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

1,2

Links

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

Formula

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

Example

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.

Prog

(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

Crossrefs

Cf. A011772, A344590, A344759.

Sequence in context: A066296 A162961 A145255 * A053626 A348990 A348968

Adjacent sequences: A344755 A344756 A344757 * A344759 A344760 A344761

Keyword

nonn

Author

Antti Karttunen, May 31 2021

Status

approved