A334409 Numbers m such that the sum of the first k divisors and the last k divisors of m is equal to 2*m for some k that is smaller than half of the number of divisors of m.

36, 152812, 6112576, 72702928, 154286848, 397955025, 15356519488, 23003680492, 35755623784, 93789539668, 302122464256, 351155553970, 1081806148665, 1090488143872, 1663167899025, 2233955122576

Offset: 1

Author

Amiram Eldar, Apr 27 2020

Keywords

• nonn
• more

Comments

If k is allowed to be equal to half of the number of divisors of m, then the perfect numbers (A000396) will be terms.

a(17) > 10^13. 3021194449732665786499072 is also a term. - Giovanni Resta, May 09 2020

Examples

			36 is a term since its divisors are {1, 2, 3, 4, 6, 9, 12, 18, 36} and the sum of the first 3 and last 3 divisors is (1 + 2 + 3) + (12 + 18 + 36) = 72 = 2 * 36.
		

Crossrefs

Subsequence of A005835 and A334405.

A variant of A194472 and A318168.

Cf. A000396, A334410.

Programs

Mathematica

seqQ[n_] := Module[{d = Divisors[n]}, nd = Length[d]; nd2 = Ceiling[nd/2] - 1; s = Accumulate[d[[1 ;; nd2]] + n/d[[1 ;; nd2]]]; MemberQ[s, 2*n]]; Select[Range[10^6], seqQ]

Python

from itertools import count, islice, accumulate
from sympy import divisors
def A334409_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        ds = divisors(n)
        if any(s==2*n for s in accumulate(ds[i]+ds[-1-i] for i in range((len(ds)-1)//2))):
            yield n
A334409_list = list(islice(A334409_gen(),2)) # Chai Wah Wu, Feb 19 2022

Extensions

a(8)-a(16) from Giovanni Resta, May 06 2020