A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

1, 1, 2, 3, 4, 2, 6, 5, 7, 4, 10, 6, 12, 6, 8, 11, 16, 7, 18, 12, 12, 10, 22, 10, 21, 12, 20, 18, 28, 8, 30, 21, 20, 16, 24, 21, 36, 18, 24, 20, 40, 12, 42, 30, 28, 22, 46, 22, 43, 21, 32, 36, 52, 20, 40, 30, 36, 28, 58, 24, 60, 30, 42, 43, 48, 20, 66, 48, 44, 24, 70, 35

Offset

1,3

Comments

For more information see the Comments in A061020.

a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a perfect square. For example, a(12) = 6 because |{1,4,5,7,8,11}|=6 and the respective GCDs with 12 are 1,4,1,1,4,1, which are squares. - Geoffrey Critzer, Feb 16 2015

If m is squarefree (A005117), then a(m) = A000010(m) where A000010 is the Euler totient function. - Michel Marcus, Nov 08 2017

Also it appears that the primorials (A002110) is the sequence of indices of minimum records for a(n)/n, and these records are A038110(n)/A060753(n). - Michel Marcus, Nov 09 2017

Also called rho(n). When rho(n) | n, then n is called k-imperfect, with k = n/rho(n), cf. A127724. - M. F. Hasler, Feb 13 2020

References

P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 25.

Links

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

László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Formula

a(n) = abs(A061020(n)).

a(n) = n*Sum_{d|n} lambda(d)/d, where lambda(n) is A008836(n). - Enrique Pérez Herrero, Sep 23 2012

Dirichlet g.f.: zeta(s - 1)*zeta(2*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015

From Michel Marcus, Nov 05 2017: (Start)

a(2^n) = A001045(n+1);

a(3^n) = A015518(n+1);

a(5^n) = A015531(n+1);

a(7^n) = A015552(n+1);

a(11^n) = A015592(n+1). (End)

a(p^k) = p^k - a(p^(k - 1)) for k > 0 and prime p. - David A. Corneth, Nov 09 2017

a(n) = Sum_{d|n, d is a perfect square} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018

a(p^k) = A071324(p^k), for k >= 0 and prime p. - Michel Marcus, Aug 11 2018

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 30. - Vaclav Kotesovec, Feb 07 2019

G.f.: Sum_{k>=1} lambda(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 23 2019

a(n) = Sum_{i=1..n} A010052(gcd(n,i)). - Ridouane Oudra, Nov 24 2019

a(p^k) = round(p^(k+1)/(p+1)). - M. F. Hasler, Feb 13 2020

Maple

a:= n-> mul(add(i[1]^(i[2]-j)*(-1)^j, j=0..i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2017

Mathematica

Table[Length[Select[Range[n], IntegerQ[GCD[n, #]^(1/2)] &]], {n, 72}] (* Geoffrey Critzer, Feb 16 2015 *)
a[n_] := n*DivisorSum[n, LiouvilleLambda[#]/#&]; Array[a, 72] (* Jean-François Alcover, Dec 04 2017, after Enrique Pérez Herrero *)
f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

Prog

(Haskell)
a206369 n = product $
   zipWith h (a027748_row n) (map toInteger $ a124010_row n) where
           h p e = sum $ take (fromInteger e + 1) $
                         iterate ((* p) . negate) (1 - 2 * (e `mod` 2))
-- Reinhard Zumkeller, Feb 08 2012
(PARI) a(n) = sum(k=1, n, issquare(gcd(n, k)));
(PARI) ak(p, e)=my(s=1); for(i=1, e, s=s*p + (-1)^i); s
a(n)=my(f=factor(n)); prod(i=1, #f~, ak(f[i, 1], f[i, 2])) \\ Charles R Greathouse IV, Dec 27 2016
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d) * issquare(d)); \\ Daniel Suteu, Jun 27 2018
(PARI) apply( {A206369(n)=vecprod([f[1]^(f[2]+1)\/(f[1]+1)|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Feb 13 2020
(Python)
from math import prod
from sympy import factorint
def A206369(n): return prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1), p+1)) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 05 2024

Crossrefs

Cf. A061020, A206368.

Cf. A027748 row, A124010, A206475 (first differences).

Cf. A078429.

Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect).

Sequence in context: A286365 A345061 A061020 * A152958 A278963 A308085

Adjacent sequences: A206366 A206367 A206368 * A206370 A206371 A206372

Keyword

nonn,mult

Author

N. J. A. Sloane, Feb 06 2012

Status

approved