A322327 -id:A322327 - cp's OEIS frontend

A355837 Dirichlet inverse of A322327.

1, -2, -2, 0, -2, 4, -2, 2, 0, 4, -2, 0, -2, 4, 4, 0, -2, 0, -2, 0, 4, 4, -2, -4, 0, 4, 2, 0, -2, -8, -2, -2, 4, 4, 4, 0, -2, 4, 4, -4, -2, -8, -2, 0, 0, 4, -2, 0, 0, 0, 4, 0, -2, -4, 4, -4, 4, 4, -2, 0, -2, 4, 0, 0, 4, -8, -2, 0, 4, -8, -2, 0, -2, 4, 0, 0, 4, -8, -2, 0, 0, 4, -2, 0, 4, 4, 4, -4, -2, 0, 4, 0, 4, 4, 4, 4

Offset: 1

Antti Karttunen, Jul 19 2022, based on Werner Schulte's comment in A322327

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A322327.

f[p_, e_] := 2 * (-1)^((e + 1)/2) * Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 19 2022 *)

A355837(n) = factorback(apply(e -> 2*(e%2)*((-1)^((1+e)/2)), factor(n)[, 2]));

Multiplicative with a(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e>0.
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA322327(n/d) * a(d).
Dirichlet g.f.: zeta(4*s)/(zeta(s)^2*zeta(2*s)). - Amiram Eldar, Dec 29 2022

A370296 Inverse Moebius transform of A322327.

Original entry on oeis.org

1, 3, 3, 7, 3, 9, 3, 13, 7, 9, 3, 21, 3, 9, 9, 21, 3, 21, 3, 21, 9, 9, 3, 39, 7, 9, 13, 21, 3, 27, 3, 31, 9, 9, 9, 49, 3, 9, 9, 39, 3, 27, 3, 21, 21, 9, 3, 63, 7, 21, 9, 21, 3, 39, 9, 39, 9, 9, 3, 63, 3, 9, 21, 43, 9, 27, 3, 21, 9, 27, 3, 91, 3, 9, 21, 21, 9, 27, 3, 63, 21, 9, 3, 63

Offset: 1

Werner Schulte, Feb 14 2024

Cf. A322327, A000005, A323308.

f[p_, e_] := e^2 + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

a(n) = factorback(apply(e->1+e+e^2,factor(n)[,2]))

Multiplicative with a(p^e) = 1 + e + e^2 for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^3 * zeta(2*s) / zeta(4*s).
Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -3 and b(p^e) = 2 * (-1)^((e+1)*(e+2)/2) for prime p and e > 1.
Dirichlet convolution of A000005 and A323308.

A227291 Characteristic function of squarefree numbers squared (A062503).

Original entry on oeis.org

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

Offset: 1

Ralf Stephan, Jul 05 2013

			a(3) = 0 because 3 is not the square of a squarefree number.
a(4) = 1 because sqrt(4) = 2, a squarefree number.

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A000196, A007427, A008836, A008683, A008966, A010052, A027750, A037213, A225546, A225569, A225817, A307430, A322327, A355448.

Absolute values of A271102.

a227291 n = fromEnum $ (sum $ zipWith (*) mds (reverse mds)) == 1
   where mds = a225817_row n
-- Reinhard Zumkeller, Jul 30 2013, Jul 07 2013

A227291 := proc(n)
    local pe;
    if n = 0 then
        1;
    else
        for pe in ifactors(n)[2] do
            if op(2,pe) <> 2 then
                return 0 ;
            end if;
        end do:
    end if;
    1 ;
end proc:
seq(A227291(n),n=1..100) ; # R. J. Mathar, Feb 07 2023

Table[Abs[Sum[MoebiusMu[n/d], {d,Select[Divisors[n], SquareFreeQ[#] &]}]], {n, 1, 200}] (* Geoffrey Critzer, Mar 18 2015 *)
Module[{nn=120,len,sfr},len=Ceiling[Sqrt[nn]];While[!SquareFreeQ[len],len++];sfr=(Select[Range[len],SquareFreeQ])^2; Table[If[MemberQ[ sfr,n],1,0],{n,nn}]] (* Harvey P. Dale, Nov 27 2024 *)

a(n)=if(n<1, 0, direuler(p=2, n, 1+X^2)[n])

A227291(n) = factorback(apply(e->(2==e), factor(n)[,2])); \\ Antti Karttunen, Jul 14 2022

A227291(n) = (issquare(n) && issquarefree(sqrtint(n))); \\ Antti Karttunen, Jul 14 2022

(define (A227291 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) 1 0) (A227291 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

