|
|
A368107
|
|
Prime powers p^m such that p | m.
|
|
1
|
|
|
4, 16, 27, 64, 256, 729, 1024, 3125, 4096, 16384, 19683, 65536, 262144, 531441, 823543, 1048576, 4194304, 9765625, 14348907, 16777216, 67108864, 268435456, 387420489, 1073741824, 4294967296, 10460353203, 17179869184, 30517578125, 68719476736, 274877906944, 282429536481
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Proper subset of A072873, which in turn is a proper subset of A342090.
This sequence represents the prime power block in A072873 and A342090.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
This sequence contains prime powers of the following form:
2^2, 2^4, i.e., 2^k such that k is even.
3^3, 3^6, 3^9, i.e., 3^k such that 3 | k.
5^5, 5^10, 5^15, i.e., 5^k such that 5 | k, etc.
|
|
MAPLE
|
N:= 10^13: # for terms <= N
R:= NULL:
for i from 1 do
p:= ithprime(i);
if p^p > N then break fi;
R:= R, seq(p^k, k=p..floor(log[p](N)), p);
od:
|
|
MATHEMATICA
|
nn = 10^12; i = 1; p = 2; While[p^p <= nn, p = NextPrime[p] ];
MapIndexed[Set[S[First[#2]], #1] &, Prime@ Range@ PrimePi[p] ];
Union@ Reap[
While[j = S[i];
While[S[i]^j < nn,
Sow[S[i]^j]; j += S[i] ]; j > 2,
i++] ][[-1, 1]]
|
|
PROG
|
(Python)
import heapq
from itertools import islice
from sympy import nextprime
def agen(): # generator of terms
v, h, m, nextp = 4, [(4, 2)], 4, 3
while True:
v, p = heapq.heappop(h)
yield v
if v >= m:
m = nextp**nextp
heapq.heappush(h, (m, nextp))
nextp = nextprime(nextp)
heapq.heappush(h, (v*p**p, p))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|