The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A111592

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A111592

Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.
(history; published version)

#59 by Michael De Vlieger at Fri Mar 29 10:22:22 EDT 2024

STATUS

proposed

approved

#58 by Michel Marcus at Fri Mar 29 10:20:57 EDT 2024

STATUS

editing

proposed

#57 by Michel Marcus at Fri Mar 29 10:20:54 EDT 2024

PROG

(PARI) for(n=1, 10^3, ap=sigma(n)-2*n; if(ap>0 && (ap%2)==0, d=ap/2; if(d!=n && (n%d)==0, print1(n", ")))) - \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

STATUS

approved

editing

#56 by Joerg Arndt at Mon Jan 27 01:46:01 EST 2020

STATUS

reviewed

approved

#55 by Michel Marcus at Mon Jan 27 01:41:50 EST 2020

STATUS

proposed

reviewed

#54 by Jon E. Schoenfield at Mon Jan 27 00:26:22 EST 2020

STATUS

editing

proposed

#53 by Jon E. Schoenfield at Mon Jan 27 00:26:12 EST 2020

COMMENTS

Odd terms are listed in A109729. For abundant non-squares, nonsquares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ... (A146768), and odd m = 13167. - M. F. Hasler, Jan 26 2020

STATUS

proposed

editing

Discussion

Mon Jan 27

00:26

Jon E. Schoenfield: (per the Style Sheet)  :-)

#52 by M. F. Hasler at Sun Jan 26 23:05:10 EST 2020

STATUS

editing

proposed

#51 by M. F. Hasler at Sun Jan 26 23:05:04 EST 2020

CROSSREFS

Cf. A000396 (perfect numbers), A005100, (deficient numbers), A000203, (sigma), A061645.

#50 by M. F. Hasler at Sun Jan 26 23:04:06 EST 2020

COMMENTS

Odd terms are listed in A109729. For abundant non-squares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ..., (A146768), and odd m = 13167. - M. F. Hasler, Jan 26 2020

CROSSREFS

Subsequence of A005101 (abundant numbers).

Cf. A000396, (perfect numbers), A005100, A000203, A061645.

Cf. A109729 (odd admirable numbers).