A234310 -id:A234310 - cp's OEIS frontend

A234388 Primes of the form 2^k*(2^{phi(m)} - 1) + 1, where k and m are positive integers, and phi(.) is Euler's totient function.

3, 5, 7, 13, 17, 31, 61, 97, 127, 193, 241, 257, 769, 1009, 1021, 2017, 4093, 7681, 8161, 8191, 12289, 15361, 16369, 16381, 32257, 61441, 64513, 65521, 65537, 131041, 131071, 523777, 524287, 786433, 1032193, 1048573, 4194301, 8257537, 8380417, 16515073, 16760833, 16776961, 16777153, 16777213, 67043329, 132120577, 134215681, 268369921, 536870401, 1073479681, 2013265921, 2113929217, 2146959361, 2147483137, 2147483647, 3221225473, 4293918721, 17175674881, 34359214081, 34359738337

Offset: 1

Zhi-Wei Sun, Dec 25 2013

nonn

Conjecture: (i) Any integer n > 1 can be written as k + m with k > 0 and m > 0 such that 2^k*(2^{phi(m)} - 1) + 1 is prime.
(ii) Each integer n > 2 can be written as k + m with k > 0 and m > 0 such that 2^k*(2^{phi(m)} - 1) - 1 is prime.
Part (i) of the conjecture implies that this sequence has infinitely many terms. See also A234399.
Note that the sequence contains all Fermat primes and Mersenne primes since 2^k + 1 = 2^k*(2^{phi(1)} - 1) + 1 and 2^p - 1 = 2*(2^{phi(p)} - 1) + 1, where k is a positive integer and p is a prime.

			a(1) = 3 since 2*(2^{phi(1)} - 1) + 1 = 3 is prime.
a(2) = 5 since 2^2*(2^{phi(1)} - 1) + 1 = 5 is prime.
a(3) = 7 since 2*(2^{phi(3)} - 1) + 1 = 7 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Cf. A000010, A000040, A000079, A000668, A019434, A152449, A234309, A234310, A234337, A234344, A234346, A234347, A234359, A234360, A234361, A234399

S:=Intersection[Union[Table[EulerPhi[k],{k,1,5000}]],Table[k,{k,1,500}]]
n=0;Do[If[MemberQ[S,k]&&PrimeQ[2^m-2^(m-k)+1],n=n+1;Print[n," ",2^m-2^(m-k)+1]],{m,1,500},{k,1,m-1}]

A234399 a(n) = |{0 < k < n: 2^k*(2^phi(n-k) - 1) + 1 is prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 2, 3, 5, 4, 4, 5, 3, 6, 5, 3, 6, 8, 4, 5, 5, 6, 4, 6, 7, 4, 5, 6, 4, 3, 4, 9, 5, 3, 8, 5, 4, 3, 8, 8, 3, 8, 6, 7, 7, 8, 8, 9, 4, 5, 8, 9, 7, 6, 10, 11, 4, 6, 6, 8, 8, 10, 4, 4, 7, 4, 12, 8, 6, 4, 9, 7, 4, 6, 10, 9, 8, 7, 7, 7, 5, 4, 10, 5, 6, 7, 9, 15, 7, 8, 10, 7, 4, 8, 6, 10, 3, 3, 10, 11

Offset: 1

Zhi-Wei Sun, Dec 25 2013

nonn

Conjecture: a(n) > 0 for all n > 1.
See also the conjecture in A234388.
The conjecture is false. a(5962) = 0. - Jason Yuen, Nov 04 2024

			a(7) = 2 since 2^1*(2^phi(6)-1) + 1 = 2*3 + 1 = 7 and 2^2*(2^phi(5)-1) + 1 = 4*15 + 1 = 61 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Cf. A000010, A000040, A000079, A234309, A234310, A234337, A234344, A234346, A234347, A234359, A234360, A234361, A234388

f[n_,k_]:=f[n,k]=2^k*(2^(EulerPhi[n-k])-1)+1
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

a(n) = sum(k=1,n-1,ispseudoprime(2^k*(2^eulerphi(n-k)-1)+1)) \\ Jason Yuen, Nov 04 2024

A239714 Primes of the form m = 4^i + 4^j - 1, where i > j >= 0.

Original entry on oeis.org

19, 67, 79, 271, 1039, 1087, 1279, 4099, 4111, 4159, 5119, 16447, 20479, 65539, 65551, 65599, 81919, 262147, 262399, 263167, 266239, 1049599, 1114111, 1310719, 4194319, 4194559, 4195327, 16842751, 17825791, 67108879, 268435459, 268435711, 272629759, 1073741827, 1073741839, 1073758207

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-4 representation of a term 4^i + 4^j - 1 has base-4 digital sum = 1 + 3*j == 1 (mod 3).
In base-4 representation the first terms are 103, 1003, 1033, 10033, 100033, 100333, 103333, 1000003, 1000033, 1000333, 1033333, 10000333, 10333333, 100000003, 100000033, 100000333, 103333333, 1000000003, 1000003333, 1000033333, ...
Numbers m which satisfy m = 4^i + 4^j + 1 are never primes, since the base-4 digital sum of m is 3, and thus, m is divisible by 3.

			a(1) = 19, since 19 = 4^2 + 4^1 - 1 is prime.
a(4) = 271, since 271 = 4^4 + 4^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..111

Cf. A234310.

A239714
  "Answer an array of the first n terms of A239714.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239714
  Answer: #(19 67 79 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 4 withOffset: -1

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A234388 Primes of the form 2^k*(2^{phi(m)} - 1) + 1, where k and m are positive integers, and phi(.) is Euler's totient function.

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A234399 a(n) = |{0 < k < n: 2^k*(2^phi(n-k) - 1) + 1 is prime}|, where phi(.) is Euler's totient function.

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A239714 Primes of the form m = 4^i + 4^j - 1, where i > j >= 0.

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