The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A233867 a(n) = |{0 < m < 2*n: m is a square with 2*n - 1 - phi(m) prime}|, where phi(.) is Euler's totient function (A000010). 8
 0, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 4, 2, 1, 3, 2, 1, 3, 3, 1, 4, 2, 1, 6, 2, 3, 4, 1, 3, 4, 2, 3, 3, 3, 2, 6, 3, 1, 6, 3, 3, 6, 2, 2, 6, 2, 4, 2, 3, 4, 5, 3, 3, 6, 4, 5, 7, 2, 3, 7, 3, 3, 3, 5, 1, 6, 2, 3, 6, 4, 5, 5, 4, 4, 7, 3, 4, 6, 4, 3, 5, 2, 2, 8, 5, 3, 5, 3, 6, 6, 4, 5, 5, 4, 4, 7, 2, 5, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: (i) a(n) > 0 for all n > 1. (ii) For any odd number 2*n - 1 > 4, there is a positive integer k < 2*n such that 2*n - 1 - phi(k) and 2*n - 1 + phi(k) are both prime. By Goldbach's conjecture, 2*n > 2 could be written as p + q with p and q both prime, and hence 2*n - 1 = p + (q - 1) = p + phi(q). By induction, phi(k^2) (k = 1,2,3,...) are pairwise distinct. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(29) = 1 since 2*29 - 1 = 37 + phi(5^2) with 37 prime. a(39) = 1 since 2*39 - 1 = 71 + phi(3^2) with 71 prime. a(66) = 1 since 2*66 - 1 = 89 + phi(7^2) with 89 prime. a(128) = 1 since 2*128 - 1 = 223 + phi(8^2) with 223 prime. a(182) = 1 since 2*182 - 1 = 331 + phi(8^2) with 331 prime. a(413) = 1 since 2*413 - 1 = 823 + phi(2^2) with 823 prime. a(171) = 3 since 2*171 - 1 = 233 + phi(18^2) = 257 + phi(14^2) = 293 + phi(12^2) with 233, 257, 293 all prime. MATHEMATICA a[n_]:=Sum[If[PrimeQ[2n-1-EulerPhi[k^2]], 1, 0], {k, 1, Sqrt[2n-1]}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A002372, A002375, A233542, A233544, A233547, A233654, A233793, A233864. Sequence in context: A135840 A228570 A173305 * A319814 A220272 A298917 Adjacent sequences: A233864 A233865 A233866 * A233868 A233869 A233870 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 17 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 20 08:57 EDT 2024. Contains 372710 sequences. (Running on oeis4.)