A076388 -id:A076388 - cp's OEIS frontend

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

James Sellers, Jan 21 2000

nonn

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)

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

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

Cf. A054372, A076388, A354933.

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 *)

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

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

A052128(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(n/d))); \\ Antti Karttunen, Jun 16 2022

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

Terms corrected by Charles R Greathouse IV, Aug 22 2013

Definition rewritten by Jianing Song, Sep 28 2022

A354933 a(1) = 1; for n > 1, a(n) = n / the largest divisor of n that is coprime to a larger divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 6, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 12, 61, 31, 9, 64, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 25, 19, 11, 13, 79, 16, 81, 41, 83, 12, 17, 43

Offset: 1

Antti Karttunen, Jun 16 2022

nonn

a(n) is the smallest divisor d of n such that d >= sqrt(n) and that gcd(d,n/d) = 1. For the proof see A052128. - Jianing Song, Sep 28 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A052128, A076388.

Differs from A346596 for the first time at n=60, where a(60) = 12, while A346596(60) = 15.

a[n_] := SelectFirst[Divisors[n], # >= n/# && CoprimeQ[#, n/#] &]; Array[a, 100] (* Amiram Eldar, Jun 16 2022 *)

A354933(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(d)));

a(n) = n / A052128(n).

a(n) = A052128(n) + A076388(n).

Definition rewritten by Jianing Song, Sep 28 2022

A354988 a(n) = A345993(n) - A345992(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 7, 8, 3, 10, 1, 12, 5, -2, 15, 16, 7, 18, 1, 4, 9, 22, -5, 24, 11, 26, -3, 28, 1, 30, 31, -8, 15, -2, 5, 36, 17, 10, 3, 40, 1, 42, -7, -4, 21, 46, 13, 48, 23, -14, 9, 52, 25, 6, 1, 16, 27, 58, -11, 60, 29, -2, 63, 8, -5, 66, 13, -20, -9, 70, 1, 72, 35, 22, -15, 4, 7, 78, 11, 80, 39, 82, 17, -12

Offset: 1

Antti Karttunen, Jun 16 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A345992, A345993, A345995 (positions of negative terms), A354989 (their characteristic function).

Absolute values differ from A076388 for the first time at n=60, where a(60) = -11, while A076388(60) = 7.

a[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), n], m++]; GCD[n, m + 1] - GCD[n, m]]; Array[a, 100] (* Amiram Eldar, Jun 16 2022 *)

A354988(n) = for(m=1, oo, if((m*(m+1))%n==0, return(gcd(n,1+m)-gcd(n,m))));
(Python 3.8+)
from math import gcd, prod
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
def A354988(n):
    if n == 1:
        return 0
    plist = tuple(p**q for p, q in factorint(n).items())
    return n-1 if len(plist) == 1 else -gcd(n,s:=int(min(min(crt((m, n//m), (0, -1))[0], crt((n//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, len(plist)//2+1) for d in combinations(plist, l))))) + gcd(n,s+1) # Chai Wah Wu, Jun 16 2022

A243981 Minimum range of sets of natural numbers with a product of n.

Original entry on oeis.org

1, 2, 0, 4, 1, 6, 0, 0, 3, 10, 1, 12, 5, 2, 0, 16, 1, 18, 1, 4, 9, 22, 1, 0, 11, 0, 3, 28, 1, 30, 0, 8, 15, 2, 0, 36, 17, 10, 3, 40, 1, 42, 7, 2, 21, 46, 1, 0, 3, 14, 9, 52, 1, 6, 1, 16, 27, 58, 2, 60, 29, 2, 0, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 2, 15, 4, 7, 78, 1, 0, 39, 82, 4, 12, 41, 26, 3, 88, 1, 6, 19, 28, 45, 14, 1, 96, 5, 2, 0

Offset: 2

Paul Richards, Nov 11 2014

The minimum difference between the largest and smallest values in the sets of positive integers with a product of n, excluding the singleton set {n}.

			For 45 the sets are {1,45}, {3,15}, {5,9}, {3,3,5} with differences of 44, 12, 4 and 2 respectively.  2 is the minimum and so a(45) = 2.

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A056737, A060794, A076388, A082119.

a(n) <= A046665(n) for all composite n, a(p) = p - 1 for primes p. - Charlie Neder, Jan 13 2019

cp's OEIS Frontend

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

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A354933 a(1) = 1; for n > 1, a(n) = n / the largest divisor of n that is coprime to a larger divisor of n.

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A354988 a(n) = A345993(n) - A345992(n).

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PARI

A243981 Minimum range of sets of natural numbers with a product of n.

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