A359433 -id:A359433 - cp's OEIS frontend

A071773 a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 10, 1, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Jun 24 2002

n is squarefree iff a(n)=1.
Product of primes dividing n more than once. - Charles R Greathouse IV, Aug 08 2013
Squarefree kernel of the square part of n. - Peter Munn, Jun 12 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A003415, A003557, A005117, A007947, A007948, A008833, A057521, A166486 (parity of terms), A359433 (Dirichlet inverse).

Cf. A065464.

Table[With[{r = Apply[Times, FactorInteger[n][[All, 1]]]}, GCD[r, n/r]], {n, 104}] (* Michael De Vlieger, Aug 20 2017 *)

a(n)=my(f=factor(n));prod(i=1,#f~,f[i,1]^(f[i,2]>1)) \\ Charles R Greathouse IV, Aug 08 2013

;; With memoization-macro definec.
(definec (A071773 n) (if (= 1 n) n (* (if (zero? (modulo n (expt (A020639 n) 2))) (A020639 n) 1) (A071773 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = gcd(A007947(n), A003557(n)).
Multiplicative with p^e -> p^ceiling((e-1)/e), p prime.
a(n) = rad(n/rad(n)) = A007947(A003557(n)). - Velin Yanev, Antti Karttunen, Aug 20 2017, Nov 28 2017
a(n) = A007947(A057521(n)). - Antti Karttunen, Nov 28 2017
a(n) = A007947(A008833(n)). - Peter Munn, Jun 12 2020
a(n) = gcd(A003415(n), A007947(n)). - Antti Karttunen, Jan 02 2023
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Nov 09 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(4*s-1) - 1/p^(4*s-2)).
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) = A065464 = Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.428249505677094440218765707581823546121298513355936144031901379532123...
f'(1) = f(1) * Sum_{p prime} 2*(3*p-2)*log(p) / (p^3-2*p+1) = f(1) * 2.939073481649229666406787986900328729326669597518287791424059647447664...
and gamma is the Euler-Mascheroni constant A001620. (End)

A359432 Dirichlet inverse of A327936, which is multiplicative sequence with a(p^e) = p if e >= p, otherwise 1.

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, 1, 1, -1, 0, -1, 1, 1, 1, -1, -1, 0, 1, -2, 1, -1, -1, -1, -1, 1, 1, 1, 0, -1, 1, 1, -1, -1, -1, -1, 1, 0, 1, -1, -1, 0, 0, 1, 1, -1, 2, 1, -1, 1, 1, -1, -1, -1, 1, 0, -1, 1, -1, -1, 1, 1, -1, -1, 0, -1, 1, 0, 1, 1, -1, -1, -1, 2, 1, -1, -1, 1, 1, 1, -1, -1, 0, 1, 1, 1, 1, 1, 1, -1, 0, 0, 0, -1, -1, -1, -1, -1, 1, -1, 2

Offset: 1

Antti Karttunen, Jan 02 2023

Multiplicative because A327936 is.

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

Cf. A327936.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
Cf. also A358216, A359433.

f[p_, e_] := Switch[Mod[e, p], 0, (1 - p)^(e/p), 1, -(1 - p)^((e - 1)/p), , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}] (* Amiram Eldar, Jan 26 2023 *)

A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
memoA359432 = Map();
A359432(n) = if(1==n,1,my(v); if(mapisdefined(memoA359432,n,&v), v, v = -sumdiv(n,d,if(dA327936(n/d)*A359432(d),0)); mapput(memoA359432,n,v); (v)));

Multiplicative with a(p^e) = (1 - p)^(e/p) if p | e, -(1 - p)^((e - 1)/p) if e == 1 (mod p), and 0 otherwise. - Amiram Eldar, Jan 26 2023

A366292 Dirichlet inverse of A353271, where A353271(n) is the numerator of n / A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -2, -3, -1, 1, -1, -5, -3, -1, -1, 0, -1, 9, -5, -9, -1, 11, -4, -11, -4, 13, -1, 5, -1, -1, -9, -15, -5, 6, -1, -17, -11, 5, -1, 21, -1, 21, -2, -21, -1, 5, -6, -8, -15, 25, -1, 22, -9, 7, -17, -27, -1, 3, -1, -29, 14, -1, -11, 11, -1, 33, -21, -3, -1, 16, -1, -35, -8, 37, -9, 13

Offset: 1

Antti Karttunen, Oct 06 2023

sign

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

Cf. A332449, A353271.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
Cf. also A209635, A342417, A354347, A354823, A359432, A359433, A359577 for other sequences that are equal modulo 2.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353271(n) = (n / gcd(n, A332449(n)));
memoA366292 = Map();
A366292(n) = if(1==n,1,my(v); if(mapisdefined(memoA366292,n,&v), v, v = -sumdiv(n,d,if(dA353271(n/d)*A366292(d),0)); mapput(memoA366292,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA353271(n/d) * a(d).

cp's OEIS Frontend

A071773 a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.

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A359432 Dirichlet inverse of A327936, which is multiplicative sequence with a(p^e) = p if e >= p, otherwise 1.

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Mathematica

PARI

Formula

A366292 Dirichlet inverse of A353271, where A353271(n) is the numerator of n / A005940(1+(3*A156552(n))).

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula