A007620 Numbers m such that every k <= m is a sum of proper divisors of m (for m>1). (Formerly M4095)

1, 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 272, 276, 280, 288, 294, 300, 304, 306

Offset

1,2

Comments

This sequence was formerly called "practical numbers (second definition)" because it was thought this was the definition used in Srinivasan's original paper. However, in Srinivasan's paper, one can read that his definition is "k < m". Stewart proves that Srinivasan's definition is equivalent to requiring every k <= sigma(m) be the sum of distinct divisors of m. This sequence is a subsequence of the practical numbers, A005153. - T. D. Noe, Apr 02 2010

A005153 without terms larger than 1 that are almost-perfect numbers (numbers k such that sigma(k) = 2*k-1, the only known such numbers are the powers of 2, A000079). - Amiram Eldar, Apr 07 2023

References

Ross Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=1..1000

A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180.

B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics, Vol. 76, No. 4 (1954), pp. 779-785.

Robert G. Wilson v, Letter to N. J. A. Sloane, date unknown.

Mathematica

DeleteCases[ A005835, q_/; (Count[ CoefficientList[ Series[ Times@@( (1+z^#)& /@ Divisors[ q ] ), {z, 0, q} ], z ], 0 ]>0) ] (* Wouter Meeussen *)

Prog

(Haskell)
a007620 n = a007620_list !! (n-1)
a007620_list = 1 : filter (\x -> all (p $ a027751_row x) [1..x]) [2..]
   where p _  0 = True
         p [] _ = False
         p ds'@(d:ds) m = d <= m && (p ds (m - d) || p ds m)
-- Reinhard Zumkeller, Feb 23 2014
(Python)
from itertools import count, islice
from sympy import divisors
def A007620_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue, 1)):
        if m == 1:
            yield 1
        else:
            c = {0}
            for d in divisors(m, generator=True):
                if d < m:
                    c |= {a+d for a in c}
            if all(a in c for a in range(m+1)):
                yield m
A007620_list = list(islice(A007620_gen(), 30)) # Chai Wah Wu, Jul 06 2023

Crossrefs

Cf. A005153 (first definition).

Cf. A000079, A027751.

Sequence in context: A023196 A204829 A005835 * A100715 A352030 A324652

Adjacent sequences: A007617 A007618 A007619 * A007621 A007622 A007623

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Robert G. Wilson v

Status

approved