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A263811 Numbers n such that n = tau(n) * phi(n-1) + 1. 1
3, 5, 17, 25, 49, 257, 289, 65537 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers n such that n = A000005(n) * A000010(n-1) + 1.
The first 5 known Fermat primes from A019434 are in sequence.
The next term, if it exists, must be greater than 2*10^7.
A prime p is in the sequence iff p is a Fermat prime (A019434) - see proof in A171271.
Observation: the known composite terms are squares of primes. - Omar E. Pol, Nov 04 2015
From Charlie Neder, Mar 02 2019: (Start)
Rearranging the definition gives (n-1)/phi(n-1) = tau(n), which means n-1 is in A007694. Since n-1 is thus 3-smooth, there are two possibilities:
1) n-1 is a power of 2 and tau(n) = 2, i.e. n is a Fermat prime,
2) n-1 is a 3-smooth number divisible by 6 and tau(n) = 3, i.e. n is a Pierpont number and the square of a prime.
In the second case, n-1 factors as (p-1)(p+1) for some p, and both parts are 3-smooth if and only if p is in {2,3,5,7,17} (2 and 3 are excluded since in those cases n-1 is not divisible by 6). Therefore, this sequence is complete if and only if there are no more Fermat primes. (End)
LINKS
EXAMPLE
17 is in this sequence because 17 = tau(17)*phi(16)+1 = 2*8+1.
MATHEMATICA
Select[Range[10^5], # == DivisorSigma[0, #] EulerPhi[# - 1] + 1 &] (* Michael De Vlieger, Nov 05 2015 *)
PROG
(Magma) [n: n in [2..1000000] | n eq NumberOfDivisors(n) * EulerPhi(n-1) + 1]
(PARI) for(n=1, 1e5, if( n-1 == numdiv(n)*eulerphi(n-1) , print1(n, ", "))) \\ Altug Alkan, Nov 05 2015
CROSSREFS
Cf. A263810 (numbers n such that n = tau(n) * phi(n-2) + 1).
Sequence in context: A079649 A354724 A255401 * A323194 A024867 A025111
KEYWORD
nonn,hard,more
AUTHOR
Jaroslav Krizek, Nov 04 2015
STATUS
approved

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Last modified June 7 00:08 EDT 2023. Contains 363151 sequences. (Running on oeis4.)