The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A182816 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A182816

Number of values b in Z/nZ such that b^n = b.
(history; published version)

#56 by Michel Marcus at Thu Sep 03 01:58:56 EDT 2020

STATUS

reviewed

approved

#55 by Joerg Arndt at Thu Sep 03 01:49:52 EDT 2020

STATUS

proposed

reviewed

#54 by Richard R. Forberg at Wed Sep 02 12:11:34 EDT 2020

STATUS

editing

proposed

#53 by Richard R. Forberg at Wed Sep 02 12:10:13 EDT 2020

COMMENTS

a(n)/n is the fraction of the integers b, within any given sufficiently large contiguous range of positive integers with any starting point, that divide b^n - b. It is 100% when n is prime or a Carmichael number. For other values of n, on ranges of 1000 integers the accuracy of the measured fraction is about +/- 0.001, and correspondingly better on larger ranges. - Richard R. Forberg, Jul 29 2020

STATUS

proposed

editing

Discussion

Wed Sep 02

12:11

Richard R. Forberg: Joerg,  I agree.  I deleted it. My apologies. -- Rick

#52 by Michael De Vlieger at Tue Sep 01 16:34:07 EDT 2020

STATUS

editing

proposed

Discussion

Wed Sep 02

04:32

Joerg Arndt: Does the comment not reword the (very clear) definition into something much less clear?

#51 by Michael De Vlieger at Tue Sep 01 16:34:02 EDT 2020

MATHEMATICA

Table[Times @@ Map[(1 + GCD[n - 1, # - 1]) &, FactorInteger[n][[All, 1]] ], {n, 113}] (* Michael De Vlieger, Sep 01 2020 *)

STATUS

proposed

editing

#50 by Richard R. Forberg at Sat Aug 01 13:33:31 EDT 2020

STATUS

editing

proposed

#49 by Richard R. Forberg at Sat Aug 01 13:28:39 EDT 2020

COMMENTS

a(n)/n is also the fraction of the integers k, b, within any given sufficiently large contiguous range of positive integers with any starting point, that will divide the expressions nb^k + k - n and - b. It is 100% when n^k - k - is prime or a Carmichael number. For other values of n. On , on ranges of 1000 integers the accuracy of the measured fraction is about +/- 0.001, and correspondingly better on larger ranges. For a(n) = n (where n is a prime or Carmichael number), all values of k will divide those expressions, on any such range. See A002997. Also related is A121707. - Richard R. Forberg, Jul 29 2020

STATUS

proposed

editing

Discussion

Sat Aug 01

13:33

Richard R. Forberg: Revised again and with symbols and content made consistent with, and building upon, prior entries already made here, rather than appearing as something new.  I apologize for some recurring confusion on my part.

#48 by Richard R. Forberg at Sat Aug 01 11:54:01 EDT 2020

STATUS

editing

proposed

#47 by Richard R. Forberg at Sat Aug 01 11:51:34 EDT 2020

COMMENTS

a(n)/n is also the fraction of the integers k, within any given sufficiently large contiguous range of positive integers with any starting point, that will divide the expression expressions n^k + k - n and n^k - k -n. On ranges of 1000 integers the accuracy of the measured fraction is about +/- 0.001, and correspondingly better on larger ranges. For a(n) = n (where n is a prime or Carmichael number), all values of k will divide those expressions, on any such range. See A002997. Also related is A121707. - Richard R. Forberg, Jul 29 2020

STATUS

proposed

editing

Discussion

Sat Aug 01

11:53

Richard R. Forberg: I was too quick to change it. The original is the correct version of this. So I reverted. I should checked the math first. - Rick