Dirichlet g.f.: zeta(2s)/zeta(4s) = prod[prime p: 1+p^(-2s) ], see A008966.
a(n) = A008966(A037213(n)), when assumed A008966(0) = 0. - Reinhard Zumkeller, Jul 07 2013
a(n) = A063524(sum(A225817(n,k)*A225817(n,A000005(n)+1-k): k=1..A000005(n))). - Reinhard Zumkeller, Aug 01 2013
Multiplicative with a(p^e) = 1 if e=2, a(p^e) = 0 if e=1 or e>2. - Antti Karttunen, Jul 28 2017
Sum_{k=1..n} a(k) ~ 6*sqrt(n) / Pi^2. - Vaclav Kotesovec, Feb 02 2019
a(n) = A225569(A225546(n)-1). - Peter Munn, Oct 31 2019
From Antti Karttunen, Jul 18 2022: (Start)
a(n) = A010052(n) * A008966(A000196(n)).
a(n) = Sum_{d|n} A008836(n/d) * A307430(d).
a(n) = Sum_{d|n} A007427(n/d) * A322327(d).
(End)

A322328 a(n) = A005361(n) * 4^A001221(n) for n > 0.

Original entry on oeis.org

1, 4, 4, 8, 4, 16, 4, 12, 8, 16, 4, 32, 4, 16, 16, 16, 4, 32, 4, 32, 16, 16, 4, 48, 8, 16, 12, 32, 4, 64, 4, 20, 16, 16, 16, 64, 4, 16, 16, 48, 4, 64, 4, 32, 32, 16, 4, 64, 8, 32, 16, 32, 4, 48, 16, 48, 16, 16, 4, 128, 4, 16, 32, 24, 16, 64, 4, 32, 16, 64, 4

Offset: 1

Werner Schulte, Dec 03 2018

Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see A322327, for k = 3 see A226602 (offset 1), and for k = 4 see this sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000007, A001221, A005361, A007427, A008836, A034444, A226602, A322327.

f:= n -> mul(4*t[2],t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Dec 07 2018

a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 4^Length[f] * Times@@f[[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)

a(n) = my(f=factor(n)); vecprod(f[,2])*4^omega(n); \\ Michel Marcus, Dec 04 2018

from math import prod
from sympy import factorint
def A322328(n): return prod(e<<2 for e in factorint(n).values()) # Chai Wah Wu, Dec 24 2022

Multiplicative with a(p^e) = 4*e for prime p and e > 0.
Dirichlet g.f.: (zeta(s))^4 / (zeta(2*s))^2.
Dirichlet inverse is b(n) = a(n) * A008836(n) for n > 0, and b(n) is multiplicative with b(p^e) = 4*e*(-1)^e for prime p and e > 0.
Equals Dirichlet convolution of A034444 with itself.
Equals Dirichlet convolution of A000005 with abs(A007427).

A252505 Number of biquadratefree (4th power free) divisors of n.

Original entry on oeis.org

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

Offset: 1

Geoffrey Critzer, Mar 21 2015

Equivalently, a(n) is the number of divisors of n that are in A046100.
a(n) is also the number of divisors d such that the greatest common square divisor of d and n/d is 1.
The number of divisors d of n such that gcd(d, n/d) is squarefree. - Amiram Eldar, Aug 25 2023

			a(16) = 4 because there are 4 divisors of 16 that are 4th power free: 1,2,4,8.
a(16) = 4 because there are 4 divisors d of 16 such that the greatest common square divisor of d and 16/d is 1: 1,2,8,16.

Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 37, Exercise 1.27.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A252505, Sequence Machine.
Eric Weisstein's World of Mathematics, Biquadratefree.

Cf. A046100 (biquadratefree numbers).
Cf. A034444 (squarefree divisors), A073184 (cubefree divisors).
Cf. A001620.
Cf. A000005, A058035, A307430.
Also obtained as a Dirichlet convolution of the following pairs: A034444 and A227291, A007427 and A286779, A008966 and A323308, A048691 and A363552, A271102 and A322327, A307445 and A370296, and A018892 and A378214 (conjectured).

Prepend[Table[Apply[Times, (FactorInteger[n][[All, 2]] /. x_ /; x > 3 -> 3) + 1], {n, 2, 100}], 1]

isA046100(n) = (n==1) || vecmax(factor(n)[, 2])<4;
a(n) = {d = divisors(n); sum(i=1, #d, isA046100(d[i]));} \\ Michel Marcus, Mar 22 2015

a(n) = vecprod(apply(x->min(x, 3) + 1, factor(n)[, 2])); \\ Amiram Eldar, Aug 25 2023

