cp's OEIS Frontend

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

%I A007620 M4095 #41 Jul 06 2023 09:05:10
%S A007620 1,6,12,18,20,24,28,30,36,40,42,48,54,56,60,66,72,78,80,84,88,90,96,
%T A007620 100,104,108,112,120,126,132,140,144,150,156,160,162,168,176,180,192,
%U A007620 196,198,200,204,208,210,216,220,224,228,234,240,252,260,264,270,272,276,280,288,294,300,304,306
%N A007620 Numbers m such that every k <= m is a sum of proper divisors of m (for m>1).
%C A007620 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
%C A007620 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
%D A007620 Ross Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
%D A007620 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007620 T. D. Noe, <a href="/A007620/b007620.txt">Table of n, a(n) for n=1..1000</a>
%H A007620 A. K. Srinivasan, <a href="https://web.archive.org/web/20200815171531/https://www.currentscience.ac.in/Downloads/article_id_017_06_0179_0180_0.pdf">Practical numbers</a>, Current Science, 17 (1948), 179-180.
%H A007620 B. M. Stewart, <a href="http://doi.org/10.2307/2372651">Sums of distinct divisors</a>, American Journal of Mathematics, Vol. 76, No. 4 (1954), pp. 779-785.
%H A007620 Robert G. Wilson v, <a href="/A007621/a007621.pdf">Letter to N. J. A. Sloane</a>, date unknown.
%t A007620 DeleteCases[ A005835, q_/; (Count[ CoefficientList[ Series[ Times@@( (1+z^#)& /@ Divisors[ q ] ), {z, 0, q} ], z ], 0 ]>0) ] (* _Wouter Meeussen_ *)
%o A007620 (Haskell)
%o A007620 a007620 n = a007620_list !! (n-1)
%o A007620 a007620_list = 1 : filter (\x -> all (p $ a027751_row x) [1..x]) [2..]
%o A007620    where p _  0 = True
%o A007620          p [] _ = False
%o A007620          p ds'@(d:ds) m = d <= m && (p ds (m - d) || p ds m)
%o A007620 -- _Reinhard Zumkeller_, Feb 23 2014
%o A007620 (Python)
%o A007620 from itertools import count, islice
%o A007620 from sympy import divisors
%o A007620 def A007620_gen(startvalue=1): # generator of terms >= startvalue
%o A007620     for m in count(max(startvalue,1)):
%o A007620         if m == 1:
%o A007620             yield 1
%o A007620         else:
%o A007620             c = {0}
%o A007620             for d in divisors(m,generator=True):
%o A007620                 if d < m:
%o A007620                     c |= {a+d for a in c}
%o A007620             if all(a in c for a in range(m+1)):
%o A007620                 yield m
%o A007620 A007620_list = list(islice(A007620_gen(),30)) # _Chai Wah Wu_, Jul 06 2023
%Y A007620 Cf. A005153 (first definition).
%Y A007620 Cf. A000079, A027751.
%K A007620 nonn,nice,easy
%O A007620 1,2
%A A007620 _N. J. A. Sloane_, _Robert G. Wilson v_