Search: id:A054216
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%I A054216
%S A054216 91,9079,9901,733674,999001,88225295,99990001,8900869208,9296908812,
%T A054216 9604060397,9999900001,326666333267,673333666734,700730927008,
%U A054216 972603739727,999999000001,34519562953737,39737862788838,49917309624956
%N A054216 Numbers n such that n^2 is a concatenation of two consecutive decreasing 
               numbers.
%C A054216 Obviously b(n) = 100^n-10^n+1 = (91,9901,999001,99990001,...) is a subsequence. 
               Are { b(2), b(4), b(6), b(8) } the only terms of this sequence which 
               are prime? - M. F. Hasler (www.univ-ag.fr/~mhasler), Mar 30 2008. 
               Answer: The smallest prime in this sequence which is not of the form 
               b(n) is A054216(155) = 811451682377384625400019885321 [Max Alekseyev, 
               Oct 08 2008]. See A145381 for further prime terms.
%C A054216 Other subsequences are: c(n) = ( 10^(6n) - 2*10^(5n) - 10^(3n) - 2*10^n 
               + 1 )/3 (n>=2) and d(n) = 33/101*(100^(404n+71)+1)+10^(404n+71) (n>
               =0) and e(n) = 33/101*(100^(404n-71)+1)+10^(404n-71) (n>=1). Primes 
               among these include c(10), c(14) and d(0). - M. F. Hasler (www.univ-ag.fr/
               ~mhasler), Oct 09 2008
%C A054216 A positive integer n is in this sequence if and only if n^2 == -1 (mod 
               10^k + 1) where k is the number of decimal digits in n. Note that 
               k cannot be odd, since in this case 11 divides 10^k + 1 while -1 
               is not a square modulo 11. [From Max Alekseyev (maxale(AT)gmail.com), 
               Oct 09 2008]
%F A054216 a(n) = sqrt(A054215(n)). - Max Alekseyev, May 14 2007
%e A054216 E.g. '8242' + '8242-1' gives 82428241 which is 9079^2.
%e A054216 Leading zeros are not allowed, which is why c(1)=266327 is not in this 
               sequence although c(1)^2 = 070930 070929.
%o A054216 isA054216(n)={ 1==[1,-1]*divrem(n^2,10^(#Str(n^2)\2)) & #Str(n^2)%2==0 
               }
%Y A054216 Cf. A054214, A054215, A030465, A030466, A030467, A020339, A020340, A145381.
%Y A054216 Sequence in context: A022253 A060078 A006244 this_sequence A109627 A095372 
               A165154
%Y A054216 Adjacent sequences: A054213 A054214 A054215 this_sequence A054217 A054218 
               A054219
%K A054216 nonn,base
%O A054216 1,1
%A A054216 Patrick De Geest (pdg(AT)worldofnumbers.com), Feb 15 2000.
%E A054216 More terms from Max Alekseyev, May 14 2007
%E A054216 Several corrections and additions. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), 
               Oct 09 2008

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