This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370209 #30 Feb 22 2024 20:07:30
%S A370209 78,820,6328,62128,5539456,155155972096,739936,69342976,431056,
%T A370209 31494016,44864128,3525354496,3788128
%N A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.
%C A370209 a(n) is the smallest number of the form described above whose symmetric representation of sigma, SRS(a(n)), consists of 2 parts that have a unimodal width pattern of type 121 and that meet at the diagonal. Since floor( (sqrt(8*a(n) + 1) - 1)/2 ) = 2^(k+1) * p, the central 0 width extent of SRS(a(n)) equals 0.
%C A370209 Conjecture: The sequence is infinite.
%H A370209 Chai Wah Wu, <a href="/A370209/b370209.txt">Table of n, a(n) for n = 2..75</a> [Notice that a(15) has 355 digits, so the graph cuts off after 14 terms. - _N. J. A. Sloane_, Feb 18 2024]
%F A370209 a(n) = min( 2^k * p * (2^(k+1) * p + 1) : p = prime(n), 2 < p < 2^(k+1), 2^(k+1) * p + 1 is prime ), n>=2.
%e A370209 a(2) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts 121 that meet at diagonal position (54, 54).
%e A370209 a(4) = 6328 = 8 * 7 * 113 = A262259(11) which demonstrates that 2^k < p < 2^(k+1) need not be true.
%e A370209 a(15) with k = 582 and p = 47, its second prime factor 2^(k+1) * p + 1 has 178 digits so that a(15) has 355 digits.
%e A370209 a(16) = 24129129742336 = 2^16 * 53 * 6946817.
%e A370209 Table of records of number of digits a(2) through a(500):
%e A370209 sequence index 2 3 4 5 6 7 15 76 419 438
%e A370209 number of digits 2 3 4 5 7 12 355 3854 5856 20049
%t A370209 minExp[p_] := Module[{k=Floor[Log[2, p]]}, NestWhile[#+1&, k+1, !PrimeQ[2^# p+1]&]-1]/;PrimeQ[p]
%t A370209 a370209[p_] := Module[{k=minExp[p]}, 2^k p(2^(k+1)p+1)]/;PrimeQ[p]
%t A370209 Map[a370209[Prime[#]]&, Range[2, 14]] (* a(15) is too large to list *)
%o A370209 (Python)
%o A370209 from itertools import count
%o A370209 from sympy import prime, isprime
%o A370209 def A370209(n):
%o A370209 p = prime(n)
%o A370209 return next((p<<k)*m for k in count(p.bit_length()-1) if isprime(m:=(p<<k+1)+1)) # _Chai Wah Wu_, Feb 17 2024
%Y A370209 Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205, A370206.
%K A370209 nonn
%O A370209 2,1
%A A370209 _Hartmut F. W. Hoft_, Feb 11 2024