A382888 -id:A382888 - cp's OEIS frontend

A382889 The largest square dividing the n-th cubefree number.

1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 25, 1, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 49, 9, 100

Offset: 1

Amiram Eldar, Apr 07 2025

Also, the powerful part of the n-th cubefree number.

All the terms are squares of squarefree numbers (A062503).

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

Cf. A002117, A004709, A008833, A057521, A062503, A371188 (positions of 1's).

Similar sequences: A382888, A382890, A382891.

f[p_, e_] := p^If[e == 1, 0, 2]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 2)), ", ")));}

a(n) = A008833(A004709(n)).
a(n) = A057521(A004709(n)).
a(n) = A382890(n)^2.
a(n) = A004709(n)/A382891(n).
a(n) = (A004709(n)/A382888(n))^2.
a(A371188(n)) = 1.
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3)^(3/2) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2)) = 1.48513488319516447978... .

A382890 The square root of the largest square dividing the n-th cubefree number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 07 2025

The product of the non-unitary prime divisors of the n-th cubefree number.
Also, the square root of the powerful part of the n-th cubefree number.
All the terms are squarefree.

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

Cf. A000188, A004709, A005117, A057521, A371188 (positions of 1's).

Similar sequences: A382888, A382889, A382891.

f[p_, e_] := p^If[e == 1, 0, 1]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 1)), ", ")));}

a(n) = A000188(A004709(n)).
a(n) = sqrt(A382889(n)).
a(n) = A004709(n)/A382888(n).
a(n) = sqrt(A004709(n)/A382891(n)).
a(A371188(n)) = 1.

A382891 The powerfree part of the n-th cubefree number.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 10, 11, 3, 13, 14, 15, 17, 2, 19, 5, 21, 22, 23, 1, 26, 7, 29, 30, 31, 33, 34, 35, 1, 37, 38, 39, 41, 42, 43, 11, 5, 46, 47, 1, 2, 51, 13, 53, 55, 57, 58, 59, 15, 61, 62, 7, 65, 66, 67, 17, 69, 70, 71, 73, 74, 3, 19, 77, 78, 79, 82, 83

Offset: 1

Amiram Eldar, Apr 07 2025

Also, the squarefree part of the n-th cubefree number.

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

Cf. A002117, A004709, A005117, A055231, A007913, A371188.

Similar sequences: A382888, A382889, A382890.

f[p_, e_] := p^If[e == 1, 1, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 1, 0)), ", ")));}

a(n) = A055231(A004709(n)).
a(n) = A007913(A004709(n)).
a(n) = A004709(n)/A382889(n) = A004709(n)/A382890(n)^2.
a(n) = A382888(n)^2/A004709(n).
a(A371188(n)) = A005117(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^5) = 0.92517253037215590197... .

cp's OEIS Frontend

A382889 The largest square dividing the n-th cubefree number.

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A382890 The square root of the largest square dividing the n-th cubefree number.

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A382891 The powerfree part of the n-th cubefree number.

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