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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 Table of n, a(n) for n=1..8. 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. A000005, A000010, A019434. Cf. A263810 (numbers n such that n = tau(n) * phi(n-2) + 1). Sequence in context: A079649 A354724 A255401 * A323194 A024867 A025111 Adjacent sequences: A263808 A263809 A263810 * A263812 A263813 A263814 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.)