A390733 a(n) is the least number k such that the sum of j-th powers of the prime factors of k, taken with multiplicity, is a prime power for j from 1 to n, but not for j = n+1.

15, 6, 48, 45, 384, 12, 2880, 200000, 40

Offset

1,1

Comments

Here "prime power" includes primes (A000961).

Disjoint from A000961, since the sum of j-th powers of the prime factors of a term of A000961 is always in A000961.

No more terms < 10^9.

Links

Table of n, a(n) for n=1..9.

Example

a(4) = 45 because 45 = 3^2 * 5 and 2 * 3^1 + 5^1 = 11, 2 * 3^2 + 5^2 = 43, 2 * 3^3 + 5^3 = 179 and 2 * 3^4 + 5^4 = 787 are primes but 2 * 3^5 + 5^5 = 3611 = 23 * 157 is not a prime power.

Maple

g:= proc(n) local F, t, d;
   F:= ifactors(n)[2];
   if nops(F) = 1 then return -1 fi;
   for d from 1 do
     if nops(ifactors(add(t[1]^d * t[2], t=F))[2]) <> 1 then return d-1 fi
   od
end proc:
V:= Vector(9): count:= 0:
for n from 2 while count < 9 do
v:= g(n);
  if v > 0 and V[v] = 0 then V[v]:= n; count:= count+1 fi
od:
convert(V, list);

Crossrefs

Cf. A000961, A390729, A390731.

Sequence in context: A380823 A040214 A364351 * A111298 A395956 A241440

Adjacent sequences: A390730 A390731 A390732 * A390734 A390735 A390736

Keyword

nonn,more,hard

Author

Will Gosnell and Robert Israel, Nov 16 2025

Status

approved