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A225903 The smallest number beginning with n whose distinct prime factors are the first n primes. 2
16, 24, 30, 420, 50820, 60060, 7147140, 87297210, 9369900540, 103515091680, 11030826957150, 126152548291770, 13387011595197240, 143910374648370330, 15372244564712285250, 162945792385950223650, 17304843151387913751630, 1876614101750511535732320 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

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

EXAMPLE

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.

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 *)

PROG

(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;

}

CROSSREFS

Cf. A000040, A002110.

Cf. A086558, A109864, A088104

Cf. A018802, A077515, A077727

Sequence in context: A271783 A114415 A320141 * A033987 A140349 A088493

Adjacent sequences:  A225900 A225901 A225902 * A225904 A225905 A225906

KEYWORD

nonn,base

AUTHOR

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

STATUS

approved

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Last modified May 20 23:05 EDT 2022. Contains 353886 sequences. (Running on oeis4.)