A225903 The smallest number beginning with n whose distinct prime factors are the first n primes.

16, 24, 30, 420, 50820, 60060, 7147140, 87297210, 9369900540, 103515091680, 11030826957150, 126152548291770, 13387011595197240, 143910374648370330, 15372244564712285250, 162945792385950223650, 17304843151387913751630, 1876614101750511535732320

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 21 2013

Keywords

• nonn
• base

Comments

a(3)=30 is the only term with fewer than 1000 digits whose superscripts are all 1.

Though counterexamples are possible, it appears that the sequence is strictly increasing (confirmed for n < 350, and counterexamples are increasingly unlikely statistically thereafter).

Examples

			For a(6), the number 60060 = 2^2 * 3 * 5 * 7 * 11 * 13. The only number smaller whose factors contains the first 6 primes is 30030, which does not begin with 6.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..349
• Christian N. K. Anderson, Table of n, the first digits of a(n), the number of digits of a(n), and all factors of a(n) whose superscript is >1

Crossrefs

Cf. A000040, A002110.

Cf. A086558, A109864, A088104

Cf. A018802, A077515, A077727

Programs

Mathematica

a[n_] := Block[{p = Prime[n], ba = Product[Prime@k, {k, n}], d = IntegerDigits@ n, mu = 1}, While[d != Take[IntegerDigits[mu*ba], Length@d] || Max[ First /@ FactorInteger[mu]] > p, mu++]; mu*ba]; Array[a, 20] (* Giovanni Resta, May 27 2013 *)

R

library(gmp); primes<-function(n) { x=as.bigz(rep(2,n)); for(i in 2:n) x[i]=nextprime(x[i-1]); as.vector(x[1:n]) }
newmin<-function(b,d) { if(d>length(b)) return();
    while(1) { b[d]=b[d]+1; if((x=prod(pr^b))>v) return()
        if(substr(x,1,ndig(i))==as.character(i)) { v<<-x; return() }
        if(b[d]==2) {b[d]=1; newmin(b,d+1); b[d]=2 }
        newmin(b,d+1)
    }
}
y=as.bigz(rep(0,50))
for(i in 1:50) {
    pr=primes(i); b=rep(1,i)
    while(substr((v=prod(pr^b)),1,ndig(i))!=as.character(i)) b[1]=b[1]+1;
    while(b[1]>1) { b[1]=b[1]-1; newmin(b,2) }
    if(y[i]>v) y[i]=v;
}