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A225120 Square numbers whose decimal representation can be divided into two or more semiprimes.

%I A225120 #45 Jan 03 2016 15:07:34
%S A225120 49,64,144,256,576,625,1156,1296,1444,1521,2209,2916,3364,3844,3969,
%T A225120 4096,4356,4489,4624,6889,7744,8649,9025,9216,9409,9604,10201,10404,
%U A225120 10609,10816,12321,12996,13456,14161,15129,15376,15625,15876,17956,18496,18769,20164
%N A225120 Square numbers whose decimal representation can be divided into two or more semiprimes.
%C A225120 For 300 < n < 10000, 12.77*n^1.86 provides an estimate of a(n) to within 10%.
%C A225120 The density of squares included in the sequence asymptotically approaches 1.
%C A225120 There are infinitely many squares that are not in the sequence. For example, no square ending in 0 can be in the sequence. Another such infinite class is given by (50k+5)^2, for k>0. Indeed, these squares all end in "025" and since the only semiprime ending in 25 is 25 itself, then the other semiprime must end in 0, but this is impossible since the only semiprime ending in 0 is 10. - _Giovanni Resta_, May 03 2013
%H A225120 Christian N. K. Anderson, <a href="/A225120/b225120.txt">Table of n, a(n) for n = 1..10000</a>
%H A225120 Christian N. K. Anderson, <a href="/A225120/a225120.txt">Table of n, a(n), sqrt(a(n)), all possible separations of a(n) into semiprimes for n = 1...10000.</a>
%e A225120 a(50) = 25921, which is 161^2, and can be separated into semiprimes three ways: 25|921, 25|9|21, and 259|21.
%o A225120 (R) issemipr<-function(n) ifelse(n<4,F,length(factorize(n))==2)
%o A225120 splithasproperty<-function(n,FUN,curdig=1,res=list(),curspl=c()) {
%o A225120     no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s}
%o A225120     s=as.character(n)
%o A225120     if(curdig>nchar(s)) return(res)
%o A225120     if(length(curspl)>0) if(FUN(as.bigz(no0(substr(s,curdig,nchar(s)))))) res[[length(res)+1]]=curspl
%o A225120     for(i in curdig:nchar(s))
%o A225120         if(FUN(as.bigz(no0(substr(s,curdig,i)))))
%o A225120             res=splithasproperty(n,FUN,i+1,res,c(curspl,i))
%o A225120     res
%o A225120 }
%o A225120 which(sapply(1:100,function(x) length(splithasproperty(x^2,issemipr))>0))^2
%Y A225120 Cf. A001358, A030459, A030461, A000290.
%K A225120 nonn,base
%O A225120 1,1
%A A225120 _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Apr 29 2013