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A271816 Deficient-perfect numbers: Deficient numbers n such that n/(2n-sigma(n)) is an integer. 1
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 Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1610.01868 [math.NT], 2016.

M. Tang, X. Z. Ren, M. Li, On Near-Perfect and Deficient-Perfect Numbers, Colloq. Math. 133 (2013), 221-226.

M. Tang and M. Feng, On Deficient-Perfect Numbers, Bull. Aust. Math. Soc. 90 (2014), 186-194.

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 deficient-perfect 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

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

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Last modified October 23 23:22 EDT 2017. Contains 293833 sequences.