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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A318286 A214646 A275382 * A157922 A015995 A068903

Adjacent sequences:  A199580 A199581 A199582 * A199584 A199585 A199586

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 08 2011

STATUS

approved

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Last modified September 26 16:43 EDT 2020. Contains 337374 sequences. (Running on oeis4.)