A225873 Squares that become prime when their most significant (or leftmost) digit is removed.

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

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 19 2013

Keywords

• nonn
• base

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=N-5 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 most-significant 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

See A249589 for the square roots. - M. F. Hasler, Nov 02 2014

Examples

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

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]

Crossrefs

Cf. A225885.

Programs

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 *)

PARI

is_A225873(n)=isprime(n%10^(#Str(n)-1))&&issquare(n)

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

Extensions

Extended by Davin Park, Dec 30 2016