A199583 a(n) is the smallest number such that the sum of the n-th powers of its distinct prime divisors is divisible by n.

2, 2, 3, 2, 5, 70, 7, 2, 3, 33, 11, 1155, 13, 78, 26, 2, 17, 2156564410, 19, 6006, 26, 114, 23, 2156564410, 5, 33, 3, 1365, 29, 110, 31, 2, 62, 15, 201, 2156564410, 37, 30, 14, 961380175077106319535, 41, 1385670, 43, 2805, 26, 266, 47, 961380175077106319535

Offset

1,1

Comments

a(n) > 1 and a(n) = n if n prime. All terms are squarefree.

Links

Table of n, a(n) for n=1..48.

Example

a(6) = 70 = 2*5*7; 2^6 + 5^6 + 7^6 = 133338 = 22223*6.
a(18)= 2*5*7*11*13*17*19*23*29 = 2156564410 because:
p^18 == 10, 9 (mod 18) for p = 2,3 respectively, and p^18 == 1 (mod 18) for p prime > 3. The minimum sum divisible by 18 is s = 2^18 + Sum_{k=3..10} prime(k)^18 whose residues sum to 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 18. Hence a(18) = 2156564410.

Maple

with(numtheory): T:=array(1..50):for n from 1 to 50 do:q:=0:for k from 2 to 7000 while(q=0)do:x:=factorset(k):s:=sum(x[j]^n , j=1..nops(x)) :if irem(s, n)=0 then printf ( "%d %d \n", n, k):q:=1:else fi:od:if q=0 then for i from 1 to n do: T[i]:=irem(ithprime(i)^n, n):od:W:=convert(T, set):n1:=nops(W):n2:=W[n1]:n3:=W[n1-1]:
s:=0:p:=1:for a from 1 to n  while(s<>n) do: if T[a]= 1 or T[a]=n2 or (T[a] = n3 and n2+n3<n) then s:=s+T[a]:p:=p*ithprime(a):else fi:if  s= n then printf ( "%d %d \n", n, p):
else fi:od:else fi:od:

Crossrefs

Cf. A005063, A005064, A005065, A199167.

Sequence in context: A214646 A340693 A275382 * A157922 A343655 A015995

Adjacent sequences: A199580 A199581 A199582 * A199584 A199585 A199586

Keyword

nonn

Author

Michel Lagneau, Nov 08 2011

Status

approved