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A225120 Square numbers whose decimal representation can be divided into two or more semiprimes. 1
49, 64, 144, 256, 576, 625, 1156, 1296, 1444, 1521, 2209, 2916, 3364, 3844, 3969, 4096, 4356, 4489, 4624, 6889, 7744, 8649, 9025, 9216, 9409, 9604, 10201, 10404, 10609, 10816, 12321, 12996, 13456, 14161, 15129, 15376, 15625, 15876, 17956, 18496, 18769, 20164 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For 300 < n < 10000, 12.77*n^1.86 provides an estimate of a(n) to within 10%.

The density of squares included in the sequence asymptotically approaches 1.

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

LINKS

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Christian N. K. Anderson, Table of n, a(n), sqrt(a(n)), all possible separations of a(n) into semiprimes for n = 1...10000.

EXAMPLE

a(50) = 25921, which is 161^2, and can be separated into semiprimes three ways: 25|921, 25|9|21, and 259|21.

PROG

(R) issemipr<-function(n) ifelse(n<4, F, length(factorize(n))==2)

splithasproperty<-function(n, FUN, curdig=1, res=list(), curspl=c()) {

    no0<-function(s){ while(substr(s, 1, 1)=="0" & nchar(s)>1) s=substr(s, 2, nchar(s)); s}

    s=as.character(n)

    if(curdig>nchar(s)) return(res)

    if(length(curspl)>0) if(FUN(as.bigz(no0(substr(s, curdig, nchar(s)))))) res[[length(res)+1]]=curspl

    for(i in curdig:nchar(s))

        if(FUN(as.bigz(no0(substr(s, curdig, i)))))

            res=splithasproperty(n, FUN, i+1, res, c(curspl, i))

    res

}

which(sapply(1:100, function(x) length(splithasproperty(x^2, issemipr))>0))^2

CROSSREFS

Cf. A001358, A030459, A030461, A000290.

Sequence in context: A216165 A065807 A038678 * A308250 A215056 A242702

Adjacent sequences:  A225117 A225118 A225119 * A225121 A225122 A225123

KEYWORD

nonn,base

AUTHOR

Kevin L. Schwartz and Christian N. K. Anderson, Apr 29 2013

STATUS

approved

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Last modified September 24 03:26 EDT 2021. Contains 347623 sequences. (Running on oeis4.)