A300717 -id:A300717 - cp's OEIS frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000

Offset: 1

N. J. A. Sloane

Numbers of the form a^2*b^3, a >= 1, b >= 1.
In other words, if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
Numbers n such that Sum_{d|n} phi(d)*phi(n/d)*mu(d) > 0; places of nonzero A300717. - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30 2011
Complement of A052485. - Reinhard Zumkeller, Sep 16 2011
The number of terms less than or equal to 10^k beginning with k = 0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, ... . - Robert G. Wilson v, Aug 15 2014
a(m) mod prime(n) > 0 for m < A258599(n); a(A258599(n)) = A001248(n) = prime(n)^2. - Reinhard Zumkeller, Jun 06 2015
From Des MacHale, Mar 07 2021: (Start)
A number m is powerful if and only if |R/Z(R)| = m, for some finite non-commutative ring R.
A number m is powerful if and only if |G/Z(G)| = m, for some finite nilpotent class two group G (Reference Aine Nishe). (End)
Numbers n such that Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)) > 0. - Richard L. Ollerton, May 09 2021

			1 is a term because for every prime p that divides 1, p^2 also divides 1.
2 is not a term since 2 divides 2 but 2^2 does not.
4 is a term because 2 is the only prime that divides 4 and 2^2 does divide 4. - _N. J. A. Sloane_, Jan 16 2022

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
Richard A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
Aine NiShe, Commutativity and Generalisations in Finite Groups, Ph.D. Thesis, University College Cork, 2000.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 9.1 Potente Zahlen, pp. 241-247.
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).
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 54, exercise 10 (in the third edition 2015, p. 63, exercise 70).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..5000 from G. C. Greubel)
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Valentin Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers II, Journal of the London Mathematical Society 71:1 (2005), pp. 69-84.
Chris K. Caldwell, Powerful Numbers.
Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1.
Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485 [math.NT], 2025.
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3.
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102. [Zahlen i-ter Art, p. 101]
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
K. Schneider, PlanetMath.org, Squarefull Number.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, The distribution of square-full integers, Ark. Mat., Volume 11, Number 1-2 (1973), 195-201.
Eric Weisstein's World of Mathematics, Powerful Number.
Eric Weisstein's World of Mathematics, Squareful.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Disjoint union of A062503 and A320966.
Cf. A007532 (Powerful numbers, definition (2)), A005934, A005188, A003321, A014576, A023052 (Powerful numbers, definition (3)), A046074, A013929, A076871, A258599, A001248, A112526, A168363, A224866, A261883, A300717.
Cf. A052485 (complement), A076446 (first differences), A376361, A376362.

a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]
(* Harvey P. Dale, Sep 18 2011; modified by Robert G. Wilson v, Aug 11 2014 *)
max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)
nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011

list(lim,mn=2)=my(v=List(),t); for(m=1,sqrtnint(lim\1,3), t=m^3; for(n=1,sqrtint(lim\t), listput(v,t*n^2))); Set(v) \\ Charles R Greathouse IV, Jul 31 2011; edited Sep 22 2015

is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012

from sympy import factorint
A001694 = [1]+[n for n in range(2,10**6) if min(factorint(n).values()) > 1]
# Chai Wah Wu, Aug 14 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A001694(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

sloane.A001694.list(54) # Peter Luschny, Feb 08 2015

A112526(a(n)) = 1. - Reinhard Zumkeller, Sep 16 2011
Bateman & Grosswald prove that there are zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}) terms up to x; see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6). - Ivan Neretin, Aug 30 2015
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1/2 (Golomb, 1970). - Amiram Eldar, Oct 02 2022

More terms from Henry Bottomley, Mar 16 2000

Definition expanded by Jonathan Sondow, Jan 03 2016

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

A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 2, 1, 7, 6, 1, 1, 1, 1, 4, 8, 11, 1, 2, 1, 13, 1, 6, 1, 14, 1, 1, 12, 17, 10, 0, 1, 19, 14, 4, 1, 20, 1, 10, 4, 23, 1, 2, 1, 1, 18, 12, 1, 1, 14, 6, 20, 29, 1, 8, 1, 31, 6, 1, 16, 32, 1, 16, 24, 34, 1, 0, 1, 37, 2, 18, 16, 38, 1, 4, 1, 41, 1, 12, 20, 43, 30, 10, 1, 4, 18, 22, 32, 47

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

Conjecture 1: After the initial zero, the positions of other zeros is given by A036785.

Conjecture 2: No negative terms. Checked up to n = 2^24.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A008683, A036785, A342001, A347232 [= a(A276086(n))].

Cf. also A300251, A300717, A347234, A347235, A347395.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A346485(n) = sumdiv(n,d,moebius(n/d)*A342001(d));

a(n) = Sum_{d|n} A008683(n/d) * A342001(d).
Dirichlet g.f.: Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 08 2022
Sum_{k=1..n} a(k) ~ c * A065464 * n^2 / 2, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, Mar 04 2023

A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 25, 11, 1, 1, 7, 1, 1, 1, 363, 1, 11, 1, 7, 1, 1, 1, 25, 19, 1, 61, 7, 1, 1, 1, 1335, 1, 1, 1, 77, 1, 1, 1, 25, 1, 1, 1, 7, 11, 1, 1, 363, 27, 19, 1, 7, 1, 61, 1, 25, 1, 1, 1, 7, 1, 1, 11, 9923, 1, 1, 1, 7, 1, 1, 1, 275, 1, 1, 19, 7, 1, 1, 1, 363, 1363, 1, 1, 7, 1, 1, 1, 25, 1, 11, 1, 7, 1, 1, 1, 1335, 1, 27, 11, 133, 1, 1, 1, 25, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Multiplicative because A003557 is.

