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A002200 Primes of the form 2^q*3^r*5^s + 1.
(Formerly M0654 N0242)
5

%I M0654 N0242

%S 2,3,5,7,11,13,17,19,31,37,41,61,73,97,101,109,151,163,181,193,241,

%T 251,257,271,401,433,487,541,577,601,641,751,769,811,1153,1201,1297,

%U 1459,1601,1621,1801,2161,2251,2593,2917,3001,3457,3889,4001,4051,4801,4861

%N Primes of the form 2^q*3^r*5^s + 1.

%D M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 53.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Harry J. Smith and Giovanni Resta, <a href="/A002200/b002200.txt">Table of n, a(n) for n = 1..10000</a> (first 200 terms from Harry J. Smith)

%t up=10^6; a=1; Sort[Reap[While[ a<up, b=a; While[ b<up, c=b; While[ c<up, If[ PrimeQ[ c+1], Sow[ c+1]]; c *= 5]; b *= 3]; a *= 2]][[2, 1]]] (* _Giovanni Resta_, Jul 18 2017 *)

%o (PARI) { default(primelimit, 16600000); n=0; forprime (p=2, 16600000, m=p-1; p2=p3=p5=0; s=m; r=0; while(r==0, q=s\2; r=s-2*q; s=q; if(r==0, p2++)); s=m; r=0; while(r==0, q=s\3; r=s-3*q; s=q; if(r==0, p3++)); s=m; r=0; while(r==0, q=s\5; r=s-5*q; s=q; if(r==0, p5++)); if (m == 2^p2*3^p3*5^p5, n++; write("b002200.txt", n, " ", p)); if (n >= 200, break); ); } \\ _Harry J. Smith_, May 25 2009

%o (PARI) { n=5000; cache=10^5; v=vector(cache); x2=2; x3=3; x5=5; i=j=k=1; v[1]=1; for(m=2,cache,v[m]=t=min(x2,min(x3,x5)); if(x2==t,x2=2*v[i++]); if(x3==t,x3=3*v[j++]); if(x5==t,x5=5*v[k++]);); i=0; c=0; while(c<n,i++;if(isprime(v[i]+1),c++; print(c" "v[i]+1))); } \\ _Jean-Marie Madiot_, Jul 17 2017

%o (MAGMA) [p: p in PrimesUpTo(5000) | forall{d: d in PrimeDivisors(p-1) | d le 5}]; // _Bruno Berselli_, Sep 24 2012

%o (GAP)

%o K:=10^7;; # to get all terms <= K.

%o A:=Filtered([1..K],IsPrime);;

%o B:=List(A,i->Factors(i-1));;

%o C:=[];; for i in B do if Elements(i)=[2] or Elements(i)=[2,3] or Elements(i)=[2,5] or Elements(i)=[2,3,5] then Add(C,Position(B,i)); fi; od;

%o A002200:=Concatenation([2],List(C,i->A[i])); # _Muniru A Asiru_, Sep 10 2017

%Y Cf. A174144, A005109, A077497.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E Better description and more terms from _Vladeta Jovovic_, May 08 2003

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Last modified August 8 05:50 EDT 2020. Contains 336290 sequences. (Running on oeis4.)