A007948 Largest cubefree number dividing n.

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 36, 73

Offset: 1

Author

R. Muller

Keywords

• nonn
• mult

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Henry Bottomley, Some Smarandache-type multiplicative functions.
• Florentin Smarandache, Only Problems, Not Solutions!, 1993.

Crossrefs

Cf. A003557, A004709, A007947, A027748, A058035, A062378, A065465, A124010, A197863.

Programs

Haskell

a007948 = last . filter ((== 1) . a212793) . a027750_row
-- Reinhard Zumkeller, May 27 2012, Jan 06 2012

Mathematica

Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> p^Min[e, 2]], {n, 73}] (* Michael De Vlieger, Jul 18 2017 *)

PARI

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = min(f[i, 2], 2)); factorback(f); \\ Michel Marcus, Jun 09 2014
(Scheme, with memoization-macro definec) (definec (A007948 n) (if (= 1 n) n (* (expt (A020639 n) (min 2 (A067029 n))) (A007948 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

Formula

Multiplicative with a(p^e) = p^(min(e, 2)). - David W. Wilson, Aug 01 2001

a(n) = max{A212793(A027750(n,k)) * A027750(n,k): k=1..A000005(n)}. - Reinhard Zumkeller, May 27 2012

a(n) = A071773(n)*A007947(n). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017

a(n) = n / A062378(n) = n / A003557(A003557(n)). - Antti Karttunen, Nov 28 2017

Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Oct 13 2022

Extensions

More terms from Henry Bottomley, Jun 18 2001