A065381 -id:A065381 - cp's OEIS frontend

A256237 Primes p such that for all 2^k < p the numbers p + 2^k, p - 2^k, p2^k + 1, and p2^k - 1 are composite.

8923, 24943, 35437, 42533, 52783, 83437, 105953, 116437, 126631, 133241, 145589, 164729, 172331, 192173, 204013, 215279, 254329, 304709, 308899, 398833, 430499, 436687, 454351, 476869, 479909, 483443, 497597, 522479, 527729, 529103, 545257, 561439, 562651

Offset: 1

Arkadiusz Wesolowski, Mar 20 2015

nonn

Subsequence of A256163.

Cf. A006285, A065381, A153352, A180247, A255967.

lst:=[]; for p in [3..562651 by 2] do if IsPrime(p) then t:=0; k:=0; while 2^k lt p do if IsPrime(p-2^k) or IsPrime(p+2^k) or IsPrime(p*2^k-1) or IsPrime(p*2^k+1) then t:=1; break; end if; k+:=1; end while; if IsZero(t) then Append(~lst, p); end if; end if; end for; lst;

A382074 a(n) is the number of solutions to phi(x) + phi(n-x) = phi(n) where 1 <= x <= floor(n/2).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 3, 2, 2, 0, 2, 2, 2, 2, 4, 0, 0, 0, 1, 3, 1, 1, 2, 0, 3, 1, 4, 0, 1, 0, 5, 3, 2, 0, 2, 0, 2, 3, 5, 0, 2, 1, 5, 2, 1, 0, 1, 0, 2, 2, 1, 2, 2, 0, 5, 2, 2, 0, 3, 0, 2, 4, 5, 1, 3, 0, 4, 0, 1, 0, 2, 2, 2, 4, 5

Offset: 1

Felix Huber, Mar 22 2025

nonn

If p is a prime and p != 3, then a(p) = 0. Proof: For p = 2, phi(1) + phi(1) = 2 > phi(2) = 1. For p > 3, phi(x) + phi(p-x) <= x - 1 + p - x - 1 = p - 2 < p - 1 = phi(p).
If a(2*i) = 0, then i is a positive odd number. Proof: If i is a positive even number, then 2*i = 2^k*(2*m-1) with integers k, m where k > 1 and m > 0. Since phi(2^k*(2*m-1)) = phi(2^k)*phi(2*m-1) = 2^(k-1)*phi(2*m-1) = 2*2^(k-2)*phi(2*m-1) = 2*phi(2^(k-1)*(2*m-1)), it follows that x = 2^(k-1)*(2*m-1) is a solution to phi(x) + phi(2^k*(2*m-1)-x) = phi(2^k*(2*m-1)).
a(2*i) = 0 is not true for every positive odd i. For example, a(2*3) = 1. It is conjectured that a(2*A065381(n)) = 0 for n > 1. However, there are positive odd numbers i not in A065381 and for which a(2*i) = 0. For example, a(2*529) = a(2*1155) = 0.

			a(20) = 3 because phi(x) + phi(20-x) = phi(20) has 3 solutions for 0 <= x <= 10:
  x = 6: phi(6) + phi(14) = 2 + 6 = 8 = phi(20).
  x = 8: phi(8) + phi(12) = 4 + 4 = 8 = phi(20).
  x = 10: phi(10) + phi(10) = 4 + 4 = 8 = phi(20).

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000010, A065381, A211225, A381747.

with(NumberTheory):
A382074:=proc(n)
    local a,x;
    a:=0;
    for x to n/2 do
        if phi(x)+phi(n-x)=phi(n) then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A382074(n),n=1..88);

a(n) = my(e=eulerphi(n)); sum(x=1, n\2, eulerphi(x) + eulerphi(n-x) == e); \\ Michel Marcus, Mar 22 2025

a(p) = 0 for primes p != 3.
a(2^k*(2*m-1)) > 0 for integers k, m where k > 1 and m > 0.
Conjecture: a(2*A065381(n)) = 0 for n > 1.

A181748 Twin primes not of the form p+2^k where p is prime and k>0.

Original entry on oeis.org

3, 149, 599, 809, 1019, 1619, 2789, 2999, 3119, 3299, 3539, 4001, 4229, 4271, 4649, 5099, 6269, 6449, 6659, 6791, 6869, 7331, 7547, 8087, 8387, 8429, 8861, 9239, 9431, 9929, 10007, 11069, 11171, 11549, 12239, 12251, 13001, 13217, 13679, 13901, 14009, 14081, 14249

Offset: 1

Juri-Stepan Gerasimov, Nov 09 2010

nonn

			149 is a twin prime, 149-2^1 through 149-2^7 are all nonprime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001359, A065381.

seqQ[p_] := PrimeQ[p] && PrimeQ[p + 2] && Module[{k = 2}, While[k < p && !PrimeQ[p - k], k *= 2]; k > p]; Select[Range[10^4], seqQ] (* Amiram Eldar, Dec 24 2019 *)

Edited and extended by D. S. McNeil, Nov 17 2010

More terms from Amiram Eldar, Dec 24 2019

A263645 Primes that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

Original entry on oeis.org

2, 52504261, 55414847, 79933129, 152485283, 166441831, 177702619, 197903207, 199013093, 220403959, 226794259, 230701763, 245215801, 266642731, 304921637, 321979283, 335035097, 355404353, 359018299, 369810769, 388048561, 412590797, 445661719, 506400173, 540426473

Offset: 1

Arkadiusz Wesolowski, Oct 22 2015

nonn

Primes p such that for all k > 0 the numbers p + 2^k and p - 2^k are nonprimes.

Except for 2, this sequence is the intersection of A065381 and A137715.

Cf. A065381, A137715, A153352, A180247, A263644.

cp's OEIS Frontend

A256237 Primes p such that for all 2^k < p the numbers p + 2^k, p - 2^k, p2^k + 1, and p2^k - 1 are composite.

Views

Author

Keywords

Crossrefs

Programs

Magma

A382074 a(n) is the number of solutions to phi(x) + phi(n-x) = phi(n) where 1 <= x <= floor(n/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A181748 Twin primes not of the form p+2^k where p is prime and k>0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A263645 Primes that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

Views

Author

Keywords

Comments

Crossrefs

cp's OEIS Frontend

A256237 Primes p such that for all 2^k < p the numbers p + 2^k, p - 2^k, p*2^k + 1, and p*2^k - 1 are composite.

Views

Author

Keywords

Crossrefs

Programs

Magma

A382074 a(n) is the number of solutions to phi(x) + phi(n-x) = phi(n) where 1 <= x <= floor(n/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A181748 Twin primes not of the form p+2^k where p is prime and k>0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A263645 Primes that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

Views

Author

Keywords

Comments

Crossrefs

A256237 Primes p such that for all 2^k < p the numbers p + 2^k, p - 2^k, p2^k + 1, and p2^k - 1 are composite.