

A007620


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


4



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
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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 < n". Stewart proves that Srinivasan's definition is equivalent to requiring every k <= sigma(n) be the sum of distinct divisors of n. This sequence is a subsequence of the practical numbers, A005153. [T. D. Noe, Apr 02 2010]


REFERENCES

R. 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), 179180.
B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics, Vol. 76, No. 4 (1954), pp. 779785.
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 !! (n1)
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


CROSSREFS

Cf. A005153 (first definition).
Cf. A027751.
Sequence in context: A023196 A204829 A005835 * A100715 A324652 A205525
Adjacent sequences: A007617 A007618 A007619 * A007621 A007622 A007623


KEYWORD

nonn,nice,easy


AUTHOR

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


STATUS

approved



