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

%I A261546 #33 Sep 08 2022 08:46:13
%S A261546 48,58,1688,2948,28338,36998,38648,96248,100308,133458,136798,187538,
%T A261546 207088,224508,253808,309738,375348,545048,598348,607688,659548,
%U A261546 672398,793958,1055648,1055688,1140008,1270408,1317808,1388398,1399098,1529488,1597008,1655338
%N A261546 Numbers k such that the five numbers k^2+1, (k+1)^2+1, ..., (k+4)^2+1 are all semiprime.
%C A261546 a(n) == 8 (mod 10).
%C A261546 a(15017) > 10^10. - _Hiroaki Yamanouchi_, Oct 03 2015
%H A261546 Hiroaki Yamanouchi, <a href="/A261546/b261546.txt">Table of n, a(n) for n = 1..15016</a>
%e A261546 48 is in the sequence because of these five semiprimes:
%e A261546 48^2+1 = 2305 = 5*461;
%e A261546 49^2+1 = 2402 = 2*1201;
%e A261546 50^2+1 = 2501 = 41*61;
%e A261546 51^2+1 = 2602 = 2*1301;
%e A261546 52^2+1 = 2705 = 5*541.
%p A261546 with(numtheory):
%p A261546   n:=5:
%p A261546   for k from 1 to 10^6 do:
%p A261546     jj:=0:
%p A261546     for m from 0 to n-1 do:
%p A261546        x:=(k+m)^2+1:d0:=bigomega(x):
%p A261546        if d0=2
%p A261546        then
%p A261546        jj:=jj+1:
%p A261546        else
%p A261546        fi:
%p A261546      od:
%p A261546         if jj=n
%p A261546         then
%p A261546         printf(`%d, `,k):
%p A261546         else
%p A261546         fi:
%p A261546     od:
%t A261546 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 *)
%o A261546 (PARI) has(n) = bigomega(n^2+1)==2;
%o A261546 isok(n) = has(n) && has(n+1) && has(n+2) && has(n+3) && has(n+4); \\ _Michel Marcus_, Aug 24 2015
%o A261546 (PARI)
%o A261546 a261546() = {
%o A261546   nterm = 0;
%o A261546   for (i = 0, 10^9,
%o A261546     if (isprime(20*i*i + 32*i + 13) &&
%o A261546       isprime(50*i*i + 90*i + 41) &&
%o A261546       isprime(50*i*i + 110*i + 61) &&
%o A261546       isprime(20*i*i + 48*i + 29) &&
%o A261546       bigomega(100*i*i + 200*i + 101) == 2,
%o A261546       nterm += 1;
%o A261546       print(nterm, " ", 10 * i + 8);
%o A261546     );
%o A261546   );
%o A261546 } \\ - _Hiroaki Yamanouchi_, Oct 03 2015
%o A261546 (PARI) issemi(n)=forprime(p=2,97, if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
%o A261546 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
%o A261546 (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
%Y A261546 Subsequence of A085722.
%Y A261546 Cf. A001358, A144255.
%K A261546 nonn,less
%O A261546 1,1
%A A261546 _Michel Lagneau_, Aug 24 2015
%E A261546 a(18)-a(33) from _Hiroaki Yamanouchi_, Oct 03 2015