cp's OEIS Frontend

A054216 Numbers m such that m^2 is a concatenation of two consecutive decreasing numbers.

%I A054216 #30 Jan 26 2024 05:00:39
%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 m such that m^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 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.
%C A054216 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
%C A054216 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
%D A054216 Luca, Florian, and Pantelimon Stănică. "Perfect Squares as Concatenation of Consecutive Integers." The American Mathematical Monthly 126.8 (2019): 728-734.
%H A054216 Giovanni Resta, <a href="/A054216/b054216.txt">Table of n, a(n) for n = 1..1000</a>
%F A054216 a(n) = sqrt(A054215(n)). - _Max Alekseyev_, May 14 2007
%e A054216 '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 (PARI) isA054216(n)={ 1==[1,-1]*divrem(n^2,10^(#Str(n^2)\2)) & #Str(n^2)%2==0 }
%Y A054216 Cf. A020339, A020340, A030465, A030466, A030467, A054214, A054215, A145381.
%K A054216 nonn,base
%O A054216 1,1
%A A054216 _Patrick De Geest_, Feb 15 2000
%E A054216 More terms from _Max Alekseyev_, May 14 2007
%E A054216 Several corrections and additions from _M. F. Hasler_, Oct 09 2008