcus, Jun 26 2014

(define (A092089 n) (cond ((= 1 n) n) ((zero? (modulo n 4)) (* 4 (+ -1 (A067029 n)) (A092089 (A000265 n)))) ((even? n) (* 2 (A092089 (/ n 2)))) (else (* (+ 1 (* 2 (A067029 n))) (A092089 (A028234 n)))))) ;; Antti Karttunen, Sep 11 2017

Conjecture: Let n = (2^k0)*(p1^k1)*(p2^k2)*...*(pm^km) be the prime factorization of n where p1, p2, ..., pm are distinct primes. Then a(n) is multiplicative and is given by a(n) = f(k0)*g(k1)*g(k2)*...*g(km), where f(0) = 1, f(1) = 2, f(k) = 4(k-1) if k>1 and g(k) = 2k+1 (This has been verified for n = 1-10000.) [Corrected by Jianing Song, Mar 24 2019]
Multiplicative with a(p^e) = 2e+1 if p is odd; a(2) = 2, a(2^e)= 4*(e-1), if e > 1. - Michel Marcus, Jun 26 2014
Dirichlet g.f.: zeta(s)^3/zeta(2*s) * (1 - 1/2^s + 1/2^(2*s-1)). - Jianing Song, Mar 25 2019 [corrected by Amiram Eldar, Dec 18 2023]
Sum_{k=1..n} a(k) ~ (3/Pi^2) * n * (log(n^2) + c_1 * log(n) + c_2), where c_1 = 6 * gamma - 2 - log(2) - 4*zeta'(2)/zeta(2) = 3.04999078122..., gamma is Euler's constant (A001620), c_2 = 2 - 6 * gamma + 6 * gamma^2 + log(2) - 3 * gamma * log(2) + 3*log(2)^2/2 - 6 *gamma_1 + 4*zeta'(2)/zeta(2) + (2 * log(2) - 12 * gamma) * zeta'(2)/zeta(2) + 8 * (zeta'(2)/zeta(2))^2 - 4 * zeta''(2)/zeta(2) = -0.1743888255..., and gamma_1 is the 1st Stieltjes constant (A082633). - Amiram Eldar, Dec 18 2023

A293227 a(n) is the number of proper divisors of n that are squarefree.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 7, 1, 2, 3, 3, 3, 4, 1, 3, 3, 4, 1, 7, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 8, 1, 3, 4, 2, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 4, 4, 3, 7, 1, 4, 2, 3, 1, 8, 3, 3, 3, 4, 1, 8, 3, 4, 3, 3, 3, 4, 1, 4, 4, 4, 1, 7, 1, 4, 7

Offset: 1

Antti Karttunen, Oct 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A005117, A008683, A008966, A034444, A293228, A293234.

Cf. A001620, A306016.

with(numtheory): seq(coeff(series(add(mobius(k)^2*x^(2*k)/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Oct 28 2018

Table[Count[Most[Divisors[n]],?SquareFreeQ],{n,110}] (* _Harvey P. Dale, Jun 15 2021 *)
a[n_] := 2^PrimeNu[n] - Boole[SquareFreeQ[n]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

PARI
```
A293227(n) = sumdiv(n, d, (d
				
```

a(n) = Sum_{d|n, dA008966(d).
a(n) = A034444(n) - A008966(n).
a(n) = 2^A001221(n) - A008683(n)^2 = 2^omega(n) - mu(n)^2.
G.f.: Sum_{k>=1} mu(k)^2*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Oct 28 2018
Sum_{k=1..n} a(k) ~ (6/Pi^2)*n*(log(n) + 2*(gamma - 1 - zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.

Original entry on oeis.org

1, 3, 11, 61, 11, 82, 171, 1951, 26133, 13424, 41273, 716656, 13871719, 4700888, 9548741, 222854273, 112857219, 3310041496, 20075905417, 628822761157, 19239404599, 9709078632, 1959180271, 73097429088, 147378388979, 445594718515, 18404305970657, 3089336006908, 133763418792581

Offset: 1

Ilya Gutkovskiy, May 26 2022

			1, 3/2, 11/6, 61/30, 11/5, 82/35, 171/70, 1951/770, 26133/10010, 13424/5005, 41273/15015, ...

