A243305 -id:A243305 - cp's OEIS frontend

A281623 Numbers of the form 2^phi(m) + 1, where phi = A000010 = Euler totient function.

3, 5, 17, 65, 257, 1025, 4097, 65537, 262145, 1048577, 4194305, 16777217, 268435457, 1073741825, 4294967297, 68719476737, 1099511627777, 4398046511105, 17592186044417, 70368744177665, 281474976710657, 4503599627370497, 18014398509481985, 72057594037927937

Offset: 1

Jaroslav Krizek, Jan 25 2017

nonn

Possible values of A243305 (2^phi(n) + 1).

The first 5 known Fermat primes from A019434 are in the sequence.

			5 = 2^2 + 1 is a term because there are 3 numbers n (3, 4 and 6) with phi(n) = 2.

Cf. A000010, A019434, A243305.

Set(Sort([2^(EulerPhi(n)) + 1: n in[1..500]]));

Union[2^EulerPhi@ Range[10^3] + 1] (* Michael De Vlieger, Jan 30 2017 *)

A281624 Numbers m such that 2^phi(m) + 1 is prime (Fermat prime).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 60

Offset: 1

Jaroslav Krizek, Jan 25 2017

Numbers m such that A243305(m) is a Fermat prime (A019434).
If there are only 5 Fermat primes, sequence is finite with 20 terms; corresponding values of Fermat primes: 3, 3, 5, 5, 17, 5, 17, 17, 17, 257, 257, 65537, 257, 257, 257, 65537, 65537, 65537, 65537, 65537.
Number of numbers k such that 2^phi(k) + 1 = A019434(n) for n = 1-5: 2, 3, 4, 5, 6.

			10 is a term because 2^phi(10) + 1 = 2^4 + 1 = 17 (prime).

Subsequence of A003401.

Cf. A000010 (phi(n)), A019434, A243305, A281623.

[n: n in[1..10000] | IsPrime(2^(EulerPhi(n)) + 1)];

Select[Range[60], PrimeQ[2^EulerPhi[#] + 1] &] (* Paolo Xausa, Jan 18 2025 *)

is(n)=isprime(2^eulerphi(n)+1) \\ Charles R Greathouse IV, Jan 27 2017

cp's OEIS Frontend

A281623 Numbers of the form 2^phi(m) + 1, where phi = A000010 = Euler totient function.

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