A284254 - OEIS

A284254

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

1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 1, 5, 1, 11, 1, 1, 1, 13, 1, 7, 1, 5, 1, 1, 11, 17, 1, 1, 1, 19, 13, 5, 1, 7, 1, 11, 1, 23, 1, 1, 1, 25, 17, 13, 1, 1, 1, 7, 19, 29, 1, 5, 1, 31, 1, 1, 1, 11, 1, 17, 23, 35, 1, 1, 1, 37, 1, 19, 1, 13, 1, 5, 1, 41, 1, 7, 1, 43, 29, 11, 1, 5, 1, 23, 31, 47, 1, 1, 1, 49, 11, 25, 1, 17, 1, 13, 1, 53, 1, 1, 1, 55

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OFFSET

1,10

LINKS

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

FORMULA

If A284252(n) = 1, then a(n) = 1, otherwise A284252(n) * a(A284253(n)).

Other identities. For all n >= 1:

n / a(n) = A284255(n).

A020639(a(n)) = A284252(n).

A001221(a(n)) = A284258(n).

A001222(a(n)) = A284256(n).

EXAMPLE

For n = 15 = 3*5, no prime factor is larger than 3^2, thus a(15) = 1. In this case the largest divisor satisfying the condition has no prime factors at all.

For n = 50 = 2*5*5, the primes larger than 2^2 are 5 and 5, thus a(50) = 5*5 = 25.

MATHEMATICA

Table[If[n == 1, 1, Function[d, Last[Select[Reverse@ First@ d, Times @@ Boole@ Map[# > Last[d]^2 &, FactorInteger[#][[All, 1]]] == 1 &] /. {} -> {1}]]@ {#, First@ Select[#, PrimeQ]} &@ Divisors@ n], {n, 108}] (* Michael De Vlieger, Mar 24 2017 *)

PROG

(Scheme, with memoization-macro definec)

(definec (A284254 n) (if (= 1 (A284252 n)) 1 (* (A284252 n) (A284254 (A284253 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(a(n), ", ")) \\ Indranil Ghosh, after David A. Corneth, Mar 24 2017

(Python)

from sympy import primefactors

def A(n):

for i in primefactors(n):

if i>min(primefactors(n))**2: return i

return 1

def a(n): return 1 if A(n) == 1 else A(n)*a(n//A(n))

print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 24 2017

CROSSREFS

Cf. A020639, A284253, A284255, A284256, A284257, A284258, A284259, A284260.

Cf. A251726 (gives the positions of ones after the initial a(1)=1).

Differs from related A284252 for the first time at n=50, where a(50) = 25, while A284252(50) = 5.

Sequence in context: A068316 A359945 A284252 * A309206 A358016 A250097

Adjacent sequences: A284251 A284252 A284253 * A284255 A284256 A284257

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 24 2017

STATUS

approved