A214408 -id:A214408 - cp's OEIS frontend

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

Phil Mason (hattrack(AT)usa.net)

nonn

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

			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.

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

Cf. A214408.

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

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

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

More terms from Michel Marcus, Oct 07 2013

A214547 Deficient numbers for which the (absolute value of) abundance is not a divisor.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106

Offset: 1

Jonathan Vos Post, Jul 20 2012

This is to A214408 as deficient numbers are to abundant numbers.
Differs from A097218, which does not contain 105, for example.
The deficient numbers which are *not* in the sequence are 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, ... the union of powers of 2 and the terms of A060326. - M. F. Hasler, Jul 21 2012

			7 is in the sequence because 7 is deficient, and its abundance is -6, and |-6| = 6 does not divide 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005100, A033880, A060326, A097218, A214408.

filter:= proc(n) local t;
t:= 2*n-numtheory:-sigma(n);
t > 0 and n mod t <> 0
end proc:
select(filter, [$1..200]); # Robert Israel, Nov 13 2019

q[n_] := Module[{def = 2*n - DivisorSigma[1, n]}, def > 0 && !Divisible[n, def]]; Select[Range[120], q] (* Amiram Eldar, Apr 07 2024 *)

is_A214547(n)={sigma(n)<2*n & n%(2*n-sigma(n))} \\ M. F. Hasler, Jul 21 2012

Terms A005100(n) such that |A033880(A005100(n))| does not divide A005100(n).

Given terms double-checked with the PARI script by M. F. Hasler, Jul 21 2012

cp's OEIS Frontend

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.

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