Robert Israel, Table of n, a(n) for n = 1..1433
Sebastian Zuniga Alterman, Explicit averages of square-free supported functions: to the edge of the convolution method, Colloquium Mathematicum, Vol. 168 (2022), pp. 1-23; arXiv preprint, arXiv:2003.05887 [math.NT], 2020.
Olivier Ramaré, Explicit average orders: news and problems, Banach Center Publications, Vol. 118 (2019), pp. 153-176.

Cf. A001008, A005117, A024451, A072980, A096795, A106830, A354418 (denominators).

Cf. A001620, A059956, A306016.

s:= 0: R:= NULL: count:= 0:
for x from 1 while count < 40 do
  if numtheory:-issqrfree(x) then
    s:= s + 1/x;
    v:= numer(s);
    R:= R, v;
    count:= count+1;
  fi;
od:
R; # Robert Israel, Mar 05 2023

Accumulate[1/Select[Range[43], SquareFreeQ]] // Numerator

a(n) = my(i=0, s=0); for(x=1, oo, if(core(x)==x, s+=1/x; i++; if(i==n, return(numerator(s))))) \\ Felix Fröhlich, May 26 2022

a(n)/A354418(n) ~ (6/Pi^2) * (log(n) + c) + O*(1.044/sqrt(n)), where f = O*(g) means |f| <= g and c = gamma + 2 * Sum_{p prime} log(p)/(p^2-1) = A001620 + 2 * A306016 = 1.71713765109059847340... (Ramaré, 2019; Alterman, 2022). - Amiram Eldar, Oct 29 2022

A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

Original entry on oeis.org

1, 5, 13, 25, 41, 65, 85, 121, 157, 205, 221, 325, 313, 425, 533, 569, 545, 785, 685, 1025, 1105, 1105, 1013, 1573, 1441, 1565, 1777, 2125, 1625, 2665, 1861, 2617, 2873, 2725, 3485, 3925, 2665, 3425, 4069, 4961, 3281, 5525, 3613, 5525, 6437, 5065, 4325, 7397, 5965

Offset: 1

Seiichi Manyama, May 21 2024

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Row sums of A350900.
Cf. A057660, A350156.
Cf. A001620, A002117, A013661, A073002, A244115, A306016.

f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 27 2024 *)

a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2);} \\ Amiram Eldar, May 27 2024

a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).
From Amiram Eldar, May 27 2024: (Start)
Multiplicative with a(p^e) = (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2.
Dirichlet g.f.: zeta(s) * zeta(s-2)^2 / zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A234307 a(n) = Sum_{i=1..n} gcd(2*n-i, i).

Original entry on oeis.org

1, 3, 6, 8, 11, 17, 16, 20, 27, 31, 26, 44, 31, 45, 60, 48, 41, 75, 46, 80, 87, 73, 56, 108, 85, 87, 108, 116, 71, 165, 76, 112, 141, 115, 158, 192, 91, 129, 168, 196, 101, 239, 106, 188, 261, 157, 116, 256, 175, 235, 222, 224, 131, 297, 256, 284, 249, 199

Offset: 1

Wesley Ivan Hurt, Dec 22 2013

Sum of the GCD's of the smallest and largest parts in the partitions of 2n into exactly two parts.

			a(6) = 17; the partitions of 2(6) = 12 into two parts are: (11,1),(10,2),(9,3),(8,4),(7,5),(6,6). Then a(6) = gcd(11,1) + gcd(10,2) + gcd(9,3) + gcd(8,4) + gcd(7,5) + gcd(6,6) = 1 + 2 + 3 + 4 + 1 + 6 = 17.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions.

Cf. A001105 (sum of parts), A002378 (differences of parts).

Cf. A001620, A018804, A306016.

