

A054216


Numbers n such that n^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
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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^(404n71)+1)+10^(404n71) (n>=1). Primes among these include c(10), c(14) and d(0).  M. F. Hasler, 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.  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): 728734.


LINKS

Giovanni Resta, Table of n, a(n) for n = 1..1000


FORMULA

a(n) = sqrt(A054215(n)).  Max Alekseyev, May 14 2007


EXAMPLE

E.g., '8242' + '82421' 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

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



