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

101, 151, 197, 251, 401, 491, 503, 601, 607, 677, 701, 727, 751, 809, 883, 907, 983, 1051, 1151, 1201, 1213, 1301, 1373, 1451, 1453, 1471, 1511, 1601, 1619, 1667, 1801, 1901, 1951, 2029, 2179, 2251, 2351, 2417, 2549, 2551, 2647, 2663, 2719, 2801, 2843, 2851, 2903, 2909

Offset: 1

Author

Jonathan Vos Post, Apr 21 2007

Keywords

• easy
• nonn

Comments

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

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

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

Links

• Ray Chandler, Table of n, a(n) for n = 1..1000
• H. Donnelly, On a problem concerning Euler's phi-function, Am. Math. Monthly 80 (9) (1973) 1029-1031.
• V. Klee, On a conjecture of Carmichael, Bull. Am. Math. Soc. 53 (1947) 1183-1186.
• V. Klee, Is there an n for which phi(x)=n has a unique solution?, Am. Math. Monthly 76 (3) (1969) 288-289.
• Florentin Smarandache, On Carmichael's Conjecture, arXiv preprint arXiv:0704.2453 [math.GM], Apr 19 2007.

Crossrefs

Cf. A229289, A289355.

Programs

Maple

isM := proc(n)
    option remember;
    local p1,pe,p,e ;
    if not isprime(n) then
        return false;
    elif n in {2,3} then
        return true;
    else
        for pe in ifactors(n-1)[2] do
            p := pe[1] ;
            e := pe[2] ;
            if p = 2 and e > 41 then
                return false;
            elif p = 3 and e > 46 then
                return false;
            elif e > 1 and p> 3 then
                return false;
            elif not procname(p) then
                return false;
            end if;
        end do:
        return true;
    end if;
end proc:
isA129563 := proc(n)
    isprime(n) and not isM(n) ;
end proc:
for n from 2 to 3000 do
    if isA129563(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jul 03 2017

Mathematica

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]]]
Select[Range[2, 3000], PrimeQ[#] && !isM[#]&] (* Jean-François Alcover, Dec 02 2017, after R. J. Mathar *)

Extensions

Definition of M clarified by R. J. Mathar, Jul 03 2017