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A257491 Least k formed by the concatenation of two numbers n and d such that d is the n-th divisor of k, or 0 if no such k exists. 2

%I #25 Jun 09 2015 04:19:00

%S 11,22,0,410,5125,620,735,816,975,0,0,1280,0,14200,0,0,17200,1872,

%T 19250,2015625,21109375,222500,0,24150,251250,262000,0,0,291450,

%U 305000,0,0,0,0,0,0,37115625,0,0,402500,0,42336,0,442750,45703125,46920,0,4850000,4912500,0,515100

%N Least k formed by the concatenation of two numbers n and d such that d is the n-th divisor of k, or 0 if no such k exists.

%C Some semiprime near-misses for a(3): 35, 323, 365, 3103, 3173, 3493, 3755, 31673, 34993, 37495, 349993, 3166673, 34999993, 31666666673. - _Michel Marcus_, May 08 2015

%H Robert Israel, <a href="/A257491/b257491.txt">Table of n, a(n) for n = 1..80</a>

%e a(4) = 410 because the divisors of 410 are {1, 2, 5, 10, 41, 82, 205, 410} and 10 is the 4th divisor of 410.

%p # this program tests k < 150000; results of 0 may be incorrect.

%p with(numtheory):nn:=51:

%p for n from 1 to nn do:

%p ii:=0:

%p for k from n to 150000 while(ii=0)do:

%p i:=length(k):p:=n*10^i+k:

%p x:=divisors(p):n0:=nops(x):

%p if n<=n0 then

%p d:=x[n]:j:=length(d):q:=n*10^j+d:

%p if p=q then

%p ii:=1:printf ( "%d %d \n",n,p):

%p else

%p fi:fi:

%p od:

%p if ii=0 then printf ( "%d %d \n",n,0):

%p else

%p fi:

%p od:

%p # Alternative

%p f:= proc(n)

%p local F, m, t, primes, enum, cands, d, divs;

%p F:= ifactors(n)[2];

%p primes:= {seq(t[1],t=F)} union {2,5};

%p for t in F do m[t[1]]:= t[2] od;

%p m[2]:= n; m[5]:= n;

%p enum:= proc(pr,t)

%p local p,r;

%p if pr = {} or t <= 1 then return [1] fi;

%p p:= pr[1];

%p r:= pr[2..-1];

%p [seq(op(map(`*`,procname(r,floor(t/(1+j))), p^j)), j=0..m[p])]

%p end proc;

%p cands:= sort(enum(primes,n));

%p for d in cands do

%p divs:= sort(convert(numtheory:-divisors(n*10^(1+ilog10(d))+d),list));

%p if nops(divs) >= n and divs[n] = d then return(n*10^(1+ilog10(d))+d) fi;

%p od;

%p 0

%p end proc:

%p seq(f(n), n=1..60); # _Robert Israel_, Jun 08 2015

%Y Cf. A000012 (1st divisor of n), A020639 (2nd divisor of n).

%K nonn,base,hard

%O 1,1

%A _Michel Lagneau_, Apr 26 2015

%E a(21), a(45) and a(48) from _Robert Israel_, Jun 08 2015

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)