cp's OEIS Frontend

A129563 Primes not in a certain recursively defined set of primes.

%I A129563 #45 Dec 02 2017 09:59:36
%S A129563 101,151,197,251,401,491,503,601,607,677,701,727,751,809,883,907,983,
%T A129563 1051,1151,1201,1213,1301,1373,1451,1453,1471,1511,1601,1619,1667,
%U A129563 1801,1901,1951,2029,2179,2251,2351,2417,2549,2551,2647,2663,2719,2801,2843,2851,2903,2909
%N A129563 Primes not in a certain recursively defined set of primes.
%C A129563 The sequence is the complement of the M-sequence constructed in Section 4 of Smarandache (2007). M is defined as follows: (a) 2, 3 are in M; and (b) if 2, 3, q_1, ..., q_n are distinct primes in M and b_m = 1 + 2^a*3^b*q_1*...*q_n is prime, where 0 <= a <= 41 and 0 <= b <= 46, then b_m is in M. - _R. J. Mathar_, Jul 03 2017
%C A129563 The restriction of the two exponents to 41 and 46 seems to be based on Smarandache's sentence "and Klee to a multiple of 2^42*3^47". This statement however is hard to locate in Klee's publications. In any case, 42 and 46 should be regarded as temporary lower bounds on the exponents, which may increase as the theory and numerical experiments continue. - _R. J. Mathar_, Jul 04 2017
%C A129563 The M-sequence in Section 3 of the arXiv paper is A229289, and its complement is A289355. - _R. J. Mathar_, _Ray Chandler_, Jul 03 2017
%H A129563 Ray Chandler, <a href="/A129563/b129563.txt">Table of n, a(n) for n = 1..1000</a>
%H A129563 H. Donnelly, <a href="http://dx.doi.org/10.2307/2318777">On a problem concerning Euler's phi-function</a>, Am. Math. Monthly 80 (9) (1973) 1029-1031.
%H A129563 V. Klee, <a href="http://dx.doi.org/10.1090/S0002-9904-1947-08940-0">On a conjecture of Carmichael</a>, Bull. Am. Math. Soc. 53 (1947) 1183-1186.
%H A129563 V. Klee, <a href="http://dx.doi.org/10.2307/2316378">Is there an n for which phi(x)=n has a unique solution?</a>, Am. Math. Monthly 76 (3) (1969) 288-289.
%H A129563 Florentin Smarandache, <a href="http://arXiv.org/abs/0704.2453">On Carmichael's Conjecture</a>, arXiv preprint arXiv:0704.2453 [math.GM], Apr 19 2007.
%p A129563 isM := proc(n)
%p A129563     option remember;
%p A129563     local p1,pe,p,e ;
%p A129563     if not isprime(n) then
%p A129563         return false;
%p A129563     elif n in {2,3} then
%p A129563         return true;
%p A129563     else
%p A129563         for pe in ifactors(n-1)[2] do
%p A129563             p := pe[1] ;
%p A129563             e := pe[2] ;
%p A129563             if p = 2 and e > 41 then
%p A129563                 return false;
%p A129563             elif p = 3 and e > 46 then
%p A129563                 return false;
%p A129563             elif e > 1 and p> 3 then
%p A129563                 return false;
%p A129563             elif not procname(p) then
%p A129563                 return false;
%p A129563             end if;
%p A129563         end do:
%p A129563         return true;
%p A129563     end if;
%p A129563 end proc:
%p A129563 isA129563 := proc(n)
%p A129563     isprime(n) and not isM(n) ;
%p A129563 end proc:
%p A129563 for n from 2 to 3000 do
%p A129563     if isA129563(n) then
%p A129563         printf("%d,",n);
%p A129563     end if;
%p A129563 end do: # _R. J. Mathar_, Jul 03 2017
%t A129563 isM[n_] := isM[n] = Module[{p, e}, Which[!PrimeQ[n], Return[False], 2 <= n <= 3, Return[True], True, Do[{p, e} = pe; Which[p == 2 && e > 41, Return[False], p == 3 && e > 46, Return[False], e > 1 && p > 3, Return[False], !isM[p], Return[False]], {pe, FactorInteger[n-1]}], True, Return[True]]]
%t A129563 Select[Range[2, 3000], PrimeQ[#] && !isM[#]&] (* _Jean-François Alcover_, Dec 02 2017, after _R. J. Mathar_ *)
%Y A129563 Cf. A229289, A289355.
%K A129563 easy,nonn
%O A129563 1,1
%A A129563 _Jonathan Vos Post_, Apr 21 2007
%E A129563 Definition of M clarified by _R. J. Mathar_, Jul 03 2017