No negative terms among the first 2^20 terms. Is the sequence nonnegative?

Antti Karttunen, Table of n, a(n) for n = 1..65537
Wikipedia, Dirichlet convolution

Cf. A003557, A046644 (denominators).

Cf. also A300717, A300719, A318317.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]--); factorback(f); }; \\ From A003557
A317935aux(n) = if(1==n,n,(A003557(n)-sumdiv(n,d,if((d>1)&&(dA317935aux(d)*A317935aux(n/d),0)))/2);
A317935(n) = numerator(A317935aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003557(n) - Sum_{d|n, d>1, d 1.

A300718 Möbius transform of A010848, number of numbers k <= n such that at least one prime factor of n is not a prime factor of k.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 2, 4, 4, 10, 4, 12, 6, 8, 4, 16, 6, 18, 8, 12, 10, 22, 8, 16, 12, 12, 12, 28, 8, 30, 8, 20, 16, 24, 10, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 36, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 16, 48, 20, 66, 32, 44, 24, 70, 20, 72, 36, 40, 36, 60, 24, 78, 32, 36, 40, 82, 24, 64, 42, 56, 40, 88, 24, 72

Offset: 1

Antti Karttunen, Mar 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A008683, A003557, A010848, A300717, A300720.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557
A010848(n) = (n - A003557(n));
A300718(n) = sumdiv(n,d,moebius(n/d)*A010848(d));

a(n) = Sum_{d|n} A008683(n/d)*A010848(d).
a(n) = A000010(n) - A300717(n).
a(n) = A010848(n) - A300720(n).

A300719 Difference between A003557 (n divided by largest squarefree divisor of n) and its Möbius transform.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 1, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 16, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 5, 2, 1, 1, 1, 8, 9, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 6, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Mar 12 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A008683, A300717.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557
A300719(n) = -sumdiv(n,d,(dA003557(d));

a(n) = -Sum_{d|n, dA008683(n/d)*A003557(d).

a(n) = A003557(n) - A300717(n).

A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 5, 4, 2, 8, 2, 4, 4, 16, 2, 10, 2, 8, 4, 4, 2, 16, 7, 4, 14, 8, 2, 8, 2, 32, 4, 4, 4, 20, 2, 4, 4, 16, 2, 8, 2, 8, 10, 4, 2, 32, 9, 14, 4, 8, 2, 28, 4, 16, 4, 4, 2, 16, 2, 4, 10, 64, 4, 8, 2, 8, 4, 8, 2, 40, 2, 4, 14, 8, 4, 8, 2, 32, 41, 4, 2, 16, 4

Offset: 1

Ilya Gutkovskiy, Oct 17 2019

Inverse Moebius transform of A003557.

Dirichlet convolution of A000203 with A097945.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A000010, A000079 (fixed points), A000203, A003557, A007947, A008683, A098108 (parity of a(n)), A191750, A300717, A335032.

Table[Sum[d/Last[Select[Divisors[d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 85}]
Table[Sum[MoebiusMu[n/d] EulerPhi[n/d] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 85}]
f[p_, e_] := 1 + (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

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

G.f.: Sum_{k>=1} (k / rad(k)) * x^k / (1 - x^k), where rad = A007947.
a(n) = Sum_{d|n} A003557(d).
a(n) = Sum_{d|n} mu(n/d) * phi(n/d) * sigma(d), where mu = A008683, phi = A000010 and sigma = A000203.
a(p) = 2, where p is prime.
From Vaclav Kotesovec, Jun 20 2020: (Start)
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 1/(p^s - p)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) + p^(-s)). (End)
Multiplicative with a(p^e) = 1 + (p^e-1)/(p-1). - Amiram Eldar, Oct 14 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*sigma(gcd(n,k)).
a(n) = Sum_{k=1..n} mu(gcd(n,k))*sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A349379 Möbius transform of A057521 (powerful part of n).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 18, 0, 0, 0, 0, 16, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative with a(p^e) = 0 if e = 1, p^2 - 1 if e = 2 and p^e - p^(e-1) otherwise. - Amiram Eldar, Nov 18 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A008683, A057521, A349441.

Cf. also A300717.

f[p_, e_] := Which[e > 2, p^e - p^(e - 1), e == 2, p^2 - 1, e == 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349379(n) = sumdiv(n,d,moebius(n/d)*A057521(d));

from math import prod
from sympy import factorint
def A349379(n): return prod(0 if e==1 else p**e - (1 if e==2 else p**(e-1)) for p,e in factorint(n).items()) # Chai Wah Wu, Nov 14 2022

a(n) = Sum_{d|n} A008683(n/d) * A057521(d).

a(n) = Sum_{d|n} A000010(n/d) * A349441(d).

cp's OEIS Frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

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

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A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

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A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

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A300718 Möbius transform of A010848, number of numbers k <= n such that at least one prime factor of n is not a prime factor of k.

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A300719 Difference between A003557 (n divided by largest squarefree divisor of n) and its Möbius transform.

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