A174113 Smallest number k such that k, k+1, and k+2 are all divisible by an n-th power.

48, 1375, 33614, 2590623, 26890623, 2372890624, 70925781248, 2889212890624, 61938212890624, 4497636425781248, 8555081787109375, 2665760081787109375, 98325140081787109375, 198816740081787109374, 11776267480163574218750, 872710687480163574218750, 50783354512519836425781248

Offset: 2

Author

Michel Lagneau, Mar 08 2010

Keywords

• nonn

Comments

Least of the smallest trio of consecutive numbers divisible by an n-th power.

Examples

			a(3) = 1375 because
  1375 =  11 * 5^3;
  1376 = 172 * 2^3;
  1377 =  51 * 3^3.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1375, p. 135, Ellipses, Paris 2008.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 2..677 (next term has 1001 digits)

Crossrefs

Cf. A068780, A068781, A068140, A068782, A068783, A068784, A045330, A059737, A063528, A051903, A051903.

Programs

Maple

with(numtheory):for n from 2 to 6 do: i:=0:for k from 1 to 3000000 while(i=0) do:j:=0:
for a from 0 to 2 do: ii:=0:for m from 1 to 4  while(ii=0) do:p:=ithprime(m)^n:if irem(k+a,p)=0 then j:=j+1:ii:=1:else fi:od:od:if j=3 then i:=1:print(k):else fi:od:od:

PARI

a(n)=my(ch,t,best=30^n);forprime(a=2, 29, forprime(b=2, 29, if(a==b,next); ch=chinese(Mod(0,a^n), Mod(-1,b^n)); if(lift(ch)>=best, next); forprime(c=2, 29, if(a==c || b==c, next); t=lift(chinese(ch, Mod(-2, c^n))); if(tCharles R Greathouse IV, Jan 16 2012

Formula

5^n < a(n) < 30^n. Can the lower bound be improved? - Charles R Greathouse IV, Jan 16 2012

Extensions

a(8)-a(18) from Charles R Greathouse IV, Jan 16 2012