On Wed, 22 Jan 2003, Richard Guy wrote:
Apologies for my wildness. Mike Speciner gives the example n = 8, which generalizes to any odd power of 2. Are these the only examples? Is the sequence
8, 32, 128, 512, 2048, ... (& including any other numbers I've overlooked)
in OEIS ? R.
ID Number: A056729 Sequence: 8,27,32,63,125,128,243,275,343,399,512,567,575,935,1127, 1331,1539,2015,2048,2187,2197,2303,2783,2915,3087,3125,4563, 4913,4991,5103,5719,5831,6399,6859,6875,6929,7055,7139,7625, 8192,8855,12167,12719,14027 Name: If p | n, then p+1 | n+1 for composite n. Comments: The Lucas-Carmichael numbers (A006972) are a subset. Math'ca: Select[ Range[ 2, 10^5 ], ! PrimeQ[ # ] && Union[ Mod[ # + 1, Transpose[ FactorInteger[ # ] ][ [ 1 ] ] + 1 ] ] == {0} & ] See also: Cf. A006972. Keywords: nonn Offset: 1 Author(s): Robert G. Wilson v (RGWv@kspaint.com), Aug 31 2000
On Wed, 22 Jan 2003, Richard Guy wrote:
I'll make a wild guess that it can be proved that no such n exists; I'll copy this to some people who may be able to confirm or deny this.
R.
On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
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