OFFSET
1,1
COMMENTS
Subsequence of A005101 but seem to be much rarer.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023), Springer Proc. Math. Stat., Vol. 474 (2025), 111-121.
EXAMPLE
The sum of the divisors of 60 is 168, and 168 = 2*60 + 48, and 48 = 4*12 and 4 and 12 are divisors of 60, so 60 is in the sequence.
MATHEMATICA
q[m_] := Module[{d = Divisors[m], ab}, ab = Total[d] - 2*m; ab > 0 && AnyTrue[Subsets[d], Times @@ # == ab &]]; Select[Range[500], q] (* Amiram Eldar, Apr 18 2025 *)
PROG
(SageMath)
def facile_candidates(n):
from itertools import combinations
divs = divisors(n)
sigma_n = sigma(n, 1)
candidates = set()
# Generate all products of distinct combinations of divisors
for r in range(2, len(divs)+1): # start from 2-element products to avoid m=n
for combo in combinations(divs, r):
product = prod(combo)
if product < sigma_n:
candidates.add(product)
return sorted(candidates)
def find_facile_perfects(x):
result = []
for n in range(1, x+1):
sig = sigma(n, 1)
if sig < 2*n:
continue
candidates = facile_candidates(n)
for m in candidates:
if sig == 2*n + m:
print(n, m)
result.append(n)
break
return result
(PARI) prodDistinctDiv(n, k, f=factor(n))=my(D=divisors([n, f])); helper(D[2..#D], k)
helper(v, k)=if(k==1, return(1)); v=select(d->k%d==0, v); if(#v<3, if(#v==2, return(v[2]==k || vecprod(v)==k)); return(#v && v[1]==k)); my(u=v[1..#v-1]); helper(u, k) || helper(u, k/v[#v])
is(n, f=factor(n))=my(t=sigma([n, f])-2*n); t>1 && prodDistinctDiv(n, t, f) \\ Charles R Greathouse IV, Apr 24 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Joshua Zelinsky, Apr 17 2025
STATUS
approved
