A289355 -id:A289355 - cp's OEIS frontend

A229289 Primes p of the form p = 2^k * m + 1, where (i) m is squarefree and odd, (ii) all primes that divide m are in the sequence, and (iii) k is 0, 1, or 2.

2, 3, 5, 7, 11, 13, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 79, 107, 131, 139, 157, 173, 211, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 421, 431, 461, 463, 547, 557, 599, 643, 659, 661, 683, 691, 709, 733, 743, 787, 827, 853, 859, 863, 911, 941

Offset: 1

José María Grau Ribas, Oct 05 2013

nonn

Taking m=1 in the definition we get the primes 2, 3, 5.
If n is in A226960, then n is a product of terms of this sequence.
If k is only allowed to be 0 or 1, we get 2, 3, 7, 43 and no more. - Jianing Song, Feb 21 2021
Also prime factors of terms in A341858. It is conjectured that this sequence is infinite. - Jianing Song, Feb 22 2021

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A227007, A226960, A007814, A229290, A229291, A289355, A341858.
For the complement, see A289355.
Proper subsequence of A066651.

fa = FactorInteger; free[n_] := n == Product[fa[n][[i, 1]], {i, Length[fa[n]]}] ; Os[b_, 1] = True; Os[b_, b_] = True; Os[b_, n_] := Os[b, n] = PrimeQ[n] && free[(n - 1)/b^IntegerExponent[n - 1, b]] &&IntegerExponent[n - 1, b] < 3 && Union@Table[Os[b, fa[n - 1][[i, 1]]], {i, Length[fa[n - 1]]}] == {True};G[b_] := Select[Prime[Range[1000]], Os[b, #] &];G[2]

is(n)=if(!isprime(n),return(0)); if(n<13,return(1)); my(k=valuation(n-1,2), m=n>>k, f); if(k>2,return(0)); f=factor(m); if(lcm(f[,2])>1, return(0)); for(i=1,#f~, if(!is(f[i,1]), return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2013

Revised definition from Charles R Greathouse IV, Nov 13 2013

Terms corrected by José María Grau Ribas, Nov 14 2013

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

Original entry on oeis.org

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

Jonathan Vos Post, Apr 21 2007

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

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.

Cf. A229289, A289355.

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

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 *)

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

cp's OEIS Frontend

A229289 Primes p of the form p = 2^k * m + 1, where (i) m is squarefree and odd, (ii) all primes that divide m are in the sequence, and (iii) k is 0, 1, or 2.

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