A234307:=n->add( gcd(2*n-i, i), i=1..n); seq(A234307(n), n=1..100);

Table[Sum[GCD[2n - i, i], {i, n}], {n, 100}]
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := (Times @@ f @@@ FactorInteger[2*n] - n)/2; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sum(i=1, n, gcd(i, 2*n-i)); \\ Michel Marcus, Dec 23 2013

a(n) = {my(f = factor(2*n)); (prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; p^(e-1)*(p+e*(p-1))) - n)/2;} \\ Amiram Eldar, Mar 30 2024

a(n) = Sum_{i=1..n} gcd(2*n-i, i).
a(n) = (A018804(2*n)-n)/2. - Sebastian Karlsson, Oct 03 2021
Conjecture: a(n) = (1/4)*Sum_{k = 1..4*n} (-1)^k *gcd(k, 8*n). - Peter Bala, Jan 01 2024
Sum_{k=1..n} a(k) ~ (Pi^2/4)*n^2 * (log(n) + 2*gamma - 1/2 + log(2)/6 - Pi^2/16 - zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024

A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 5, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 5, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 9, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 5, 10, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

The number of unitary divisors of n that are squares is A056624(n) and their sum is A358347(n).

The unitary analog of A069290.

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

Cf. A001620, A056624, A069290, A306016, A358347.

f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 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]/2) + 1)); }

a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} sqrt(d).
Multiplicative with a(p^e) = p^(e/2) + 1 if e is even, and 1 if e is odd.
Dirichlet g.f.: zeta(s)*zeta(2*s-1)/zeta(3*s-1).
Sum_{k=1..n} a(k) ~ (3*n/Pi^2)*(log(n) + 3*gamma - 1 - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 3, 8, 1, 16, 1, 18, 1, 4, 5, 22, 1, 4, 3, 2, 3, 28, 2, 30, 1, 20, 4, 24, 1, 36, 9, 8, 1, 40, 1, 42, 5, 8, 11, 46, 1, 6, 1, 32, 3, 52, 1, 8, 3, 4, 7, 58, 1, 60, 15, 4, 1, 48, 5, 66, 2, 44, 6, 70, 1, 72, 9, 8, 9, 60, 2, 78

Offset: 1

Amiram Eldar, Jun 01 2024

Numbers that are unitarily divided by n are numbers k such that n is a unitary divisor of k, or equivalently, numbers of the form m*n, with gcd(m, n) = 1.

			Fractions begin with: 1, 1/4, 2/9, 1/8, 4/25, 1/18, 6/49, 1/16, 2/27, 1/25, 10/121, 1/36, ...
For n = 2, the numbers that are unitarily divided by 2 are the numbers of the form 4*k+2 whose asymptotic density is 1/4. Therefore a(2) = numerator(1/4) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A000010, A003277, A034444, A064549, A076512, A077610, A109395, A173557, A373319(denominators), A373320.
Cf. A001620, A013661, A306016.
Numbers that are unitarily divided by k: A000027 (k=1), A016825 (k=2), A016051 (k=3), A017113 (k=4), A051062 (k=8), A051063 (k=9).

a[n_] := Numerator[EulerPhi[n]/n^2]; Array[a, 100]

PARI
```
a(n) = numerator(eulerphi(n)/n^2);
```

a(n) = 1 if and only if n is in A090778.
a(n) = A000010(n) if and only if n is a cyclic number (A003277).
Let f(n) = a(n)/A373319(n). Then:
f(n) = A000010(n)/n^2 = A076512(n)/(n*A109395(n)).
f(n) = A173557(n)/A064549(n).
f(n) is multiplicative with f(p^e) = (1 - 1/p)/p^e.
Sum_{k=1..n} f(k) = (log(n) + gamma - zeta'(2)/zeta(2)) / zeta(2), where gamma is Euler's constant (A001620).

A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 16 2025

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

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

The unitary analog of A035218.
Cf. A007310, A034444, A065330, A385045.
Cf. A001620, A013661, A306016.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
a(n) = A034444(n)/A382488(n).
a(n) <= A034444(n), with equality if and only if n is 5-rough.
a(n) <= A035218(n).
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A045771 Number of similar sublattices of index n^2 in root lattice D_4.

