A284255 Largest divisor of n such that all its prime factors are less than the square of the smallest prime factor of n, a(1) = 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 11, 12, 13, 2, 15, 16, 17, 18, 19, 4, 21, 2, 23, 24, 25, 2, 27, 4, 29, 6, 31, 32, 3, 2, 35, 36, 37, 2, 3, 8, 41, 6, 43, 4, 45, 2, 47, 48, 49, 2, 3, 4, 53, 54, 55, 8, 3, 2, 59, 12, 61, 2, 63, 64, 65, 6, 67, 4, 3, 2, 71, 72, 73, 2, 75, 4, 77, 6, 79, 16, 81, 2, 83, 12, 85, 2, 3, 8, 89, 18, 91, 4, 3, 2, 95, 96, 97, 2, 9, 4, 101, 6

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..10001

Formula

a(n) = n / A284254(n).

Other identities. For all n >= 1:

A006530(a(n)) = A284260(n).

A020639(a(n)) = A020639(n).

A001221(a(n)) = A284259(n).

A001222(a(n)) = A284257(n).

Example

For n = 50 = 2*5*5, only prime less than 2^2 is 2, thus a(50) = 2.
For n = 90 = 2*3*3*5, the primes less than 2^2 are 2, 3 and 3, thus a(90) = 2*3*3 = 18.

Mathematica

Table[If[n == 1, 1, Function[d, First[Select[Reverse@ First@ d, Times @@ Boole@ Map[# < Last[d]^2 &, FactorInteger[#][[All, 1]]] == 1 &] /. {} -> {1}]]@ {#, First@ Select[#, PrimeQ]} &@ Divisors@ n], {n, 102}] (* Michael De Vlieger, Mar 24 2017 *)

Prog

(Scheme) (define (A284255 n) (/ n (A284254 n)))
(PARI)
A(n) = if(n<2, return(1), my(f=factor(n)[, 1]); for(i=2, #f, if(f[i]>f[1]^2, return(f[i]))); return(1));
a(n) = if(A(n)==1, 1, A(n)*a(n/A(n)));
for(n=1, 150, print1(n/a(n), ", ")) \\ Indranil Ghosh, after David A. Corneth, Mar 24 2017
(Python)
from sympy import primefactors
def A(n):
    pf = primefactors(n)
    if pf: min_pf2 = min(pf)**2
    for i in pf:
        if i > min_pf2: return i
    return 1
def a(n): return 1 if A(n)==1 else A(n)*a(n//A(n))
print([n//a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 24 2017

Crossrefs

Cf. A001221, A001222, A006530, A020639, A284252, A284254, A284256, A284257, A284258, A284259, A284260.

Differs from A284253 for the first time at n=50, where a(50) = 2, while A284253(50) = 10.

Sequence in context: A122481 A110805 A356234 * A284253 A318492 A061499

Adjacent sequences: A284252 A284253 A284254 * A284256 A284257 A284258

Keyword

nonn

Author

Antti Karttunen, Mar 24 2017

Status

approved