A085844 Numbers equal to a permutation (or rearrangement) of the digits of the sum of their proper divisors. Rearrangements which cause leading zeros are excluded.

6, 28, 411, 496, 604, 1305, 3664, 4086, 4672, 4896, 5046, 7785, 8128, 8739, 9331, 14535, 16012, 18342, 18585, 19648, 20634, 21534, 21628, 22365, 25911, 27568, 28108, 29160, 29188, 31185, 32091, 32271, 34956, 35898, 35925, 36172, 37698, 40182, 54945, 56970

Offset

1,1

Comments

Perfect numbers (A000396) are a subsequence. Perfect numbers are defined as numbers equal to the sum of their proper divisors. The numbers in this sequence are not perfect, but are "permutably perfect" since they are equal to a permutation of the sum of their proper divisors.

Note a(79)=195921 and a(80)=199251 and these are both permutations of each other. The same is true of a(90)=226944 and a(93)=244692.

Links

V. Raman, Table of n, a(n) for n = 1..10000

Example

604 is a term because the divisors of 604 are {1, 2, 4, 151, 302, 604} and 1 + 2 + 4 + 151 + 302 = 460.
16012 is included because it has the proper divisors 1, 2, 4, 4003, 8006. These divisors sum to 12016, which is a rearrangement of the digits of 16012.

Prog

(PARI) is_A085844(n)={vecsort(digits(n))==vecsort(digits(sigma(n)-n))} \\ M. F. Hasler, Feb 24 2014
(Python)
from sympy import divisors
def ok(n):
    divs = divisors(n)[:-1]
    return sorted(str(n)) == sorted(str(sum(divs)))
print(list(filter(ok, range(1, 57000)))) # Michael S. Branicky, Jun 28 2021

Crossrefs

Cf. A000396.

Sequence in context: A332465 A335290 A173360 * A325638 A331752 A083387

Adjacent sequences: A085841 A085842 A085843 * A085845 A085846 A085847

Keyword

base,nonn

Author

Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 05 2003; corrected Jul 07 2003

Extensions

Typo in example corrected by Alan Frank, Jan 20 2010

Status

approved