Original entry on oeis.org

1, 1, 8, 1, 12, 8, 16, 1, 41, 12, 24, 8, 28, 16, 96, 1, 36, 41, 40, 12, 128, 24, 48, 8, 97, 28, 176, 16, 60, 96, 64, 1, 192, 36, 192, 41, 76, 40, 224, 12, 84, 128, 88, 24, 492, 48, 96, 8, 177, 97, 288, 28, 108, 176, 288, 16, 320, 60, 120, 96, 124, 64, 656, 1

Offset: 1

Michael Baake (baake(AT)miles.math.ualberta.ca)

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math., Vol. 51, No. 6 (1999), 1258-1276.
Michael Baake and Peter Zeiner, "Similar Sublattices", Ch. 3.5 in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity, Cambridge, 2017, see page 105.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Index entries for sequences related to D_4 lattice.
Index entries for sequences related to sublattices.

Cf. A001620, A013662, A035292, A261506, A306016.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[1 <= p <= 2, 1, (e + 1) p^e + (2 (1 + (e p - e - 1)*p^e))/((p - 1)^2)]] &, 64] (*  Michael De Vlieger, Mar 02 2018 *)

fp(p, e) = if (p % 2, (e+1)*p^e + 2*(1-(e+1)*p^e+e*p^(e+1))/(p-1)^2, 1);
a(n) = { my(f = factor(n)); prod(i=1, #f~, fp(f[i, 1], f[i, 2]));} \\ Michel Marcus, Mar 03 2014

Multiplicative with a(2^p) = 1, a(p^e) = (e+1)*p^e + (2*(1+(e*p-e-1)*p^e))/((p-1)^2), p>2. - Christian G. Bower, May 21 2005
From Amiram Eldar, May 26 2025: (Start)
Dirichlet g.f.: (zeta(s-1)^2 * zeta(s)^2 / zeta(2*s)) * (1 - 1/2^(s-1))^2/(1 + 1/2^s).
Sum_{k=1..n} a(k) ~ (n^2/4)*(log(n) + 2*gamma - 1/2 + 11*log(2)/5 + 2*zeta'(2)/zeta(2) - 2*zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)

More terms from Michel Marcus, Mar 03 2014

A064368 Number of 2 X 2 symmetric singular matrices with entries from {0,...,n}.

Original entry on oeis.org

1, 4, 7, 10, 15, 18, 21, 24, 29, 36, 39, 42, 47, 50, 53, 56, 65, 68, 75, 78, 83, 86, 89, 92, 97, 108, 111, 118, 123, 126, 129, 132, 141, 144, 147, 150, 163, 166, 169, 172, 177, 180, 183, 186, 191, 198, 201, 204, 213, 228, 239, 242, 247, 250, 257, 260, 265, 268

Offset: 0

Vladeta Jovovic, Sep 27 2001

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000010, A000188, A064276, A059306, A062801, A059976, A039623.

Cf. A001620, A306016.

f[p_, e_] := p^Floor[e/2]; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 100}, 1 + Range[0, max] + 2 * Accumulate[Array[a, max + 1, 0]]] (* Amiram Eldar, Nov 07 2024 *)

a(n) = n + 1 + 2*sum(k=1, n, sumdiv(k, d, issquare(d)*eulerphi(sqrtint(d)))) \\ Michel Marcus, Jun 17 2013

a(n) = n + 1 + 2*Sum_{k=1..n} Sum_{d^2|k} phi(d), where phi = Euler totient function A000010.

a(n) ~ (n/zeta(2)) * (log(n) + 3*gamma - 1 + zeta(2) - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 07 2024

cp's OEIS Frontend

A092089 Number of odd-length palindromes among the k-tuples of partial quotients of the continued fraction expansions of n/r, r = 1, ..., n.

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A293227 a(n) is the number of proper divisors of n that are squarefree.

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A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.

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A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

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PARI

PARI

Formula

A234307 a(n) = Sum_{i=1..n} gcd(2*n-i, i).

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PARI

PARI

Formula

A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares.

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Formula

A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

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