

A271816


Deficientperfect numbers: Deficient numbers n such that n/(2nsigma(n)) is an integer.


7



1, 2, 4, 8, 10, 16, 32, 44, 64, 128, 136, 152, 184, 256, 512, 752, 884, 1024, 2048, 2144, 2272, 2528, 4096, 8192, 8384, 12224, 16384, 17176, 18632, 18904, 32768, 32896, 33664, 34688, 49024, 63248, 65536, 85936, 106928, 116624, 117808, 131072, 262144, 524288, 526688, 527872, 531968, 556544, 589312, 599072, 654848, 709784
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Every power of 2 is part of this sequence, with 2n  sigma(n) = 1.
Any odd element of this sequence must be a perfect square with at least four distinct prime factors (Tang and Feng). The smallest odd element of this sequence is a(74) = 9018009 = 3^2 * 7^2 * 11^2 * 13^2, with 2n  sigma(n) = 819.
a(17) = 884 = 2^2 * 13 * 17 is the smallest element with three distinct prime factors.
For all n > 1 in this sequence, 5/3 <= sigma(n)/n < 2.  Charles R Greathouse IV, Apr 15 2016


LINKS

Giovanni Resta, Table of n, a(n) for n = 1..273 (terms < 2*10^12)
Jose A. B. Dris, Conditions Equivalent to the DescartesFrenicleSorli Conjecture on Odd Perfect Numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017), pp. 1220, arXiv preprint, arXiv:1610.01868 [math.NT], 2016.
Jose Arnaldo Bebita Dris, DoliJane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199205.
CuiFang Sun, ZhaoCheng He, On odd deficientperfect numbers with four distinct prime divisors, arXiv:1908.04932 [math.NT], 2019.
Judy Holdener and Emily Rachfal, Perfect and Deficient Perfect Numbers, The American Mathematical Monthly, Vol. 126, No. 6 (2019), pp. 541546.
M. Tang, X. Z. Ren, M. Li, On NearPerfect and DeficientPerfect Numbers, Colloq. Math. 133 (2013), 221226.
M. Tang and M. Feng, On DeficientPerfect Numbers, Bull. Aust. Math. Soc. 90 (2014), 186194.


FORMULA

2^k is always an element of this sequence.
If 2^(k+1) + 2^t  1 is an odd prime and t <= k, then n = 2^k(2^(k+1) + 2^t  1) is deficientperfect with 2n  sigma(n) = 2^t. In fact, these are the only terms with two distinct prime factors. (Tang et al.)


EXAMPLE

When n = 1, 2, 4, 8, 2n  sigma(n) = 1.
When n = 10, sigma(10) = 18 and so 2*10  18 = 2, which divides 10.


MATHEMATICA

ok[n_] := Block[{d = DivisorSigma[1, n]}, d < 2*n && Divisible[n, 2*n  d]]; Select[Range[10^5], ok] (* Giovanni Resta, Apr 14 2016 *)


PROG

(PARI) isok(n) = ((ab = (sigma(n)2*n))<0) && (n % ab == 0); \\ Michel Marcus, Apr 15 2016


CROSSREFS

Cf. A000203, A033879.
Deficient analog of A153501. Contains A000079.
Sequence in context: A291165 A083655 A335404 * A097210 A097214 A045579
Adjacent sequences: A271813 A271814 A271815 * A271817 A271818 A271819


KEYWORD

nonn


AUTHOR

Carlo Francisco E. Adajar, Apr 14 2016


STATUS

approved



