A271816 Deficient-perfect numbers: Deficient numbers n such that n/(2n-sigma(n)) is an integer.

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

Offset: 1

Author

Carlo Francisco E. Adajar, Apr 14 2016

Keywords

• nonn

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

Examples

			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.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..273 (terms < 2*10^12)
• BYU Computational Number Theory Group, Odd, spoof perfect factorizations, arXiv:2006.10697 [math.NT], 2020.
• Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017), pp. 12-20, arXiv preprint, arXiv:1610.01868 [math.NT], 2016.
• Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
• Hùng Việt Chu, Divisibility of Divisor Functions of Even Perfect Numbers, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
• Cui-Fang Sun and Zhao-Cheng He, On odd deficient-perfect 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. 541-546.
• M. Tang, X. Z. Ren, and 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.

Crossrefs

Cf. A000203, A005100, A033879.

Deficient analog of A153501. Setwise difference A097498\A153501.

Contains A000079.

Programs

Maple

q:= k-> (s-> s>0 and irem(k, s)=0)(2*k-numtheory[sigma](k)):
select(q, [$1..500000])[];  # Alois P. Heinz, Aug 26 2023

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 *)

PARI

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

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.)