A344590 Number of divisors 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, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2

Offset

1,6

Links

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

Formula

a(n) = Sum_{d|n} [A011772(d) = A011772(n)], where [ ] is the Iverson bracket.

a(n) = A000005(n) - A344589(n).

a(n) <= A344770(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, therefore a(36) = 4.

Mathematica

A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
a[n_] := With[{m = A011772[n]}, Count[Divisors[n], d_ /; A011772[d] == m]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

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
A344590(n) = { my(x=A011772(n)); sumdiv(n, d, A011772(d)==x); };
(Python)
from itertools import combinations
from functools import reduce
from operator import mul
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 (reduce(mul, d) for l in range(1, len(plist)//2+1) for d in combinations(plist, l)))
def A344590(n):
    m = A011772(n)
    return sum(1 for d in divisors(n) if A011772(d) == m) # Chai Wah Wu, Jun 02 2021

Crossrefs

Cf. A000005, A011772, A344588 (positions of records), A344589, A344758, A344770, A344881 (positions of ones), A344882 (of terms > 1).

Sequence in context: A031230 A355032 A244226 * A111616 A299152 A332770

Adjacent sequences: A344587 A344588 A344589 * A344591 A344592 A344593

Keyword

nonn

Author

Antti Karttunen, May 31 2021

Status

approved