A293008 Primes of the form 2^q * 3^r * 7^s + 1.

2, 3, 5, 7, 13, 17, 19, 29, 37, 43, 73, 97, 109, 113, 127, 163, 193, 197, 257, 337, 379, 433, 449, 487, 577, 673, 757, 769, 883, 1009, 1153, 1297, 1373, 1459, 2017, 2269, 2593, 2647, 2689, 2917, 3137, 3457, 3529, 3889, 7057, 8233, 10369, 10753, 12097, 12289, 14407, 15877, 17497, 18433

Offset: 1

Author

Muniru A Asiru, Sep 28 2017

Keywords

• nonn

Comments

Fermat prime exponents q occur in the case when q = 0, 1, 2, 4, 8, 16.

Examples

			With n = 1,  a(1) = 2^0 * 3^0 * 7^0 + 1 = 2.
With n = 5,  a(5) = 2^2 * 3^1 * 7^0 + 1 = 13.
list of (q, r, s): (0, 0, 0), (1, 0, 0), (2, 0, 0), (1, 1, 0), (2, 1, 0), (4, 0, 0), (1, 2, 0), (2, 0, 1), (2, 2, 0), (1, 1, 1), ...
		

Crossrefs

Cf. A002200 (Primes of the form 2^q * 3^r * 5^s + 1).

Programs

GAP

K:=10^7+1;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;    I:=[3,7];;
B:=List(A,i->Elements(Factors(i-1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A293008:=Concatenation([2],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

Mathematica

With[{n = 19000}, Union@ Select[Flatten@ Table[2^p1*3^p2*7^p4 + 1, {p1, 0, Log[2, n/(1)]}, {p2, 0, Log[3, n/(2^p1)]}, {p4, 0, Log[7, n/(2^p1*3^p2)]}], PrimeQ]] (* Michael De Vlieger, Sep 30 2017 *)