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

%I A174144 #30 Feb 16 2025 08:33:12
%S A174144 2,3,5,7,11,13,17,19,29,31,37,41,43,61,71,73,97,101,109,113,127,151,
%T A174144 163,181,193,197,211,241,251,257,271,281,337,379,401,421,433,449,487,
%U A174144 491,541,577,601,631,641,673,701,751,757,769,811,883,1009,1051,1153,1201
%N A174144 Primes of the form 2^p*3^q*5^r*7^s + 1.
%C A174144 Restricting to r=s=0 gives the Pierpont primes (A005109); s = 0 gives A002200.
%H A174144 Charles R Greathouse IV, <a href="/A174144/b174144.txt">Table of n, a(n) for n = 1..10000</a>
%H A174144 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierpontPrime.html">Pierpont Prime</a>
%e A174144 6301 = 2^2 * 3^2 * 5^2 * 7 + 1.
%p A174144 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:
%t A174144 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],
%t A174144 ClassMinusNbr[ Prime[ # ]] == 1 &]] Select[Prime /@ Range[10^5], Max @@ First /@ FactorInteger[ # - 1] < 5 &]
%o A174144 (Sage)
%o A174144 A174144 = list(p for p in primes(2000) if set(prime_factors(p-1)) <= set([2,3,5,7]))
%o A174144 (PARI) list(lim)={
%o A174144     lim\=1;
%o A174144     my(v=List([2]),s,t,p);
%o A174144     for(i=0,log(lim\2+.5)\log(7),
%o A174144         t=2*7^i;
%o A174144         for(j=0,log(lim\t+.5)\log(5),
%o A174144             s=t*5^j;
%o A174144             while(s < lim,
%o A174144                 p=s;
%o A174144                 while(p < lim,
%o A174144                     if(isprime(p+1),listput(v,p+1));
%o A174144                     p <<= 1
%o A174144                 );
%o A174144                 s *= 3;
%o A174144             )
%o A174144         )
%o A174144     );
%o A174144     vecsort(Vec(v))
%o A174144 }; \\ _Charles R Greathouse IV_, Sep 21 2011
%o A174144 (Magma) [p: p in PrimesUpTo(2000) | forall{d: d in PrimeDivisors(p-1) | d le 7}]; // _Bruno Berselli_, Sep 24 2012
%o A174144 (GAP)
%o A174144 K:=10^7;; # to get all terms <= K.
%o A174144 A:=Filtered([1..K],IsPrime);;    I:=[3,5,7];;
%o A174144 B:=List(A,i->Elements(Factors(i-1)));;
%o A174144 C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));
%o A174144 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
%Y A174144 Cf. A002200, A005109.
%K A174144 nonn
%O A174144 1,1
%A A174144 _Michel Lagneau_, Mar 09 2010
%E A174144 Corrected and edited by _D. S. McNeil_, Nov 20 2010