cp's OEIS Frontend

A085118 Primes together with twice the odd primes.

%I A085118 #41 Feb 16 2025 08:32:50
%S A085118 2,3,5,6,7,10,11,13,14,17,19,22,23,26,29,31,34,37,38,41,43,46,47,53,
%T A085118 58,59,61,62,67,71,73,74,79,82,83,86,89,94,97,101,103,106,107,109,113,
%U A085118 118,122,127,131,134,137,139,142,146,149,151,157,158,163,166,167,173,178
%N A085118 Primes together with twice the odd primes.
%C A085118 Probably the same sequence as: numbers n such that phi(n)+1 divides n.
%C A085118 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
%H A085118 Vincenzo Librandi, <a href="/A085118/b085118.txt">Table of n, a(n) for n = 1..1000</a>
%H A085118 G. L. Cohen and S. L. Segal, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-3/cohen.pdf">A note concerning those n for which phi(n)+1 divides n</a>, Fibonacci Quarterly, Vol. 27, No. 3 (1989), pp. 285-286.
%H A085118 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LehmersTotientProblem.html">Lehmer's Totient Problem</a>
%t A085118 With[{nn=40},Take[Sort[Join[Prime[Range[2nn]],2Prime[Range[2,nn]]]],2nn]] (* _Harvey P. Dale_, Oct 03 2013 *)
%o A085118 (Python)
%o A085118 from sympy import primepi
%o A085118 def A085118(n):
%o A085118     def bisection(f,kmin=0,kmax=1):
%o A085118         while f(kmax) > kmax: kmax <<= 1
%o A085118         while kmax-kmin > 1:
%o A085118             kmid = kmax+kmin>>1
%o A085118             if f(kmid) <= kmid:
%o A085118                 kmax = kmid
%o A085118             else:
%o A085118                 kmin = kmid
%o A085118         return kmax
%o A085118     def f(x): return int(n+x-primepi(x)-primepi(x>>1)+(x>=4))
%o A085118     return bisection(f,n,n) # _Chai Wah Wu_, Oct 17 2024
%Y A085118 Cf. A000010, A068422.
%Y A085118 Equals A001751\{4}.
%K A085118 nonn,easy
%O A085118 1,1
%A A085118 _Leroy Quet_, Apr 25 2004
%E A085118 More terms from _David Wasserman_, Jan 27 2005