A174144 Primes of the form 2^p*3^q*5^r*7^s + 1.

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 61, 71, 73, 97, 101, 109, 113, 127, 151, 163, 181, 193, 197, 211, 241, 251, 257, 271, 281, 337, 379, 401, 421, 433, 449, 487, 491, 541, 577, 601, 631, 641, 673, 701, 751, 757, 769, 811, 883, 1009, 1051, 1153, 1201

Offset: 1

Author

Michel Lagneau, Mar 09 2010

Keywords

• nonn

Comments

Restricting to r=s=0 gives the Pierpont primes (A005109); s = 0 gives A002200.

Examples

			6301 = 2^2 * 3^2 * 5^2 * 7 + 1.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Pierpont Prime

Crossrefs

Cf. A002200, A005109.

Programs

GAP

K:=10^7;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;    I:=[3,5,7];;
B:=List(A,i->Elements(Factors(i-1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));
A174144:=Concatenation([2],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])], j->Positions(B,C[i][j]))))),i->A[i])); # Muniru A Asiru, Sep 12 2017

Magma

[p: p in PrimesUpTo(2000) | forall{d: d in PrimeDivisors(p-1) | d le 7}]; // Bruno Berselli, Sep 24 2012

Maple

with(numtheory):T:=array(0..50000000):U=array(0..50000000 ):k:=1:for a from 0 to 25 do:for b from 0 to 16 do:for c from 0 to 16 do:for d from 0 to 16 do: p:= 2^a*3^b*5^c*7^d + 1:if type(p, prime)=true then T[k]:=p:k:=k+1: else fi: od :od:od:od:mini:=T[1]:ii:=1:for p from 1 to k-1 do:for n from 1 to k-1 do: if T[n] < mini then mini:= T[n]:ii:=n: indice:=U[n]: else f i:od:print(mini):T[ii]:= 10^30: ii:=1:mini:=T[1] :od:

Mathematica

Take[ Select[ Sort[ Flatten[ Table[2^a*3^b*5^c*7^d + 1, {a, 0, 25}, {b, 0, 16},{c, 0, 16},{d, 0, 16}]]], PrimeQ[ # ] &], 100] (* or *) PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[3, 6300],
ClassMinusNbr[ Prime[ # ]] == 1 &]] Select[Prime /@ Range[10^5], Max @@ First /@ FactorInteger[ # - 1] < 5 &]

PARI

list(lim)={
    lim\=1;
    my(v=List([2]),s,t,p);
    for(i=0,log(lim\2+.5)\log(7),
        t=2*7^i;
        for(j=0,log(lim\t+.5)\log(5),
            s=t*5^j;
            while(s < lim,
                p=s;
                while(p < lim,
                    if(isprime(p+1),listput(v,p+1));
                    p <<= 1
                );
                s *= 3;
            )
        )
    );
    vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Sep 21 2011

Sage

A174144 = list(p for p in primes(2000) if set(prime_factors(p-1)) <= set([2,3,5,7]))

Extensions

Corrected and edited by D. S. McNeil, Nov 20 2010