

A225873


Squares that become prime when their most significant (or leftmost) digit is removed.


12



25, 289, 361, 441, 529, 729, 841, 961, 1089, 1521, 2401, 2601, 2809, 4761, 5041, 5929, 6241, 7569, 8281, 9409, 20449, 21609, 22801, 24649, 25281, 26569, 29241, 29929, 34969, 36481, 39601, 40401, 52441, 53361, 54289, 57121, 58081, 59049, 61009, 63001, 71289
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OFFSET

1,1


COMMENTS

a(1)=25 is the only term in the sequence that ends in 5. Proof: Any number ending in 5 is divisible by 5, and no square ending in 5 can have all 0 internal digits. Let N=A+B where A=N5 and B=5. Then N^2 = A^2 + 2AB + B^2. B^2 is 25, and because A ends in a zero, A^2 and 2AB ends in two zeros; therefore the sum ends in 25.
All other terms end in 1 or 9, because no square ends in 3 or 7.
Observation: The sequence often experiences large gaps when the mostsignificant digit is square. For example, there is a gap of over 10^8 between a(764)=99420841 and a(765)=200307409, and over 10^9 between a(9156)=39980402401 and a(9157)=50000984881.
These gaps occur because if n^2 = (10^k*d+r)^2 = 10^(2k)d^2+r*(2*10^k+r) with d=1, 2, or 3 and r small enough so that the first digit of n^2 is d^2, then removing that digit d^2 we are left with r*(2*10^k+r) which is divisible by r and thus cannot be prime if r>1.  Giovanni Resta, May 23 2013


LINKS

Christian N. K. Anderson and Davin Park, Table of n, a(n) for n = 1..20000 [Terms 1 through 10000 were computed by Christian N. K. Anderson and terms 10001 through 20000 were computed by Davin Park]


EXAMPLE

2401 = 49^2 becomes the prime number 401 when 2 is removed. 5041 = 71^2 becomes the prime number 41 when 5 is removed.


MATHEMATICA

b^2 /. Flatten[Outer[Solve[a + #2*10^#1 == b^2 && 0 <= a < 10^#1 && Sqrt[#2*10^#1] <= b < Sqrt[10^(#1 + 1)] && a \[Element] Primes, {a, b}, Integers] &, Range[0, 10], Range[9]], 2] (* Davin Park, Dec 30 2016 *)


PROG

(R)no0<function(s){ while(substr(s, 1, 1)=="0" & nchar(s)>1) s=substr(s, 2, nchar(s)); s};
issquare<function(x) ifelse(as.bigz(x)<2, T, all(table(as.numeric(gmp::factorize(x)))%%2==0));
which(sapply(1:200, function(x) isprime(no0(substr(x^2, 2, ndig(x^2)))))>0)^2
(PARI) is_A225873(n)=isprime(n%10^(#Str(n)1))&&issquare(n)


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



