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A054216
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Numbers n such that n^2 is a concatenation of two consecutive decreasing numbers.
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10
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91, 9079, 9901, 733674, 999001, 88225295, 99990001, 8900869208, 9296908812, 9604060397, 9999900001, 326666333267, 673333666734, 700730927008, 972603739727, 999999000001, 34519562953737, 39737862788838, 49917309624956
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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.
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
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]
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FORMULA
| a(n) = sqrt(A054215(n)). - Max Alekseyev, May 14 2007
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EXAMPLE
| E.g. '8242' + '8242-1' gives 82428241 which is 9079^2.
Leading zeros are not allowed, which is why c(1)=266327 is not in this sequence although c(1)^2 = 070930 070929.
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PROG
| isA054216(n)={ 1==[1, -1]*divrem(n^2, 10^(#Str(n^2)\2)) & #Str(n^2)%2==0 }
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CROSSREFS
| Cf. A054214, A054215, A030465, A030466, A030467, A020339, A020340, A145381.
Sequence in context: A060078 A202564 A006244 * A109627 A095372 A165154
Adjacent sequences: A054213 A054214 A054215 * A054217 A054218 A054219
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KEYWORD
| nonn,base
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AUTHOR
| Patrick De Geest (pdg(AT)worldofnumbers.com), Feb 15 2000.
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EXTENSIONS
| More terms from Max Alekseyev, May 14 2007
Several corrections and additions. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Oct 09 2008
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