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 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. 0
 8, 12, 140, 64, 2236, 196, 1300, 1600, 6256, 5084, 248756, 246196, 484400, 36680, 887884, 821836, 1559116, 104120, 126072244, 9586736, 4156840 (list; graph; refs; listen; history; text; internal format)
 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, ... 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 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) && (q=f) && (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 STATUS approved

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Last modified October 21 18:54 EDT 2019. Contains 328308 sequences. (Running on oeis4.)