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

1,1

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

Links

Table of n, a(n) for n=1..16.

Example

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.

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]

Prog

(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

Crossrefs

Subsequence of A005835 and A334405.

A variant of A194472 and A318168.

Cf. A000396, A334410.

Sequence in context: A133015 A203270 A185960 * A216832 A013839 A271386

Adjacent sequences: A334406 A334407 A334408 * A334410 A334411 A334412

Keyword

nonn,more

Author

Amiram Eldar, Apr 27 2020

Extensions

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

Status

approved