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A286708 Powerful numbers (A001694) that are not prime powers (A000961). 48
36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 900, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500, 2592, 2601, 2700, 2704, 2744 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
If a prime p divides a(n) then p^2 must also divide a(n) and number of distinct primes dividing a(n) > 1.
Intersection of A001694 and A024619.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 5997 terms from Robert Israel)
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Powerful Number.
FORMULA
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{p prime} 1/(p*(p-1)) - 1 = A082695 - A136141 - 1 = 0.17043976777096407719... - Amiram Eldar, Feb 12 2021
EXAMPLE
-------------------------------
| n | a(n) | prime |
| | | factorization |
|------------------------------
| 1 | 36 | {{2, 2}, {3, 2}} |
| 2 | 72 | {{2, 3}, {3, 2}} |
| 3 | 100 | {{2, 2}, {5, 2}} |
| 4 | 108 | {{2, 2}, {3, 3}} |
| 5 | 144 | {{2, 4}, {3, 2}} |
| 6 | 196 | {{2, 2}, {7, 2}} |
| 7 | 200 | {{2, 3}, {5, 2}} |
| 8 | 216 | {{2, 3}, {3, 3}} |
| 9 | 225 | {{3, 2}, {5, 2}} |
-------------------------------
a(n) = p_1^e_1*p_2^e_2*... : {{p_1, e_1}, {p_2, e_2}, ...}.
MAPLE
N:= 10000:
S:= {1}: P:= {1}:
p:= 1:
do
p:= nextprime(p);
if p^2 > N then break fi;
S:= map(s -> (s, seq(s*p^k, k = 2 .. floor(log[p](N/s)))), S);
P:= P union {seq(p^k, k=2..floor(log[p](N)))}:
od:
sort(convert(S minus P, list)); # Robert Israel, May 14 2017
MATHEMATICA
Select[Range@2750, Min@FactorInteger[#][[All, 2]] > 1 && ! PrimePowerQ[#] &]
(* Second program *)
nn = 2^25; Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], ! PrimePowerQ[#] &] (* Michael De Vlieger, Jun 22 2022 *)
PROG
(Python)
from sympy import primefactors, factorint
print([n for n in range(4, 2745) if len(primefactors(n)) > 1 and min(list(factorint(n).values())) > 1]) # Karl-Heinz Hofmann, Feb 07 2023
CROSSREFS
Sequence in context: A036785 A338539 A347960 * A355462 A363216 A363169
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 13 2017
STATUS
approved

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Last modified June 17 11:50 EDT 2024. Contains 373445 sequences. (Running on oeis4.)