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

1,1

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

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A214408.

Sequence in context: A085582 A058310 A005720 * A200448 A124852 A220923

Adjacent sequences: A060323 A060324 A060325 * A060327 A060328 A060329

Keyword

nonn

Author

Phil Mason (hattrack(AT)usa.net)

Extensions

More terms from Michel Marcus, Oct 07 2013

Status

approved