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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

LINKS

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

EXAMPLE

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.

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 *)

PROG

(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

(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

CROSSREFS

Subsequence of A085722.

Cf. A001358, A144255.

Sequence in context: A255267 A259037 A231469 * A114821 A108098 A114505

Adjacent sequences:  A261543 A261544 A261545 * A261547 A261548 A261549

KEYWORD

nonn,less

AUTHOR

Michel Lagneau, Aug 24 2015

EXTENSIONS

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

STATUS

approved

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Last modified February 26 11:40 EST 2020. Contains 332279 sequences. (Running on oeis4.)