A258780 a(n) is the least k such that k^2 + 1 is a semiprime p*q, p < q, and (q - p)/2^n is prime.

8, 12, 140, 64, 2236, 196, 1300, 1600, 6256, 5084, 248756, 246196, 484400, 36680, 887884, 821836, 1559116, 104120, 126072244, 9586736, 4156840, 542759984, 1017981724, 2744780140, 405793096, 148647496, 1671024916

Offset

2,1

Comments

The corresponding primes are 2, 3, 71, 7, 1069, 7, 5, 5, 59, 2, 368471, 180463, 12421, 2, 29, 125683, 226169, 5, 369704891, 197, 5, 263, 7444559, 239621423, 594271, 2, 474359, ...

All terms are even, in order for k^2+1 to be odd. Otherwise, with k^2+1 being even, p-q would be odd and hence not a multiple of 2^n. - Michel Marcus, Apr 13 2019

Links

Table of n, a(n) for n=2..28.

Example

a(2)=8 because 8^2+1 = 5*13 and (13-5)/2^2 = 2 is prime. The number 8 is the first term of the sequence 8, 22, 34, 46, 50, 58, ...
a(3)=12 because 12^2+1 = 5*29 and (29-5)/2^3 = 3 is prime. The number 12 is the first term of the sequence 12, 28, 44, 52, 76, 80, ...
a(4)=140 because 140^2+1 = 17*1153 and (1153-17)/2^4 = 71 is prime. The number 140 is the first term of the sequence 140, 296, 404, 604, ...

Mathematica

lst={}; Do[k=2; While[!(Plus@@Last/@FactorInteger[k^2+1]==2&&PrimeQ[(FactorInteger[k^2+1][[-1, 1]]-FactorInteger[k^2+1][[1, 1]])/2^n]), k=k+2]; Print[n, " ", k], {n, 2, 19}]; lst

Prog

(PARI) isok(k, n) = my(kk=k^2+1, f=factor(kk)[, 1]~); (bigomega(kk) == 2) && (#f == 2) && (p=f[1]) && (q=f[2]) && (qq=(q-p)/2^n) && !frac(qq) && isprime(qq);
a(n) = my(k=2); while (!isok(k, n), k+=2); k; \\ Michel Marcus, Apr 13 2019

Crossrefs

Cf. A085722, A144255.

Sequence in context: A083128 A196077 A067923 * A228663 A076028 A281256

Adjacent sequences: A258777 A258778 A258779 * A258781 A258782 A258783

Keyword

nonn,more

Author

Michel Lagneau, Jun 10 2015

Extensions

Name edited by Jon E. Schoenfield, Sep 12 2017

a(20)-a(22) from Daniel Suteu, Apr 13 2019

a(23)-a(28) from Daniel Suteu, Nov 09 2019

Status

approved