A227105 Smallest number k such that the n prime distinct divisors of k are also n consecutive divisors of k.

2, 6, 30, 1155, 15015, 969969, 37182145, 1078282205, 33426748355, 7244053893505, 311494317420715, 14640232918773605, 775932344695001065, 64092011671807087969, 3909612711980232366109, 261944051702675568529303, 18598027670889965365580513, 1357656019974967471687377449

Offset

1,1

Comments

Property of the prime divisors: Let p(1), p(2), ..., p(n) the n prime divisors of k. Necessarily, p(1)*p(2) > p(n) for n > 2. Hence another definition of this sequence: smallest number k such that the product of n distinct primes p(1), p(2), ..., p(n) is minimized with the condition p(1)*p(2) > p(n).

Links

Table of n, a(n) for n=1..18.

Example

a(4) = 1155 because the 4 prime divisors are {3, 5, 7, 11}and the divisors of 1155 are {1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385, 1155} with 4 prime consecutive divisors and the property 3*5 > 11.
Factorizations of the first several terms are as follows:
n=1 => 2;
n=2 => 2 * 3 = 6;
n=3 => 2 * 3 *  5 = 30;
n=4 => 3 * 5 *  7 * 11 = 1155;
n=5 => 3 * 5 *  7 * 11 * 13 = 15015;
n=6 => 3 * 7 * 11 * 13 * 17 * 19 = 969969;
n=7 => 5 * 7 * 11 * 13 * 17 * 19 * 23 = 37182145;
...

Maple

with(numtheory): printf ( "%d %d \n", 2, 6):T:=array(1..2):i:=1:for n from 3 to 20 do:for it from 1 to 2 do:ii:=0:for j from i+1 to i+3 while(ii=0) do:if ithprime(i)*ithprime(j)> ithprime(j+n-2) then pp:=product('ithprime(a)', 'a'=j..j+n-2):ii:=1:pp:=pp*ithprime(i):T[it]:=pp:i:=i+1:else fi:od:od: ppp:=min(T[1], T[2]): printf(`%d, `, ppp):if T[2]> T[1] then i:=i-2:else i:=i-1:fi:od:

Crossrefs

Sequence in context: A102927 A376399 A376400 * A127295 A352127 A153046

Adjacent sequences: A227102 A227103 A227104 * A227106 A227107 A227108

Keyword

nonn

Author

Michel Lagneau, Jul 01 2013

Status

approved