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

1,6

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

Formula

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

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
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

Crossrefs

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

Sequence in context: A327372 A328323 A090751 * A337820 A322127 A282496

Adjacent sequences: A344756 A344757 A344758 * A344760 A344761 A344762

Keyword

nonn

Author

Antti Karttunen, Jun 01 2021

Status

approved