A261546 Numbers k such that the five numbers k^2+1, (k+1)^2+1, ..., (k+4)^2+1 are all semiprime.

48, 58, 1688, 2948, 28338, 36998, 38648, 96248, 100308, 133458, 136798, 187538, 207088, 224508, 253808, 309738, 375348, 545048, 598348, 607688, 659548, 672398, 793958, 1055648, 1055688, 1140008, 1270408, 1317808, 1388398, 1399098, 1529488, 1597008, 1655338

Offset: 1

Author

Michel Lagneau, Aug 24 2015

Keywords

• nonn
• less

Comments

a(n) == 8 (mod 10).

a(15017) > 10^10. - Hiroaki Yamanouchi, Oct 03 2015

Examples

			48 is in the sequence because of these five semiprimes:
48^2+1 = 2305 = 5*461;
49^2+1 = 2402 = 2*1201;
50^2+1 = 2501 = 41*61;
51^2+1 = 2602 = 2*1301;
52^2+1 = 2705 = 5*541.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..15016

Crossrefs

Subsequence of A085722.

Cf. A001358, A144255.

Programs

Magma

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [1..3*10^5] | IsSemiprime(n^2+1) and IsSemiprime(n^2+2*n+2)and IsSemiprime(n^2+4*n+5)and IsSemiprime(n^2+6*n+10)and IsSemiprime(n^2+8*n+17)]; // Vincenzo Librandi, Aug 24 2015

Maple

with(numtheory):
  n:=5:
  for k from 1 to 10^6 do:
    jj:=0:
    for m from 0 to n-1 do:
       x:=(k+m)^2+1:d0:=bigomega(x):
       if d0=2
       then
       jj:=jj+1:
       else
       fi:
     od:
        if jj=n
        then
        printf(`%d, `,k):
        else
        fi:
    od:

Mathematica

PrimeFactorExponentsAdded[n_]:=Plus@@Flatten[Table[#[[2]], {1}]&/@FactorInteger[n]]; Select[Range[2 10^5], PrimeFactorExponentsAdded[#^2+1] == PrimeFactorExponentsAdded[#^2 + 2 # + 2]== PrimeFactorExponentsAdded[#^2 + 4 # + 5]== PrimeFactorExponentsAdded[#^2 + 6 # + 10]== PrimeFactorExponentsAdded[#^2 + 8 # + 17] == 2 &] (* Vincenzo Librandi, Aug 24 2015 *)

PARI

has(n) = bigomega(n^2+1)==2;
isok(n) = has(n) && has(n+1) && has(n+2) && has(n+3) && has(n+4); \\ Michel Marcus, Aug 24 2015

PARI

a261546() = {
  nterm = 0;
  for (i = 0, 10^9,
    if (isprime(20*i*i + 32*i + 13) &&
      isprime(50*i*i + 90*i + 41) &&
      isprime(50*i*i + 110*i + 61) &&
      isprime(20*i*i + 48*i + 29) &&
      bigomega(100*i*i + 200*i + 101) == 2,
      nterm += 1;
      print(nterm, " ", 10 * i + 8);
    );
  );
} \\ - Hiroaki Yamanouchi, Oct 03 2015

PARI

issemi(n)=forprime(p=2,97, if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
list(lim)=my(v=List()); forstep(k=48,lim,[10,30,10], if(issemi(k^2+1) && issemi((k+1)^2+1) && issemi((k+3)^2+1) && issemi((k+4)^2+1) && issemi((k+2)^2+1), listput(v,k))); Vec(v) \\ Charles R Greathouse IV, Jul 06 2017

Extensions

a(18)-a(33) from Hiroaki Yamanouchi, Oct 03 2015