Dirichlet g.f.: zeta(s)^2/zeta(4*s).
Sum_{k=1..n} a(k) ~ 90*n/Pi^4 * (log(n) - 1 + 2*gamma - 360*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} mu(gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
Multiplicative with a(p^e) = min(e, 3) + 1. - Amiram Eldar, Sep 19 2020
From Antti Karttunen, May 14 2025: (Start)
Following formulas have been generated for this sequence by Sequence Machine:
a(n) = A000005(A058035(n)).
a(n) = Sum_{d|n} A307430(d).
a(n) = Sum_{d|n} A034444(d)*A227291(n/d).
a(n) = Sum_{d|n} A007427(d)*A286779(n/d).
a(n) = Sum_{d|n} A008966(d)*A323308(n/d).
a(n) = Sum_{d|n} A048691(d)*A363552(n/d).
a(n) = Sum_{d|n} A271102(d)*A322327(n/d).
a(n) = Sum_{d|n} A307445(d)*A370296(n/d).
a(n) = Sum_{d|n} A018892(d)*A378214(n/d). [Conjectured]
(End)

A360908 Multiplicative with a(p^e) = 2*e - 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 7, 7, 1, 1, 3, 1, 1

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A048691, A322327, A361430, A367822.

a[n_] := Times @@ ((2*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+(1+1/X)/(1-1/X)^2))[n], ", "))

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]=2*f[k,2]-1; f[k,2]=1); factorback(f); \\ Michel Marcus, Feb 25 2023

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/(p^s*(p^s-1))).
Sum_{k=1..n} a(k) ~ c*n, where c = A367822 = Product_{p prime} (1 + 2/(p*(p-1))) = 3.279577150984783607372919498914633983999130708105267540952619534539808381...
a(n) = A361430(n^2). - Amiram Eldar, Feb 11 2024

A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 26 2023

The terms of A005361 that are powers of 2 (A000079).

The first position of 2^k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

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

Cf. A000079, A005361, A085629, A138302, A271727.
Cf. A366538, A367169.
Similar sequences: A322327, A368472, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[p == 2^IntegerExponent[p, 2], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p >> valuation(p,2) == 1, print1(p, ", ")));}

a(n) = A005361(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=0} 2^k/p^(2^k)) = 1.504710204899266020302..., where d = A271727 is the asymptotic density of the exponentially 2^n-numbers.

A360909 Multiplicative with a(p^e) = 3*e + 2.

Original entry on oeis.org

1, 5, 5, 8, 5, 25, 5, 11, 8, 25, 5, 40, 5, 25, 25, 14, 5, 40, 5, 40, 25, 25, 5, 55, 8, 25, 11, 40, 5, 125, 5, 17, 25, 25, 25, 64, 5, 25, 25, 55, 5, 125, 5, 40, 40, 25, 5, 70, 8, 40, 25, 40, 5, 55, 25, 55, 25, 25, 5, 200, 5, 25, 40, 20, 25, 125, 5, 40, 25, 125

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), this sequence (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+3*X-X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)), (with a product that converges for s=1).
Sum_{k=1..n} a(k) ~ c * n * log(n)^4 / 24, where c = Product_{primes p} (1 - 7/p^2 + 11/p^3 - 6/p^4 + 1/p^5) = 0.091414252314317101861531055690354339957600046..., more precise (but very complicated) asymptotics can be obtained (in Mathematica notation) as Residue[Zeta[s]^5 * f[s] * n^s / s, {s, 1}], where f[s] = Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)).

A360997 Multiplicative with a(p^e) = e + 3.

Original entry on oeis.org

1, 4, 4, 5, 4, 16, 4, 6, 5, 16, 4, 20, 4, 16, 16, 7, 4, 20, 4, 20, 16, 16, 4, 24, 5, 16, 6, 20, 4, 64, 4, 8, 16, 16, 16, 25, 4, 16, 16, 24, 4, 64, 4, 20, 20, 16, 4, 28, 5, 20, 16, 20, 4, 24, 16, 24, 16, 16, 4, 80, 4, 16, 20, 9, 16, 64, 4, 20, 16, 64, 4, 30, 4, 16

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

Cf. A000005, A007425, A007426, A074816.

g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A361264(n)).
a(n) = A074816(n)*A007426(n)/A007425(n). (End)

A368472 Product of exponents of prime factorization of the exponentially odd numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Dec 26 2023

The odd terms of A005361.

The first position of 2*k-1, for k = 1, 2, ..., is 1, 7, 24, 91, 154, 1444, 5777, 610, 92349, ..., which is the position of A085629(2*k-1) in A268335.

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

Cf. A005361, A013661, A065463, A085629, A098198, A268335.
Cf. A366438, A366439.
Similar sequences: A322327, A368473, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[OddQ[p], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p%2, print1(p, ", ")));}

a(n) = A005361(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2/d) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^5) = 1.38446562720473484463..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

cp's OEIS Frontend

A355837 Dirichlet inverse of A322327.

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A370296 Inverse Moebius transform of A322327.

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A227291 Characteristic function of squarefree numbers squared (A062503).

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A322328 a(n) = A005361(n) * 4^A001221(n) for n > 0.

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A252505 Number of biquadratefree (4th power free) divisors of n.

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A360908 Multiplicative with a(p^e) = 2*e - 1.

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A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

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