A068422 -id:A068422 - cp's OEIS frontend

A085118 Primes together with twice the odd primes.

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163, 166, 167, 173, 178

Offset: 1

Leroy Quet, Apr 25 2004

Probably the same sequence as: numbers n such that phi(n)+1 divides n.

Cohen and Segal showed that in case there were other solutions to this problem (which appeared to be posed by Schinzel), then they should have at least 15 distinct prime factors. Moreover, there is a connection with the Lehmer's totient problem which asks whether there is a composite n such that phi(n)|(n-1). If no such composite exists, then p and 2p are the only members for Leroy's sequence. - Francisco Salinas (franciscodesalinas(AT)hotmail.com), Apr 25 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
G. L. Cohen and S. L. Segal, A note concerning those n for which phi(n)+1 divides n, Fibonacci Quarterly, Vol. 27, No. 3 (1989), pp. 285-286.
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem

Cf. A000010, A068422.

Equals A001751\{4}.

With[{nn=40},Take[Sort[Join[Prime[Range[2nn]],2Prime[Range[2,nn]]]],2nn]] (* Harvey P. Dale, Oct 03 2013 *)

from sympy import primepi
def A085118(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-primepi(x)-primepi(x>>1)+(x>=4))
    return bisection(f,n,n) # Chai Wah Wu, Oct 17 2024

More terms from David Wasserman, Jan 27 2005

A055164 (n - phi(n)) | sigma(n) for composite n not congruent to 2 (mod 4).

Original entry on oeis.org

99, 168, 187, 493, 637, 780, 943, 1273, 1537, 1836, 2183, 2225, 2976, 3103, 3589, 4183, 5353, 5928, 6201, 6468, 6667, 8881, 9553, 9727, 13393, 13888, 14453, 15397, 17587, 19897, 24253, 24883, 30883, 33667, 36259, 36853, 37523, 43657, 45901

Offset: 1

Robert G. Wilson v, Jun 30 2000

Let p be an odd prime. If d > p+1 is a divisor of p^2 - p - 2 such that d+1-p is prime, then p*(d+1-p) is in the sequence. - Robert Israel, May 03 2019

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A000010, A000203, A068422.

filter:= proc(n) uses numtheory;
  if isprime(n) then return false fi;
  type(sigma(n)/(n-phi(n)), integer)
end proc:
select(filter, [seq(seq(4*i+j,j=[0,1,3]),i=1..20000)]); # Robert Israel, May 03 2019

Do[ If[ !PrimeQ[ n ], If[ Mod[ n, 4 ] != 2, If[ Mod[ DivisorSigma[ 1, n ], n-EulerPhi[ n ] ] == 0, Print[ n ] ] ] ], {n, 2, 50000} ]

[n for n in (1..50000) if not mod(n, 4)==2 and not is_prime(n) and mod(sigma(n), n - euler_phi(n))==0] # G. C. Greubel, May 03 2019

cp's OEIS Frontend

A085118 Primes together with twice the odd primes.

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