A344758 Smallest divisor d of n for which A011772(d) = A011772(n), where A011772(n) is the smallest number m such that m(m+1)/2 is divisible by n.

1, 2, 3, 4, 5, 2, 7, 8, 9, 5, 11, 12, 13, 14, 15, 16, 17, 9, 19, 20, 7, 22, 23, 8, 25, 13, 27, 4, 29, 30, 31, 32, 33, 17, 35, 9, 37, 38, 13, 8, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 51, 52, 53, 54, 11, 56, 19, 29, 59, 20, 61, 62, 63, 64, 65, 22, 67, 17, 69, 70, 71, 72, 73, 37, 25, 76, 77, 13, 79, 80, 81, 41, 83

Offset: 1

Author

Antti Karttunen, May 31 2021

Keywords

• nonn

Examples

			36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A011772 is applied to them, one obtains values [1, 3, 2, 7, 3, 8, 8, 8, 8], thus there are four divisors that obtain the maximal value 8 obtained at 36 itself, of which divisor 9 is the smallest, and therefore a(36) = 9.
		

Links

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

Crossrefs

Cf. A011772, A344590, A344759.

Programs

PARI

A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344758(n) = { my(x=A011772(n)); fordiv(n,d,if(A011772(d)==x, return(d))); };
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
def A011772(n):
    plist = [p**q for p, q in factorint(2*n).items()]
    if len(plist) == 1:
        return n-1 if plist[0] % 2 else 2*n-1
    return min(min(crt([m,2*n//m],[0,-1])[0],crt([2*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)))
def A344758(n):
    m = A011772(n)
    for d in divisors(n):
        if A011772(d) == m:
            return d # Chai Wah Wu, Jun 03 2021

Formula

a(n) = n / A344759(n).