The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A054216 Numbers m such that m^2 is a concatenation of two consecutive decreasing numbers. 10
 91, 9079, 9901, 733674, 999001, 88225295, 99990001, 8900869208, 9296908812, 9604060397, 9999900001, 326666333267, 673333666734, 700730927008, 972603739727, 999999000001, 34519562953737, 39737862788838, 49917309624956 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 that are prime? - M. F. Hasler, Mar 30 2008. Answer: The smallest prime in this sequence that 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), 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, Oct 09 2008 A positive integer m is in this sequence if and only if m^2 == -1 (mod 10^k + 1) where k is the number of decimal digits in m. Note that k cannot be odd, since in this case 11 divides 10^k + 1 while -1 is not a square modulo 11. - Max Alekseyev, Oct 09 2008 REFERENCES Luca, Florian, and Pantelimon Stănică. "Perfect Squares as Concatenation of Consecutive Integers." The American Mathematical Monthly 126.8 (2019): 728-734. LINKS Giovanni Resta, Table of n, a(n) for n = 1..1000 FORMULA a(n) = sqrt(A054215(n)). - Max Alekseyev, May 14 2007 EXAMPLE '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. PROG (PARI) isA054216(n)={ 1==[1, -1]*divrem(n^2, 10^(#Str(n^2)\2)) & #Str(n^2)%2==0 } CROSSREFS Cf. A020339, A020340, A030465, A030466, A030467, A054214, A054215, A145381. Sequence in context: A060078 A202564 A006244 * A109627 A095372 A165154 Adjacent sequences: A054213 A054214 A054215 * A054217 A054218 A054219 KEYWORD nonn,base AUTHOR Patrick De Geest, Feb 15 2000 EXTENSIONS More terms from Max Alekseyev, May 14 2007 Several corrections and additions from M. F. Hasler, Oct 09 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 12 09:11 EDT 2024. Contains 375085 sequences. (Running on oeis4.)