A383148 k-facile numbers: Numbers m such that the sum of the divisors of m is equal to 2*m+s where s is a product of distinct divisors of m.

12, 18, 20, 24, 30, 40, 42, 54, 56, 60, 66, 78, 84, 88, 90, 102, 104, 114, 120, 132, 138, 140, 168, 174, 186, 196, 204, 222, 224, 234, 246, 252, 258, 264, 270, 280, 282, 308, 312, 318, 348, 354, 360, 364, 366, 368, 380, 402, 414, 420, 426, 438, 440, 456, 464, 468, 474, 476

Offset: 1

Author

Joshua Zelinsky, Apr 17 2025

Keywords

• nonn

Comments

Subsequence of A005101 but seem to be much rarer.

Examples

			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.
		

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.

Crossrefs

Subsequence of A005101.

Cf. A000203, A000396, A181595.

Programs

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 *)

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

Sage

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