A060326 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.

10, 44, 136, 152, 184, 752, 884, 2144, 2272, 2528, 8384, 12224, 17176, 18632, 18904, 32896, 33664, 34688, 49024, 63248, 85936, 106928, 116624, 117808, 526688, 527872, 531968, 556544, 589312, 599072, 654848, 709784, 801376, 879136, 885928, 1090912

Offset: 1

Author

Phil Mason (hattrack(AT)usa.net)

Keywords

• nonn

Comments

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

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

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

Examples

			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.
		

Links

• R. J. Mathar and Donovan Johnson, Table of n, a(n) for n = 1..200 (first 42 terms from R. J. Mathar)

Crossrefs

Cf. A214408.

Programs

Mathematica

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

PARI

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

Formula

{ m in A005100 \ A000079 : A033879(m) divides m }. - M. F. Hasler, Jul 21 2012

Extensions

More terms from Michel Marcus, Oct 07 2013