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A054416 Numbers k such that 9090...9091 (with k-1 copies of 90 and one copy of 91) is prime.

%I A054416 #22 Jan 14 2022 05:24:54
%S A054416 2,3,9,15,26,33,146,320,1068,1505,134103,800393
%N A054416 Numbers k such that 9090...9091 (with k-1 copies of 90 and one copy of 91) is prime.
%C A054416 Numbers k such that 10*(10^(2k)-1)/11 + 1 is prime.
%D A054416 J. A. H. Hunter and J. S. Madachy, Mathematical Diversions, New York: Dover Publications, Inc., 1974, pp. 4-5. Originally published by Van Nostrand, 1963.
%H A054416 David Broadhurst, <a href="http://groups.yahoo.com/group/primenumbers/message/783">Proof that 1505 term is prime</a> [BROKEN LINK]
%H A054416 David Broadhurst, <a href="/A054416/a054416.txt">Unique record</a>, digest of 3 messages in primenumbers Yahoo group, Apr 8-9, 2001. [Cached copy]
%F A054416 a(n) = (A001562(n)-1)/2.
%e A054416 The first 3 numbers are 9091, 909091, 909090909090909091.
%t A054416 Do[ If[ PrimeQ[ 10*(10^(2n) - 1)/11 + 1], Print[ n ] ], {n, 0, 1505} ]
%t A054416 Position[Table[FromDigits[PadLeft[{9,1},2n,{9,0}]],{n,1510}], _?PrimeQ]// Flatten (* _Harvey P. Dale_, Nov 02 2017 *)
%o A054416 (Python)
%o A054416 from sympy import isprime, prime
%o A054416 def afind(limit, startk=1):
%o A054416     s = "90"*(startk-1)
%o A054416     for k in range(startk, limit+1):
%o A054416         if isprime(int(s+"91")):
%o A054416             print(k, end=", ")
%o A054416         s += "90"
%o A054416 afind(400) # _Michael S. Branicky_, Jan 13 2022
%Y A054416 Cf. A001562.
%K A054416 nonn,more,hard
%O A054416 1,1
%A A054416 Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 22 2000
%E A054416 More terms from _Michael Kleber_ and Harvey Dubner (harvey(AT)dubner.com), May 22 2000
%E A054416 _Ignacio Larrosa Cañestro_ reports that the 1068 term has now been established to be a prime using Titanix 1.01, Oct 23 2000
%E A054416 a(11)-a(12) from _Michael S. Branicky_, Jan 13 2022 using A001562