login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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), 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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 17 16:32 EDT 2021. Contains 347487 sequences. (Running on oeis4.)