|
Search: id:A005153
|
|
|
| A005153 |
|
Practical numbers (first definition): numbers n such that every k <= sigma(n) is a sum of distinct divisors of n. Also called panarithmic numbers.
(Formerly M0991)
|
|
+0
9
|
|
| 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
Equivalently, numbers n such that every number k <= n is a sum of distinct divisors of n.
2^r is a member for all r as every number < = sigma(2^r) = 2^(r+1)-1 is a sum of a distinct subset of divisors {1,2,2^2,...2^n}. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 23 2004
Also, numbers n such that A030057(n) > n. This is a consequence of the following lemma, found at the McLeman link: An integer m >= 2 with factorization Product_{i=1}^k p_i^e_i with the p_i in ascending order is practical if and only if p_1 = 2 and, for 1 < i <= k, p_i <= sigma(Product_{j < i} p_j^e_j) + 1. - Franklin T. Adams-Watters, Nov 09 2006
Comment from R. J. Mathar, Nov 27 2006: this definition is used in http://www.dm.unipi.it/gauss-pages/melfi/public_html/articoli/jnt.ps but the same author uses the definition in A007620 in his web page http://members.unine.ch/giuseppe.melfi/pratica.html. See also http://citeseer.ist.psu.edu/285.html and http://arXiv.org/abs/math.NT/0404555.
|
|
REFERENCES
|
H. Heller, Mathematical Buds, Vol. 1 Chap. 2 pp. 10-22, Mu Alpha Theta OK 1978.
M. R. Heyworth, More on Panarithmic Numbers. New Zealand Math. Mag. 17, 28-34 (1980) [ ISSN 0549-0510 ].
H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 1980, pp. 41-45.
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
E. J. Scourfield, J. Number Theory 62 (1) (1997) p. 163 uses this definition.
A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180.
B. M. Stewart, Sums of distinct divisors, Amer. J. Math., 76 (1954), 779-785.
See also Math. Rev. 96i:11106.
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=1..1000
C. McLeman, PlanetMath.org, Practical number
G. Melfi, Practical Numbers
G. Melfi, practical Numbers
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wikipedia, Practical number
|
|
CROSSREFS
|
Cf. A007620 (second definition), A030057.
Adjacent sequences: A005150 A005151 A005152 this_sequence A005154 A005155 A005156
Sequence in context: A103288 A125225 A092903 this_sequence A068563 A124240 A068997
|
|
KEYWORD
|
nonn,nice,easy
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
EXTENSIONS
|
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
|
|
|
Search completed in 0.002 seconds
|
