

A271816


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


4



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
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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.
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: A226816 A291165 A083655 * A097210 A097214 A045579
Adjacent sequences: A271813 A271814 A271815 * A271817 A271818 A271819


KEYWORD

nonn


AUTHOR

Carlo Francisco E. Adajar, Apr 14 2016


STATUS

approved



