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A225135 Squares that are a concatenation of primes.

%I A225135 #22 Jan 03 2016 15:07:03
%S A225135 25,225,289,361,529,729,2025,2401,2601,2809,3025,4761,5041,5329,5929,
%T A225135 7225,7569,11025,11449,11881,13225,15129,19881,20449,21609,22801,
%U A225135 23409,24649,25281,26569,27225,29241,29929,31329,32041,32761,34969,36481,39601,47089
%N A225135 Squares that are a concatenation of primes.
%C A225135 Lim inf a(n)/n^2 >= 2. Is it finite? - _Charles R Greathouse IV_, Apr 30 2013
%H A225135 Christian N. K. Anderson, <a href="/A225135/b225135.txt">Table of n, a(n) for n = 1..10000</a>
%H A225135 Christian N. K. Anderson, <a href="/A225135/a225135.txt">Table of n, a(n), sqrt(a(n)), all possible separations of a(n) into primes for n = 1...10000</a>
%e A225135 25 = 5^2 and can be separated into two prime numbers: 2|5.
%e A225135 231361 = 481^2 and can be separated into prime numbers in six ways: 2|3|1361, 2|3|13|61, 2|31|3|61, 2|313|61, 23|1361, and 23|13|61.
%e A225135 Leading zeros are allowed: 2025 = 2|02|5.
%t A225135 r[d_] := Catch@ Block[{z = Length@d}, z<1 || Do[ If[ PrimeQ@ FromDigits@ Take[d, i] && r@ Take[d, i-z], Throw@ True], {i, z}]]; Select[ Range[1000]^2, r@ IntegerDigits@ # &] (* _Giovanni Resta_, Apr 30 2013 *)
%o A225135 (R) library(gmp); isprime2=function(x) isprime(x)>0
%o A225135 splithasproperty<-function(n,FUN,curdig=1,res=list(),curspl=c()) {
%o A225135 no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s}
%o A225135     s=as.character(n)
%o A225135     if(curdig>nchar(s)) return(res)
%o A225135     if(length(curspl)>0) if(FUN(as.bigz(no0(substr(s,curdig,nchar(s)))))) res[[length(res)+1]]=curspl
%o A225135     for(i in curdig:nchar(s))
%o A225135         if(FUN(as.bigz(no0(substr(s,curdig,i)))))
%o A225135             res=splithasproperty(n,FUN,i+1,res,c(curspl,i))
%o A225135     res
%o A225135 }
%o A225135 which(sapply(1:100,function(x) length(splithasproperty(x^2,isprime2)))>0)^2
%o A225135 (PARI) has(n)=if(isprime(n),return(1)); if(n<202,return(isprime(n%10) && isprime(n\10))); my(k=n%10,v);if(k==5||k==2,return(if(n<6,1,n\=10;has(n/10^valuation(n,10)))));if(k%2==0,return(0));v=digits(n);for(i=1,#v,if(isprime(n%10^i) && has(n\10^i), return(1))); 0
%o A225135 forstep(n=5,1e3,2,if(has(n^2),print1(n^2", ")))
%o A225135 \\ _Charles R Greathouse IV_, Apr 30 2013
%Y A225135 Cf. A019549, A030459, A106582.
%K A225135 base,nonn
%O A225135 1,1
%A A225135 _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Apr 29 2013