A344759 a(n) = n divided by the 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, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 1, 7, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 3, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 3, 1, 2, 1, 4, 1, 1, 1, 1, 3

Offset: 1

Author

Antti Karttunen, Jun 01 2021

Keywords

• nonn

Comments

It seems that A006516 gives the positions of records after its initial zero.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A006516, A011772, A344758, A344881 (positions of ones), A344882 (of terms > 1).

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
A344759(n) = { my(x=A011772(n)); fordiv(n,d,if(A011772(d)==x, return(n/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 A344759(n):
    m = A011772(n)
    for d in divisors(n):
        if A011772(d) == m:
            return n//d # Chai Wah Wu, Jun 03 2021

Formula

a(n) = n / A344758(n).