A060326 - OEIS

%I #30 Feb 17 2023 20:59:20

%S 10,44,136,152,184,752,884,2144,2272,2528,8384,12224,17176,18632,

%T 18904,32896,33664,34688,49024,63248,85936,106928,116624,117808,

%U 526688,527872,531968,556544,589312,599072,654848,709784,801376,879136,885928,1090912

%N Numbers m such that 2*m - sigma(m) is a divisor of m and greater than one, where sigma = A000203 is the sum of divisors.

%C For m=2^k, sigma(m)=2m-1, so that 2m-sigma(m)=1 would trivially divide m. These m are excluded. All abundant numbers (with sigma(m) > 2m) are also excluded, even when sigma(m) - 2m divides m, as for m=12 which is a multiple of 2m - sigma(m) = -4. - _M. F. Hasler_, Jul 21 2012

%C The sequence can also be obtained by looking for numbers whose abundancy sigma(m)/m is of the form (2*k-1)/k (hence deficient), while excluding powers of 2. - _Michel Marcus_, Oct 07 2013

%C Contains 2^(p-1)*(2^p + 2^q - 1) whenever 0 < q < p and 2^p + 2^q - 1 is prime. - _Michael R Peake_, Feb 01 2023

%H R. J. Mathar and Donovan Johnson, <a href="/A060326/b060326.txt">Table of n, a(n) for n = 1..200</a> (first 42 terms from R. J. Mathar)

%F { m in A005100 \ A000079 : A033879(m) divides m }. - _M. F. Hasler_, Jul 21 2012

%e m=10 is a term because the divisors of 10 are 1,2,5,10, with sum 18 and 2*m-18 = 2, which divides 10. Or sigma(10)/10 = 9/5 = (2*k-1)/k with k=5.

%t sdnQ[n_]:=Module[{c=2n-DivisorSigma[1,n]},c>1&&Divisible[n,c]]; Select[ Range[600000],sdnQ] (* _Harvey P. Dale_, Jul 23 2012 *)

%o (PARI) for(n=1,6e5,(t=2*n-sigma(n))>1 & !(n%t) & print1(n","))  \\ _M. F. Hasler_, Jul 21 2012

%Y Cf. A214408.

%K nonn

%O 1,1

%A Phil Mason (hattrack(AT)usa.net)

%E More terms from _Michel Marcus_, Oct 07 2013