A052128 a(1) = 1; for n > 1, a(n) is the largest divisor of n that is coprime to a larger divisor of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 1, 4, 1, 5, 1, 1, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 7, 3, 2, 1, 5, 1, 2, 7, 1, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 7, 5, 2, 3, 8, 1, 9, 7, 4, 3, 2, 5, 3, 1, 2, 9, 4, 1, 6

Offset

1,6

Comments

Least k > 0 such that the resultant of the k-th cyclotomic polynomial and the n-th cyclotomic polynomial is not 1. - Benoit Cloitre, Oct 13 2002

From Jianing Song, Sep 28 2022: (Start)

a(n) is the largest divisor d of n such that d <= sqrt(n) and that gcd(d,n/d) = 1.

Proof: write n = Product_{1<=i<=r} (p_i)^(e_i), let d be the largest divisor of n such that d <= sqrt(n) and that gcd(d,n/d) = 1. Obviously we have a(n) >= d. Suppose that a(n) = Product_{1<=i<=s} (p_i)^(m_i) for s <= r, 1 <= m_i <= e_i, then the larger divisor to which a(n) is coprime is a divisor of Product_{s+1<=i<=r} (p_i)^(e_i), so by definition we have a(n) <= min{Product_{1<=i<=s} (p_i)^(e_i), Product_{s+1<=i<=r} (p_i)^(e_i)} <= d. Thus a(n) = d. (End)

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) = n / A354933(n) = A354933(n) - A076388(n). - Antti Karttunen, Jun 16 2022

Example

a(6) = 6 / 3^1 = 2.

Mathematica

Table[best = 1; d = Divisors[n]; While[Length[d] > 1, e = d[[1]]; d = Rest[d]; If[Min[GCD[e, d]] == 1, best = e]]; best, {n, 102}] (* T. D. Noe, Aug 23 2013 *)

Prog

(PARI) a(n) = my(i, j, d = divisors(n)); forstep (i = #d-1, 1, -1, for (j = i+1, #d, if (gcd(d[i], d[j]) == 1, return (d[i])))); 1 \\ Michel Marcus, Aug 22 2013
(PARI) a(n)=my(f=factor(n), v=[1]); for(i=1, #f~, v=concat(v, f[i, 1]^f[i, 2] *v)); v=vecsort(v); forstep(i=#v\2, 2, -1, for(j=i+1, #v-1, if(gcd(v[i], v[j])==1, return(v[i])))); 1 \\ Charles R Greathouse IV, Aug 22 2013
(PARI) A052128(n) = fordiv(n, d, if((d>=(n/d)) && 1==gcd(d, n/d), return(n/d))); \\ Antti Karttunen, Jun 16 2022

Crossrefs

Cf. A054372, A076388, A354933.

Sequence in context: A025865 A085091 A345994 * A284600 A114536 A330692

Adjacent sequences: A052125 A052126 A052127 * A052129 A052130 A052131

Keyword

nonn

Author

James A. Sellers, Jan 21 2000

Extensions

Terms corrected by Charles R Greathouse IV, Aug 22 2013

Definition rewritten by Jianing Song, Sep 28 2022

Status

approved